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1. File Tools Window Help e _ New Fixed Term Test New Interim Test Z Plot Power vs Sample Size LU Open Two Proportions Non Inferiorit PY i Riel 2 3 4 Test significance level a 0 05 Test Type Unpooled Likelihood score z Likelihood score z Likelihood score z Higher Proportions Better Worse Better x Better x Better x Better z Control Group Proportion p2 0 5 Solve using p1 p2 p1 or p1 p2 Differences z Differences z Differences z Differences z Non inferiority Test Statistic 0 1 Actual Value of Test Statistic 0 05 Intracluster Correlation ICC 0 001 Clusters in Treatment Group K1 Clusters in Control Group K2 K1 Cluster Sample Size in Treatment Group M1 100 Cluster Sample Size in Control Group M2 100 so Cost per sample Total study cost Le gt Calculate attainable power with the given sample size and number of clusters X Run All columns Figure 5 4 3 Values entered for CRT Two Proportions Non Inferiority study design 6 Select Calculate required treatment group clusters K1 given power and sample size from the dropdown menu beside the Run button Then click Run This will give a result of 14 for K1 and K2 as in Figure 5 4 4 246 N nQuen nlerim File Edit View Assistants Plot Tools Window Help i _ New Fixed Term Test _ New Interim Test Z Plot Power vs Sample Size LL
2. Two Proportions Equivalence 1 Test Type Likelihood score z Likelihood score z Likelihood score z Likelihoodscore x Control Group Proportion p2 _ Solve using p1 p2 p1 or p1 p2 Differences Differences iz Differences z Differences EA Test Statistic for Upper Equivalence Margin Test Statistic for Lower Equivalence Margin Actual Value of Test Statistic Intracluster Correlation ICC Clusters in Treatment Group K1 Clusters in Control Group K2 Cluster Sample Size in Treatment Group M1 Cluster Sample Size in Control Group M2 Power Cost per sample Total study cost DEE gt Calculate attainable power with the given sample size and number of clusters Run All columns Figure 5 3 2 CRT Two Proportions Equivalence Completely Randomized Test Table The first calculation will be for the Clusters in Treatment Group K1 This example is can be undertaken using the following steps 3 First enter 0 05 for the Test Significance level row Next select Pooled from the Test Type option and enter a control group proportion of 0 5 then select Differences from the Solve Using dropdown option 4 Enter 0 1 for the Upper Equivalence Margin 0 1 for the Lower Equivalence Margin and 0 for the Actual Value of Test Statistic Enter 0 001 for the intracluster correlation Enter 100 for both cluster sample size variables and 8
3. J Compute Effect Size Assistant ui Specify Multiple Factors ui Output Figure 4 1 9 Compute Effect Size Assistant Table 6 Once you enter a value for the number of levels M the Compute Effect Size Assistant table automatically updates as shown in Figure 4 1 10 7 In order to calculate a value for Effect Size two parameters need to be calculated first the Contrast C and Scale D 8 The mean for each level and the corresponding coefficient value need to be entered in the Compute effect Size Assistant table 9 For the Mean values for each level enter 6 for level 1 3 for level 2 and 3 for level 3 10 For the Coefficient values for each level enter 2 for level 1 1 for level 2 and 1 for level 3 The sum of these values must always equate to zero This is illustrated in Figure 4 1 11 below 65 66 ag ee File Edit View Assistants Plot Tools Window Help gt New Fixed Term Test New Interim Test Plot Power vs Sample Size Scale D SQRT Sci2 Standard deviation at each level o Between level correlation p Effect size A C D o SQRT 1 p Power Group size N Cost per sample unit Total study cost 7m a Calculate required sample sizes for given power Compute Effect Size Assistant Scale D SQRT Sci2 Figure 4 1 10 Automatically Updated Compute Effect Size Assistant Table 11 Once the table in Figure 4 1 11 is co
4. Figure 4 2 5 Highlight desired columns for plotting 15 To highlight the desired columns click on the column title for Column 1 and drag across to Column 2 as illustrated in Figure 4 2 5 16 Then click on the Plot Power vs Sample Size button on the menu bar The multiple column plot is displayed in Figure 4 2 6 77 78 Im et i a A Power vs Sample Size Column 1 rn Column 2 Power 82 67 Sample Size N1 N2 Sample Size 102 Figure 4 2 6 Power vs Sample Size Plot It can be seen from the legend on the left hand side legend can be altered manually that the blue line represents Column 1 and the orange line represents Column 2 The cross on the graph illustrates how the user can identify what the sample size is for a corresponding power value for each column In the bottom right corner of the plot indicated the exact values for Power and Sample Size for each identifier on the graph It can be seen in Figure 4 2 6 that Column 1 reaches an acceptable power level faster than the design in Column 2 The researcher can now make an assessment as to which design they would prefer to use Example 2 Differences in Power and Between Level Correlations In this example we investigate how a change in Power and a change in Between Level Correlation has an effect on sample size The following steps outline the procedure for Example 2 1 Open nTerim through the Start Menu or by double clicking
5. Don t Calculate O Brien Fleming x off 5 z Equally Spaced 1 O Brien Fleming z No Don t Calculate O Brien Fleming Calculate attainable power with the given sample sizes v E All columns z Spending Function x Spending Function l z off 5 z Equally Spaced 1 x O Brien Fleming x No Don t Calculate O Brien Fleming B 3 Figure 3 2 7 Completed Two Proportions Test using Power Family Spending Function 8 Also the boundary values will be recalculated and boundary plot will automatically be plotted as shown in Figure 3 2 8 and 3 2 9 below Looks 0 4 Lower bound 3 00000 3 00000 Upper bound 3 00000 3 00000 Nominal alpha 0 00270 0 00270 Incremental alpha 0 00270 0 00222 Cumulative alpha 0 00270 0 00492 Exit probability 441 9 19 Cumulative exit probability 441 13 60 Nominal beta Cumulative beta Exit probability under HO 0 8 0 6 2 67717 2 31962 2 67717 2 31962 0 00742 0 02036 0 00588 0 01480 0 01080 0 02560 21 18 27 13 34 78 61 91 2 05069 2 05069 0 04030 0 02440 0 05000 19 26 81 17 u Looks ud Specify Multiple Factors u Output Figure 3 2 8 Boundary Table for Power Family Spending Function 42 BoundariesGraph i ial Power Family Boundaries with Phi 3 and Alpha 0 05 1 2 3 4 Figure 3 2 9 Boundary Plot for Power Family Spending Fun
6. Equivalence Agreement Regression y One sample t test Paired t test for difference in Means Univariate one way repeated measures analysis of variance Hg One way repeated measures contrast Univariate one way repeated measures analysis of variance Greenhouse Geisser Cancel Figure 4 1 7 Study Goal and Design Window Once the correct test has been selected click OK and the test window will appear There are two main tables required for this test the main test table illustrated in Figure 4 1 8 and the effect size assistant table shown in Figure 4 1 9 Enter 0 05 for alpha the desired significance level and enter 3 for the number of levels M as shown in Figure 4 1 10 Now you are required to complete the Compute Effect Size Assistant table in order to calculate values for the Contrast C and Scale D parameters File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size RM Contrast 1 ee Number of levels M Contrast C ici pi Scale D SQRT Sci Standard deviation at each level o Between level correlation p Effect size A C D o SQRT 1 p Power Group size N Cost per sample unit Total study cost i gt Calculate required sample sizes for given power X All columns Figure 4 1 8 One way Repeated Measures Contrast Test Table
7. Number of looks Information times Max Times Determine bounds Spending function Truncate bounds Truncate at f Futility boundaries Spending function Lied 2 0 05 2 220 200 20 30 30 0 667 57 57 1 90 33 250 28500 5 Equally Spaced 1 Spending Function Pocock No Don t Calculate O Brien Fleming 2 5 Equally Spaced mi 1 Spending Function EJE O Brien Fleming No E Don t Calculate O Brien Fleming EJE z 2 5 z Equally Spaced 1 x Spending Function x O Brien Fleming x No x Don t Calculate x O Brien Fleming X X X JE 5 Equally Spaced 1 Spending Function O Brien Fleming No Don t Calculate O Brien Fleming Calculate required sample sizes for given power gt E All columns Figure 3 1 7 Complete Two Means Test Table 4 Select Run and the sample size along with the boundary values will be calculated 5 The boundaries that are calculated will be automatically plotted Clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation Sample sizes of 57 in group 1 and 57 in group 2 are required to achieve 90 33 power to detect a difference in means of 20 the difference between group 1 mean u1 of 220 and group 2 mean u2 of 200
8. This is a measure of the variability between subjects within a group and is assumed to be the same for all groups The user must then also enter the number covariates c to be used in the study along with the average r squared value between the response and the covariates R In order to calculate power a value for the total sample size N must be entered remember this can also be read in from the effect size assistant nTerim then calculates the power of the design by first determining the critical value Fg_1 N G c The non centrality parameter A is then calculated using the equation __V A nG 4 5 3 O where N a 4 5 4 G and og 1 p o 4 5 5 111 im where g is the within group variance after considering the covariates and p is the coefficient of multiple determination estimated by R Using these two values nTerim calculates the power of this design as the probability of being greater than FG 1 N G c a ON a non central F distribution with non centrality parameter A In order to calculate sample size nTerim does not use a closed form equation Instead a search algorithm is used This search algorithm calculates power at various sample sizes until the desired power is reached 112 4 5 3 Examples Example 1 Calculating Attainable Power given Sample Size In this example we are going to calculate the attainable power for a given sample size for an ANCOVA design The follo
9. _ 80 0115952771 Cost per sample Total study cost 4 m gt cacuite required sample size foraiven power Ram an columns Figure 4 9 4 Completed Negative Binomial study design 6 The next calculation is a sensitivity analysis for sample size where we change the mean rate of the event for the control to investigate the impact this has on the sample size estimate To do this copy the same values across to columns 2 to 4 and delete the values for sample size Then change the values for columns 2 3 and 4 to 1 1 2 and 1 4 respectively This will give a table as per Figure 4 9 5 Query nTer File Edit View Assistants Plot Tools Window Help i E New Fixed Term Test i New Interim Test Plot Power vs Sample Size LU Of Negative Binomial 1 a 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 Mean Rate of Event for Control r0 0 8 1 1 2 1 4 Rate Ratio r1 r0 0 85 0 85 0 85 0 85 Average Exposure Time pT 0 75 0 75 0 75 0 75 Dispersion Parameter k 0 7 0 7 0 7 07 Rates Variance Reference Group Rate z Reference Group Rate z Reference Group Rate z Reference Group Rate z Sample Size Ratio N1 N0 1 1 1 Control Group Sample Size N0 80 80 80 I gt Calculate required sample size for given power X Figure 4 9 5 Sensitivity analysis around the Mean Rate of Event for Control 7 Select Calculate required sample size for given power from the dropdown men
10. e 14 2 2b A b A 4 12 15 pa a ET ee A ER 12 b B A b 4 Binomial Distribution for X4 The calculation requires the proportion of the variable p e g p 0 5 if 50 of variable were treatment and 50 controls These are used to calculate Vp and V as follows eel p 1 p 1 1 p pe 4 12 16 4 12 17 A closed form equation is not used to calculate the response rate ratio Instead a search algorithm is used This search algorithm calculates power at various values of the rate ratio until the desired power is reached 4 12 3 Examples Example 1 Validation example calculating required sample size for a given power The following examples are taken from Signorini 1991 where a sample size calculation problem is conducted for a normally distributed independent variable followed by a sensitivity analysis for the effect of changing response rate ratio and of differently distributed independent variables Finally an example of the calculation for the rate ratio is introduced The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 4 12 1 below eere O OOO Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval 5 Survival
11. 2 In order to select the CRT Two Proportions Superiority Completely Randomized design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Cluster Randomized as the Goal Two as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 5 5 2 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Two Proportions Superiority 1 a E a Test Type Likelihnoodscore Likelihood score x Likelihood score x Likelihood score v Higher Proportions Better Worse Better x Better x Better x Better Control Group Proportion p2 4 Solve using p1 p2 p1 or p1 p2 Differences Differences z Differences Differences Superiority Test Statistic Actual Value of Test Statistic Intracluster Correlation ICC Clusters in Treatment Group K1 Clusters in Control Group K2 Cluster Sample Size in Treatment Group M1 Cluster Sample Size in Control Group M2 __ Power Cost per sample Total study cost Moo oom gt Calculate attainable power with the given sample size and number of clusters Zi Run C All columns Figure 5 5 2 CRT Two Proportions Superiority Completely Randomized Test Table The first calculation will be for the Clusters in Treatment Group K1 This example is
12. Study Goal and Design window will appear ia Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval Survival gt Two D Equivalence Agreement Regression One sample t test Paired t test for difference in Means i Univariate one way repeated measures analysis of variance One way repeated measures contrast Univariate one way repeated measures analysis of variance Greenhouse Geisser OK Cancel Figure 4 1 13 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear 3 There are two main tables required for this test the main test table illustrated in Figure 4 1 14 and the effect size assistant table shown in Figure 4 1 15 4 Enter 0 05 for alpha the desired significance level and enter 4 for the number of levels M as shown in Figure 4 1 16 5 Now you are required to complete the Compute Effect Size Assistant table in order to calculate values for the Contrast C and Scale D parameters File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size RM Contrast 1 a E 2 3 4 5 Lol Number of levels M Contrast C ici pi Scale D SQRT Sci Standard deviation at each level o Between level correlation p
13. cancer Figure 4 2 13 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear This test table is illustrated in Figure 4 2 14 3 Enter 0 05 for alpha the desired significance level and enter 5 for the number of levels M as shown in Figure 4 2 15 4 Two sided test is the default setting in nTerim as well as a Ratio value of 1 for the group sizes 5 In this example we will examine a study where the difference in means is 40 and the standard deviation at each level is 80 Therefore enter a value of 40 in the Difference in Means row and a value of 80 in the Standard deviation at each level row as shown in Figure 4 2 15 83 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size 1 or 2 sided test Number of levels M Difference in means pi p2 Standard deviation at each level o __ Between level correlation p Group 1 size ni Group 2 size n2 Ratio n2 I ni _ Power Cost per sample unit Total study cost l 7n Calculate required sample sizes for given power X E All columns Figure 4 2 14 Repeated Measures for Two Means Test Table 6 The between level correlation is estimated as 0 5 so enter 0 5 in the Between level correlation row 7 We want to calculate the required sample size to obtain a powe
14. 181 182 Melim 10 Select the first column by clicking the 1 at the top of column 1 Then hold down Shift and click the 4 at the top of column 4 All four columns will now be highlighted 11 Click the Plot Power vs Sample Size button at the top of the screen This will give you a plot of Power vs Sample Size as displayed Figure 4 11 7 This plot highlights the relationship between power and sample size for each column Right click to add features such as a legend to the graph and double click elements for user options and editing Power vs Sample i Power vs Sample Size 410 Event Rate 0 1 Event Rate 0 3 Event Rate 0 5 Event Rate 1 0 10 60 110 160 210 260 310 360 410 460 510 560 610 Power 91 03 Sample Size Sample Size 119 Figure 4 11 7 Power vs Sample Size Plot 12 Finally by clicking on any of the columns and going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 4 11 8 Output x OUTPUT STATEMENT A Cox regression of the log hazard ratio on a covariate with a standard deviation of 1 based on a sample of 248 observations achieves 80 power at a 0 05 significance level to detect a regression coefficient equal to 0 5 The sample size was adjusted since a multiple regression of the variable of interest on the other covariates in the Cox regression is expected to have an R Squa
15. 2 0 2 0 4 4 87688 3 35695 Upper bound 4 87688 3 35695 Futility bound Nominal alpha 0 00000 0 00079 Incremental alpha 0 00000 0 00079 Cumulative alpha 0 00000 0 00079 Exit probability 0 03 9 98 Cumulative exit probability 0 03 10 01 Nominalbeta Incremental beta Cumulative beta Cumulative exit probability under HO a 0 6 0 8 1 2 68026 2 28979 2 03100 2 68026 2 28979 2 03100 0 00736 0 02203 0 04226 0 00683 0 01681 0 02558 0 00762 0 02442 0 05000 34 73 29 96 15 36 44 75 74 71 90 07 i Looks u Specify Multiple Factors i Output Figure 3 3 4 Boundary Table for Column 1 9 In the second column enter the same parameters as above but change the Group 2 proportion to 0 40 Select Run nQuery nTerim 26 File Edit View Assistants Plot Tools Window Help _ New Fixed Term Test _ New Interim Test a Plot Power vs Sample Size LUJ Open Manual Statistical Solutions Support GST Survival 1 xX A 1 2 3 4 Test significance level a 0 05 0 05 3 1 or 2 sided test 2 2 me i Group 1 proportion ni at time t 03 03 Group 2 proportion n2 at time t 0 45 0 4 Hazard ratio h In n1 In n2 1 508 1314 Survival time assumption Exponential Survival Exponential Survival Exponential Survival Exponential Survival Total sample size N 409 888 Power E 90 07 90 02 Number of e
16. 5 x 2 dose groups giving a total sample of 110 subjects achieves 90 5379299894 power to detect an increase in relative potency of 1 1 using a one sided t test at the 0 05 significance level The common slope between the probits and the log doses was set as 23 25 i Specify Multiple Factors ui Probit Regression Side Table ui Output Figure 4 13 13 Study design Output statement Finally we will use the side table to calculate the regression slope This will be done by specifying both the Target Response Proportions and the Number of Doses rows in the side table 209 210 Melim 17 Delete columns 2 to 4 Then in column 1 delete the slope sum of weights and sample size values 18 In the side table enter the 0 05 0 275 0 5 0 725 and 0 95 in the Target Response Proportions row Enter 1 to 5 in the Number of Doses row 19 Click the Compute button and this will transfer a value 4 364 for the Slope of Probit Regression and 2 201 for the Sum of Weights into the main table Probit Regression Side Table x 4 Specify Multiple Factors ij Probit Regression Side Table Output Figure 4 13 14 Completed Probit Regression Side Table 20 Run the sample size calculation with the updated values for the slope and you get a Sample Size per Group of 305 as in Figure 4 13 15 File Edit View Assistants Plot Tools Window Help Z Plot Power vs Sample Size E New
17. DIF n 1 p IF ee pA PIF pA PIF 5 5 15 K M KM2 where K M p K M p 1M1P 2M2P2 5 5 16 K M K M A closed form equation is not used to calculate the other parameters Instead a search algorithm is used The search algorithm calculates power at various values for the relevant parameter until the desired power is reached 253 254 Im 5 5 3 Examples Example 1 Validation example calculating required number of clusters for a given power The following example will look at a number of clusters calculation problem is conducted and then a sensitivity analysis is conducted to show the effect of changing control group number of clusters The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 5 5 1 below cos ns a Design Goal No of Groups Analysis Method Fixed Term O Means One Test Interim Proportions Two Confidence Interval Survival gt Two Equivalence Agreement Regression Cluster Randomized CRT Two Means Completely Randomized CRT Two Proportions Inequality Completely Randomized CRT Two Proportions Equivalence CRT Two Proportions Non Inferiority CRT Two Proportions Superiority OK Cancel Figure 5 5 1 Study Goal and Design Window
18. Figure 4 12 17 Results from second Sensitivity analysis 196 Example 2 Calculating required Response Rate Ratio for given power and sample size Finally an example is provided for calculating the response rate ratio given the sample size and power 20 To do this set sample size to 100 and power to 80 in column 1 Remove the two Variance of b1 variable values from the main table and the response rate ratio value Delete all other columns 21 Next select Response Rate Ratio gt 1 given sample size and power from the dropdown menu beside the Run button 22 Next enter a mean of 0 and a standard deviation of 1 in the Normal Distribution side table Then click Compute Note that no figure is transferred into the main table as the Variance of b1 Null Hypothesis and Variance of b1 Alternative Hypothesis statistics require the response rate ratio aJ Normal Side Table Exponential Side Table u Uniform Side Table a Binomial Side Table ui Specify Multiple Factors wij Output Figure 4 12 18 Normal Distribution Side Table 23 Click Run The response rate ratio will update to 1 281 as in Figure 4 12 19 Pe Quen nrenm 3 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test i New Interim Test Plot Power vs Sample Size Ww Poisson Regression 2 4 ee 2 3 4 Test significance level a 0 05 1or 2 sided test 1 z z z x Ba
19. R d VX X2p where d t ty 149 150 3 Log Unconstrained Maximum Likelihood Estimate In MLE Testing the natural log for the unconstrained MLE can yield a null hypothesis of Hy U ln y1 y2 ln R The variance of U is given by of 1 tyy 1 t2Y2 Similar re arrangement as for the MLE above yields the following test statistic In X X2 Ind InR _ In X X Inp J A JIU 4 Log Constrained Maximum Likelihood Estimate In CMLE Similar arguments for the above equation can be used to yield the following test statistic for the log of the constrained estimators Wine Xa X2 4 8 5 In X1 X2 lnd ln R V a R R d X aan In X X2 Inp Winccmie Xi X2 V 24 1 p p X X2 Win cMLE Xi X2 K 5 Variance Stabilized Estimate The variance stabilizing transformation of Huffman 1984 can accelerate the convergence to normality This transformation uses the following test statistic 2 X 3 8 p X 3 8 vy1l p Wys X1 X2 4 8 7 The sample size calculation for two Poisson rates is taken from Gu et al 2008 This table can be used to calculate the power the sample size and the rate ratio under the alternative hypothesis given all other terms in the table are specified To calculate power and sample size the user must specify the test significance level the ratio of the two samples Poisson rates under the null hypothesis Rg the ratio of the two
20. The Assistants tables are a new feature added to nTerim to aid the user in calculating various additional components of certain study designs These tables are only associated with certain design tables With nTerim we know which Assistant table is associated with each test so they automatically pop up once a design table is opened LLJ Open Manuat gt Statistical Solutions Support Compute Effect Size Randomisation Distribution Function Windows Calculator Figure 2 6 1 Assistants Menu Options The full list of Assistants tables is given in the menu bar as shown in Figure 2 6 1 including Compute Effect Size and Specify Multi Factor table A very common Assistant table that is regularly required is the Compute Effect Size table Once the appropriate information is entered nTerim will calculate the values required for the main test table Once the user is happy with the values entered and calculated they can click Transfer and the required values from the Assistant table will be transferred up to the main design table An example of the Compute Effect Size assistant table is shown below in Figure 2 6 2 Compute Effect Size Assistant Variance of means V Total sample size N N as multiple of n1 Sri Ini ni Figure 2 6 2 Example of Effect Size Assistant Table The Specify Covariance Matrix assistant table where available can be utilised by the user to manually define the
21. row as shown in Figure 4 6 12 134 File Edit View Assistants Plot Tools Window Help gt E New Fixed Term Test New Interim Test Z Plot Power vs Sample Size LU Open Manual Statistical Solutions Support Figure 4 6 11 Multivariate Analysis of Variance Table 5 The next step in this process is to specify the number of levels per factor This can be done using the Factor Level Assistant table illustrated in Figure 4 6 13 6 In this example we are going to specify 3 levels for Factor A 3 levels for Factor B and 3 levels for Factor C 7 We can also alter the default settings of 0 05 for the alpha value This represents the significance level for each factor In this example we will leave it at 0 05 8 Finally the as we are calculating attainable power the Power is where our output power values for each factor will appear thus we leave this column empty 135 Edit View Assistants Plot Tools Window Help I New Fixed Term Test New Interim Test Z Plot Power vs Sample Size LU Open Manual Statistical Solutions Support Figure 4 6 12 Enter Number of Response variables 9 Once the number of levels for each factor has been specified the next step is to populate the Means Matrix The Means Matrix is displayed in Figure 4 6 14 10 Depending on the values entered into the Factor Level table the size of the means matrix will be created wij Factor Level Table u Means M
22. 0 8 0 8 0 8 Rate Ratio ri r0 0 85 0 85 0 85 Average Exposure Time pT 0 75 0 75 0 75 Dispersion Parameter k 07 0 7 0 7 Rates Variance Reference Group Rate z True Rates z Maximum Likelihood z Reference Group Rate z Sample Size Ratio N1 N0 1 1 1 Control Group Sample Size N0 1433 1494 1490 Treatment Group Sample Size N1 1433 1494 1490 80 0115952771 80 0002694664 80 0254850802 Cost per sample Total study cost Calculate required sample size for given power X Run 7 All columns Figure 4 9 10 Result from comparing Rate Variance options This will give the resultant sample sizes of 1433 1494 and 1490 sequentially as in Figure 4 9 10 A similar graph and output statement can be generated for this example as for the above example 167 168 4 10 Two Incidence Rates 4 10 1 Introduction Incidence data is often obtained in a variety of clinical and field research activities such as the number of accidents at a junction or number of occurrences of a disease in a year Researchers often plan studies in terms of years of exposure rather than directly through sample size This two sample test is used to test hypotheses about the difference between two incidence rates in terms of person years of exposure This table facilitates the calculation of the power and sample size for hypothesis tests comparing two incidence rates Power and sample size is computed using the method outlined by Smith and Morrow 1
23. 1 1 1 1 Treatment Group Incidence Rate A2 0 8 0 9 11 1 2 Difference in Rates A1 A2 0 2 0 1 0 1 0 2 Sample Size per Group in Person Years 354 1492 1649 432 80 80 80 80 Cost per sample Total study cost Mal oomoo gt Calculate required sample size for given power X All columns Figure 4 10 6 Results from Sensitivity analysis The effect of changing the test statistic on the relationship between Power and Sample size can be explored further using the Plot Power vs Sample Size button at the top of the screen 8 Select the first column by clicking the 1 at the top of column 1 Then hold down Shift and click the 4 at the top of column 4 All four columns will now be highlighted 9 Click the Plot Power vs Sample Size button at the top of the screen This will give you a plot as displayed in Figure 4 10 7 which will show the relationship between power and sample size for each column Right click to add features such as a legend to the graph and double click elements for user options and editing 173 Power vs Sample Size 2170 3170 Treatment Group Rate 0 8 Treatment Group Rate 0 9 Treatment Group Rate 1 1 Treatment Group Rate 1 2 1170 1670 2170 2670 3170 3670 Sample Size Figure 4 10 7 Power vs Sample Size plot 10 Finally by clicking on the desired study design column and going to the Output tab at the bottom of the screen you can get an output statement f
24. 5 Now it can be seen from Figure 4 4 10 that there is a change in Effect Size and ultimately Power due to both increasing and decreasing the Common Standard Deviation It s easy to compare the implications of a slight increase or decrease in the Common Standard Deviation 6 Another feature that enables us to compare designs side by side is by using the Power vs Sample Size plot Multiple columns can be plotted together by simply highlighting the desired columns and clicking on the Plot Power vs Sample Size button on the menu bar 7 To highlight the desired columns click on the column title for Column 1 and drag across to Column 3 Then click on the Plot Power vs Sample Size button on the menu bar The multiple column plot is displayed in Figure 4 4 11 Power vs Sample Size Oo a ee Power vs Sample Size 10 30 50 70 90 110 Tog 999 a so e a p id 90 804 80 ig Std Dev 6 Std Dev 4 Std Dev 8 Power 10 20 30 40 50 60 70 80 90 100 110 120 Power 82 62 Total sample size N Sample Size 59 Figure 4 4 11 Multiple Column Power vs Sample Size Plot It can be seen from the legend on the left hand side legend can be altered manually that the blue line represents Column 1 the orange line represents Column 2 and the red line represents Column 3 The cross on the graph illustrates how the user can identify what the sample size is for a corresponding power value fo
25. Boundary Plot for Column 2 Likewise by clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation Output Statement Column 2 Sample sizes of 43 in group 1 and 86 in group 2 are required to achieve 90 5 power to detect a difference in means of 20 the difference between group 1 mean 1 of 220 and group 2 mean u2 of 200 assuming that the common standard deviation is 30 using a 2 sided z test with 0 05 significance level These results assume that 5 sequential tests are made and the Pocock spending function is used to determine the test boundaries Drift 3 56942 31 32 Im 3 2 Two Proportions 3 2 1 Introduction nTerim 3 0 is designed for the calculation of Power and Sample Size for both Fixed Period and Group Sequential design In relation to Group Sequential designs calculations are performed using the Lan DeMets alpha spending function approach DeMets amp Lan 1984 DeMets amp Lan 1994 for estimating boundary values Using this approach boundary values can be estimated for O Brien Fleming O Brien amp Fleming 1979 Pocock Pocock 1977 Hwang Shih DeCani Hwang Shih amp DeCani 1990 and the Power family of spending functions Calculations follow the approach of Reboussin et al 1992 and Jennsion amp Turnbull 2000 Calculations can be performed for studies that involve comparisons of means comparisons of proportions and survival studies as
26. Click the Plot Power vs Sample Size button at the top of the screen This will give you a plot as displayed in Figure 4 9 7 which will show the relationship between power and sample size for each column Right click to add features such as a legend to the graph and double click elements for user options and editing 165 Power vs Sample Size Mean Rate ro 0 8 Mean Rate ro 1 0 Mean Rate ro 1 2 Mean Rate ro 1 4 Figure 4 9 7 Power vs Sample Size plot 10 Additionally by clicking on the desire study design column and going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 4 9 8 Output OUTPUT STATEMENT The comparison of a negative binomial rate for a control group and a negative binomial rate for a treatment group using a control group sample of 1433 and a treatment group sample of 1433 would achieve 80 0115952771 power at the 0 05 significance level when the mean rate for the control group was 0 8 and the rate ratio was 0 85 The mean exposure time was 0 75 and the two groups had the common dispersion parameter of 0 7 a Specify Multiple Factors a Output Figure 4 9 8 Study design Output statement 11 Finally we investigate the effect on the choice of Rates Variance on sample size To do this delete the columns 2 to 4 and copy the co
27. Confidence Interval Survival lt Equivalence Agreement Regression Cluster Randomized One sample t test One sample t test Finite Population Paired t test for difference in Means Paired t test for difference in Means Finite Population Univariate one way repeated measures analysis of variance One way repeated measures contrast Univariate one way repeated measures analysis of variance Greenhouse Geisser Poisson One Mean Figure 4 7 1 Study Goal and Design Window 2 In order to select the Poisson One Mean design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Means as the Goal One as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 4 7 2 143 144 Im File Edit View Assistants Plot Tools Window Help i E New Fixed Term Test gt New Interim Test Plot Power vs Sample Size Poisson One Mean 1 P 2 3 4 Null or Baseline Mean Poisson Rate A0 Alternative Mean Poisson Rate A1 Sample Size N Power Cost per sample Total study cost poe e gt Calculate attainable power with the given sample size xX l Run E All columns Figure 4 7 2 Poisson One Mean Test Table The first calculation will be for Sample Size to begin we enter the values as follows 3 First enter 0 05 for the Test S
28. E All columns Figure 4 2 16 Fill Right Function Shortcut 10 Once all the parameter information has been entered right click on the Column 1 heading and select Fill Right from the drop down menu as shown in Figure 4 2 16 11 As illustrated in Figure 4 2 17 all columns have been filled in with the same parameter information contained in Column 1 We want to alter the other columns Columns 2 to 4 to see how the sample size is affected by various parameter changes 12 In this example we want to investigate how the sample size will be affected by a change in the Ratio between the two groups sample sizes To do this we will enter Ratio values of 2 3 and 4 in Columns 2 3 and 4 respectively 85 Melim 13 As we want to calculated the required sample size to obtain the given power select Calculate required sample sizes for given power from the drop down menu below the test table File Edit View Assistants Plot Tools Window Help i New Fixed Term Test New Interim Test Z Plot Power vs Sample Size RM Two Means 1 EEaee 1 2 J Test significance level a 0 05 0 05 0 05 1 or 2 sided test 2 x2 x2 _ Number of levels M J5 5 5 Difference in means p1 p2 40 40 40 Standard deviation at each levego 80 80 80 Between level correlation p 0 5 0 5 0 5 Group 1 size n1 i Group 2 size n2 Ratio n2 ni 1 2 3 Power 85 85 85 Cost per sample unit i 75 T
29. M1 Cluster Sample Size in Control Group M2 Power Cost per sample Total study cost EJE Differences x Differences E Differences E Run All columns Calculate attainable power with the given sample size and number of clusters Figure 5 4 2 CRT Two Proportions Non Inferiority Completely Randomized Test Table The first calculation will be for the Clusters in Treatment Group K1 This example is can be undertaken using the following steps 3 First enter 0 05 for the Test Significance level row Next select Unpooled from the Test Type option Better from the Higher Proportions Better Worse option and enter a control group proportion of 0 5 4 Next select Differences from the Solve Using dropdown option and enter 0 1 for the Non Inferiority Test Statistic and 0 05 for the Actual Value of Test Statistic Enter 0 001 for the intracluster correlation Enter 100 for both cluster sample size variables and 80 for power 5 Finally enter K1 in the Clusters in Control Group K2 row This will solve so that K1 and K2 must be equal Other ratios between K1 and K2 can be calculated by using arguments for K2 such as 2 K1 or K1 2 to have K2 be twice as large as K1 for example The table will appear as in Figure 5 4 3 245 Edit View Assistants Plot
30. No Don t Calculate O Brien Fleming Calculate required sample sizes for given power J C Figure 3 2 2 Two Proportions Test Table x off z Equally Spaced Xxx z x Spending Function Spending Function x x O Brien Fleming No Don t Calculate O Brien Fleming i B Lz All columns 8 Once all the values have been entered select Calculate required sample size for given power from the drop down menu and click Run 38 File Edit View Assistants Plot Tools New Fixed Term Test __ New Interim Test Window Help Z Plot Power vs Sample Size GST Two Proportions 1 Xx _ O g F z j Test significance level a 0 05 1 or 2 sided test fi BE z z Sro proportio 0 4 _ Group 2 proportions n2 0 6 Odds ratio W n2 1 n1 n1 1 n2 2 25 Group 1 size ni 129 Group 2 size n2 129 Ratio N2 N1 l1 1 i i Continuity correction off x off x J off off z Power 90 12 Cost per sample unit 180 Total study cost 46440 Number of looks 5 5 5 5 Information Times Equally Spaced z Equally Spaced iz Equally Spaced z Equally Spaced z Max times l1 1 1 1 Determine bounds Spending Function Spending Function Spending Function Spending Function Spending function Pocock a H O Brien Fleming E O Brien Fleming E
31. Open Two Proportions Non Inferiorit A e 2 3 4 Test significance level a 0 05 Test Type Unpooled Likelihood score z Likelihood score z Likelihood score z Higher Proportions Better Worse Better Better x Better x Better z Control Group Proportion p2 0 5 Solve using p1 p2 p1 or p1 p2 Differences z Differences z Differences z Differences z Non inferiority Test Statistic 0 1 Actual Value of Test Statistic 0 05 Intracluster Correlation ICC 0 001 Clusters in Treatment Group K1 14 Clusters in Control Group K2 14 Cluster Sample Size in Treatment Group M1 100 Cluster Sample Size in Control Group M2 100 Cost per sample Total study cost MA e E gt Calculate required treatment group clusters K1 given power and sample size v All columns Figure 5 4 4 Completed CRT Two Proportions Non Inferiority study design The next calculation is a sensitivity analysis for the treatment group number of clusters when the control group number of clusters is changed 7 Delete the values for K1 and K2 in the first column then replace the updated power with 80 and enter K1 in the control group clusters row Then copy the first column into columns 2 to 4 Other ratios other than K1 and K2 being same between K1 and K2 can be calculated by using arguments for K2 such as 2 K1 or K1 2 to have K2 be twice as large as K1 for example 8 Enter K1 2 in column 2 for Clusters in Control Group K2
32. Univariate one way repeated measures analysis of variance Greenhouse Geisser OK Cancel Figure 4 1 1 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear 3 There are two main tables required for this test the main test table illustrated in Figure 4 1 2 and the effect size assistant table shown in Figure 4 1 3 4 Enter 0 05 for alpha the desired significance level and enter 3 for the number of levels M as shown in Figure 4 1 4 5 Now you are required to complete the Compute Effect Size Assistant table in order to calculate values for the Contrast C and Scale D parameters nQuery nTerin File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Contrast 1 Number of levels M Contrast C Icirpi Scale D SQRT Sci Standard deviation at each level o Between level correlation p Effect size A C D o SQRT 1 p Power Group size N Cost per sample unit Total study cost OT gt Calculate required sample sizes for given power All columns Figure 4 1 2 One way Repeated Measures Contrast Test Table u Compute Effect Size Assistant Specify Multiple Factors i Output Figure 4 1 3 Compute Effect Size Assistant Table 6 Once you enter a value for the number of levels M the Co
33. a Truncate bounds No No x No No z Truncate at Futility boundaries Don t Calculate z Don t Calculate x Don t Calculate Don t Calculate z Spending function z O Brien Fleming ix O Brien Fleming m pata x Phi e 1k Figure 3 2 3 Completed Two Proportions Test Table 9 The boundaries calculated are shown in Figure 3 2 4 Looks Lower bound 0 2 8 00000 Cumulative exit probability under HO Upper bound 2 17621 Nominal alpha 0 01477 Incremental alpha 0 01477 Cumulative alpha 0 01477 Cumulative exit probability 22 98 Nominal beta Incremental beta Cumulative beta 4 0 4 8 00000 2 14371 0 01603 0 01139 0 02616 25 92 48 90 1 0 6 0 8 8 00000 8 00000 8 00000 2 11322 2 08952 2 07091 0 01729 0 01833 0 01918 0 00927 0 00782 0 00676 0 03543 0 04324 0 05000 20 00 13 21 8 00 68 90 82 11 90 12 ui Looks jus Specify Multiple Factors g Output Figure 3 2 4 Boundary Table for Pocock Spending Function 39 Melim 40 10 Finally the boundaries calculated in the table in Figure 3 2 4 are automatically plotted as illustrated in Figure 3 2 5 oraison O a Pocock Boundaries with Alpha 0 05 2 a 4 Figure 3 2 5 Boundary Plot for Two Proportions one sided Test By clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation Sample sizes of at l
34. and oz are known e Equation 3 1 1 can be expressed as a quadratic in 44 or u2 The roots give the unknown LL By default nTerim assumes that u4 lt fz and will select the appropriate root 23 24 3 1 3 Examples Example 1 O Brien Fleming Spending Function This example is adopted from Reboussin et al 1992 using the O Brien Fleming spending function 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Interim Test from the tool bar at the top of the window A Study Goal and Design window will appear as shown below Select the options as mapped out in Figure 3 1 1 then Click OK es Study Goal And Design ax Design No of Groups D Fixed Term Interim Group Sequential Test of Two Means Figure 3 1 1 Study Goal and Design Window Now you have opened the test table as illustrated in Figure 3 1 2 you can begin entering values Enter 0 05 for alpha 2 sided 220 for Group 1 mean 200 for Group 2 mean The difference in means is calculated as 20 Enter 30 for Standard Deviation for Group 1 and Group 2 We are interested in solving for sample size given 90 power so enter 90 in the Power row This study planned for 4 interim analyses Including the final analysis this requires Number of Looks to be 5 The looks will be equally spaced and the O Brien Fleming spending function is to be used There will be no t
35. gt Two Equivalence Agreement Regression Cluster Randomized Logistic Regression one normal covariate Logistic Regression one normal covariate adjusted for others Linear Regression one covariate Linear Regression multiple covariates Linear Regression multiple covariates adjusted for others Linear Regression test of coefficient Cox Regression Poisson Regression Probit Regression ok cane Figure 4 12 1 Study Goal and Design Window 2 In order to select the Poisson Regression design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Regression as the Goal One as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 4 12 2 187 188 File Edit View Assistants Plot Tools Window Help I New Fixed Term Test New Interim Test Zz Plot Power vs Sample Size 1 or 2 sided test Baseline Response Rate eB0 Response Rate Ratio eB1 eB0 Mean Exposure Time pT Overdispersion Parameter Distribution of X1 sidetable required Variance of b1 Null Hypothesis Variance of b1 Alternative Hypothesis Calculate attainable power with the given sample size Run An columns Figure 4 12 2 Poisson Regression Test Table At the bottom will be the Normal S
36. r Calculate group size using Hotelling Lawley trace Calculate power using Wilks lambda a columns X Figure 4 6 8 Selecting Type of Test to Run wi Covariance Matrix u Specify Multiple Factors Output 22 In order to view the results for Power for each level the power values are displayed in the Factor Level Assistants table as illustrated below in Figure 4 6 9 Factor Level Table x u Factor Level Table u Means Matrix Group Sizes ui Covariance Matrix ui Specify Multiple Factors ui Output Figure 4 6 9 Output Power values calculated 23 Finally the output statement can be obtained by clicking on the Output tab on the bottom of the nTerim window Output Statement A multivariate analysis of variance design with 2 factors and 2 response variables has 12 groups When the total sample size across the 12 groups is 61 distributed across the groups as specified a multivariate analysis of variance will have 95 41 power to test Factor A if a Pillai Bartlett Trace test statistic is used with 0 05 significance level 73 99 power to test Factor B if a Pillai Bartlett Trace test statistic is used with 0 05 significance level 99 69 power to test Factor AB if a Pillai Bartlett Trace test statistic is used with 0 05 significance level 133 im Example 2 Wilks Lambda In this example we will calculate the attainable power given a specified sample size using the Wilks Lambda meth
37. the cost per sample unit of 100 the overall cost of sample size required has amounted to 45 600 By clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation When the group sample size n is 152 the test of a single contrast at the 0 05 level in a one way repeated measures analysis of variance with 3 levels will have 89 95 power to detect a contrast C dci pi of 2 with a scale D SQRT Dci of 1 41421 assuming a standard deviation at each level of 6 and a between level correlation of 0 2 63 64 im Example 2 Examining M Period Crossover Design This design may require treatments to appear an equal number of times per each sequence It can be assumed these sequences are chosen in order to prevent confounding from occurring between treatment and period effects Therefore this is ensuring the design is balanced In this example we will investigate a three period two treatment design of ABB and BAA The following steps outline the procedure for Example 2 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear E Study Goal And Design x Design Goal No of Groups Analysis Method Fixed Term Means Test Interim Proportions Confidence Interval 5 Survival
38. 100 Cluster Sample Size in Control Group M2 100 so Cost per sample Total study cost 4 alll all gt Calculate required treatment group clusters K1 given power and sample size v Run E All columns Figure 5 2 3 Values entered for CRT Two Proportions Completely Randomized study design 6 Select Calculate required treatment group clusters K1 given power and sample size from the dropdown menu beside the Run button Then click Run This will give a result of 38 for K1 and K2 as in Figure 5 2 4 nQuery nTerim File Edit View Assistants Plot Tools Window Help gt I New Fixed Term Test gt New Interim Test Z Plot Power vs Sample Size LLI Open Two Proportions Inequality Co CO m m Test significance level a 0 05 1 or 2 sided test 2 2 2 z 2 z Test Type Unpooled x Likelihood score x Likelihood score z Likelihood score z Control Group Proportion p2 0 06 Solve using p1 p2 p1 or p1 p2 Proportions z Differences z Differences z Differences z Test Statistic under HO 0 06 Test Statistic under H1 0 04 Intracluster Correlation ICC 0 01 Custers in Treatment Group K1 38 Clusters in Control Group K2 38 Cluster Sample Size in Treatment Group M1 100 Cluster Sample Size in Control Group M2 Ea Cost per sample Total study cost Calculate required treatment group clusters K1 given power and sample si
39. 4 K1 in column 3 and 30 in column four This will give a table as per Figure 5 4 5 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size LU Oper 7 Two Proportions Non Inferiorit 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 Test Type Unpooled Unpooled x Unpooled Unpooled X Higher Proportions Better Worse Better Better Better Better m Control Group Proportion p2 0 5 0 5 0 5 0 5 Solve using p1 p2 p1 or p1 p2 Differences v Differences v Differences x Differences X Non inferiority Test Statistic 0 1 0 1 0 1 0 1 Actual Value of Test Statistic 0 05 0 05 0 05 0 05 Intracluster Correlation ICC 0 001 0 001 0 001 0 001 Clusters in Treatment Group K1 Clusters in Control Group K2 K1 K1 2 4 K1 30 Cluster Sample Size in Treatment Group M1 100 100 100 100 Cluster Sample Size in Control Group M2 100 100 100 100 wer a 80 80 Cost per sample Total study cost Malo m gt Calculate required treatment group clusters K1 given power and sample size v Run V All columns Figure 5 4 5 Sensitivity analysis around the Control Group Number of Clusters 9 Select Calculate required treatment group clusters K1 given power and sample size from the dropdown menu beside the Run button Then tick the box for All Columns The
40. 47 48 Im 3 3 3 Examples Example 1 O Brien Fleming Spending function with Power vs Sample Size Plot 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Interim Test from the menu bar at the top of the window A Study Goal and Design window will appear as shown below Select the options as mapped out in Figure 3 3 1 then Click OK GB gt Study Goal And Design A Design No of Groups Analysis Method Fixed Term e Test Interim Proportions Two Survival Group Sequential Test of Two Survivals Figure 3 3 1 Study Goal and design Window 2 Enter 0 05 for alpha 2 sided 0 3 for Group 1 proportion this is the proportion surviving until time t and 0 45 for Group 2 proportion The hazard ratio is calculated as 1 508 3 Select Exponential Survival for the Survival time assumption 4 We are interested in solving for sample size given 90 power so enter 90 in the Power row 5 This study planned for 4 interim analyses Including the final analysis this requires Number of Looks to be 5 6 The looks will be equally spaced and the O Brien Fleming spending function is to be used There will be no truncation of bounds 7 It is estimated that the cost per unit is roughly 100 so enter 100 in the Cost per sample unit row File Edit New Fixed Term Test View Assistants Plot New Inte
41. 4701829073 Total Number of Subjects 110 80 98 70 Cost per sample Total study cost 8 a All columns Figure 4 13 11 Results from Sensitivity analysis The effect of changing these values on the relationship between Power and Sample size can be explored further using the Plot Power vs Sample Size button at the top of the screen 14 Select the first column by clicking the 1 at the top of column 1 Then hold down the Shift key and click the 4 at the top of column 4 All four columns will now be highlighted 15 Click the Plot Power vs Sample Size button at the top of the screen This will give you a plot as displayed in Figure 4 13 12 which will show the relationship between power and sample size for each column Right click to add features such as a legend to the graph and double click elements for user options and editing 208 Power vs Sample Size Column 1 Column 2 Column 3 Column 4 Power 92 35 Sample Size 12 Figure 4 13 12 Power vs Sample Size plot 16 Finally by clicking on the desired study design column and going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 4 13 13 Output OUTPUT STATEMENT In a study using probit analysis to compare the poteng in a treatment group and a control group a sample size of 11 subjects from each of the
42. 5 ly Spaced 1 k z No z Dont Calculate z Equally Spaced Spending Function z Spending Function fafaa Function z Spending Function z Pocock Calculate O Brien Fleming x O Brien Fieming z z 1 1 5 5 z Equally Spaced z Equally Spaced 1 1 O Brien Fleming x O Brien Fleming Dont Calculate Dont Calculate O Brien Fleming y O Brien Fleming E B zno z no E B Calculate required sample sizes for given power All columns Figure 3 1 8 Comparison of two separate Means Tests 8 Also the boundary values will be recalculated and boundary plot will automatically be plotted as shown in Figure 3 1 9 and 3 1 10 below 2 43798 Upper bound 2 43798 Nominal alpha 0 01477 Incremental alpha 0 01477 Cumulative alpha 0 01477 Exit probability 20 00 Cumulative exit probability 20 00 Nominal beta Incremental beta Cumulative beta Cumulative exit probability under HO 04 2 42677 2 42677 0 01523 0 01139 0 02616 26 18 46 19 41 0 6 08 2 41014 2 39658 2 38591 2 41014 2 39658 2 38591 0 01595 0 01655 0 01704 0 00927 0 00782 0 00676 0 03543 0 04324 0 05000 21 44 14 34 8 54 67 62 81 96 90 50 u Looks u Specify Multiple Factors ua Output Figure 3 1 9 Boundary Table for Column 2 Pocock Boundaries with Alpha 0 05 2 3 4 Figure 3 1 10
43. 90 in the Power row and the table will appear as per Figure 4 13 4 203 204 Im File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Z Plot Power vs Sample Size L Probit Regression 1 _ a 2 3 4 Test significance level a 0 05 Number of Dose Levels S 5 5 5 Sum of Weights Slope of Probit Regression B1 23 25 Relative Potency p Sample Size per Group N Power 90 11 Cost per sample Total study cost Mab ooa bl Calculate attainable power with the given sample size Run Al columns Figure 4 13 4 Values entered for Probit Regression study design 5 Next we need to fill in the side table To do this we will fill the Target Response Proportions row with 0 05 0 275 0 5 0 725 0 95 This gives Figure 4 13 5 Probit Regression Side Table x 0 275 0 5 Slope of Probit Regression B1 u Specify Multiple Factors wij Probit Regression Side Table a Output Figure 4 13 5 Values entered for Probit Regression Assistant table 6 Next click the Compute button at the bottom of the Probit Regression side table and this will calculate and transfer the Sum of Weights value to the main table N Select Calculate required sample size for given power from the dropdown menu beside the Run button Then click Run This will give a result of 11 as per Kodell et al 2010 for the sample size per g
44. Assistants Plot Tools Window Help f _ New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Test significance level a Observation Time for Group 2 t2 Mean Poisson Rate in Group 1 y1 2 2 0 0005 Sample Size Allocation Ratio N2 N1 0 5 0 5 0 5 0 5 0 5 Sample Size in Group 1 N1 8565 6889 6685 6685 8590 Sample Size in Group 2 N2 4283 3445 3343 3343 4295 90 0034557027 2 2 0 0005 90 0021326631 2 2 0 0005 90 0030831823 Null Poisson Rate Ratio RO y0 y1 1 1 1 1 1 Alt Poisson Rate Ratio R1 y0 y1 4 4 4 4 4 Test Statistic W1 MLE w2 cme w3 tamie w W4 Ln CMLE ie WS Variance Stabili w1 2 2 0 0005 90 0030831823 Calculate required Group 1 and 2 sample sizes for given power and sample size allocation ratio v v All columns 2 2 0 0005 90 0014651088 Figure 4 8 6 Results from Sensitivity analysis The effect of changing the test statistic on the relationship between Power and Sample size can be explored further using the Plot Power vs Sample Size button at the top of the screen For this particular calculation W3 and W4 give the same answer and thus only W3 was plotted in this example 156 8 Select the first column by clicking the 1 at the top of column 1 Then hold down Ctrl and click the 2 at the top of column 2 the 3 at the top of column 3 and the 5 at t
45. C ay 2 3 4 Test significance level a 0 05 Number of levels M 4 Contrast C Scirpi 15 Scale D SQRT Zci 4 47214 Standard deviation at each level o 10 Between level correlation p 07 Effect size A C D o SQRT 1 p 0 51237 Power 90 32 29 Cost per sample unit __ Total study cost Le gt Calculate required sample sizes for given power z C All columns Figure 4 1 18 Completed One way Repeated Measures Contrast Table It can be seen from Figure 4 1 18 that a sample size of 29 per group for each of the three groups thus a total sample size N of 116 is required to obtain a power of 90 32 By clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation When the group sample size n is 29 the test of a single contrast at the 0 05 level in a one way repeated measures analysis of variance with 4 levels will have 90 32 power to detect a contrast C Sci pi of 15 with a scale D SQRT Sci of 4 47214 assuming a standard deviation at each level of 10 and a between level correlation of 0 7 71 72 im 4 2 Repeated Measures Design for Two Means 4 2 1 Introduction A repeated measures design is an experimental design in which multiple measurements are taken on one or more groups of subjects over time or under different conditions This type of design leads to a more precise estimate of an endpoint and can avoid
46. Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test gh Plot Power vs Sample Size ww Open Manual Statistical Solutions Support GST Survival 1 x i a Asse 2 i 3 E 4 Test significance level a 0 05 1 or 2 sided test 2 z z z Group 1 proportion n1 attimet 0 3 Group 2 proportion n2 at time t 0 45 Hazard ratio h in n1 in n2 1 508 Survival time assumption Exponential Surviva Exponential Surviva a Exponential Survival Exponential Surviva Total sample size N 1409 Power 90 07 Number of events 256 Cost per sample unit 100 Total study cost 40900 Number of Looks 5 5 5 5 Information times Equally Spaced z Equally Spaced z Equally Spaced z Equally Spaced z Max Times 1 1 1 1 Determine bounds Spending Function Spending Function Spending Function x Spending Function Spending function O Brien Fleming x O Brien Fleming w O Brien Fleming w O Brien Fleming x Phi Truncate bounds No z No z No x No x Truncate at Futility boundaries Dont Calculate y Don t Calculate Don t Calculate v Spending function O Brien Fleming O Brien Fleming O Brien Fleming u Phi eo a gt Calculate required sample sizes for given power X All columns Figure 3 3 3 Complete Survival Table for One test In addition to the sample size and cost output for Column 1 the boundary calculations are also presented as shown below Looks
47. Effect size A C D o SQRT 1 p Power Group size N Cost per sample unit Total study cost Calculate required sample sizes for given power All columns Figure 4 1 14 One way Repeated Measures Contrast Test Table Compute Effect Size Assistant x Conti iJ Compute Effect Size Assistant u Specify Multiple Factors ui Output Figure 4 1 15 Compute Effect Size Assistant Table 6 Once you enter a value for the number of levels M the Compute Effect Size Assistant table automatically updates as shown in Figure 4 1 16 7 In order to calculate a value for Effect Size two parameters need to be calculated first the Contrast C and Scale D 8 The mean for each level and the corresponding coefficient value need to be entered in the Compute effect Size Assistant table 9 For the Mean values for each level enter 55 for level 1 56 5 for level 2 58 for level 3 and 59 5 for level 4 10 For the Coefficient values for each level enter 3 for level 1 1 for level 2 1 for level 3 and 3 for level 4 The sum of these values must always equate to zero This is illustrated in Figure 4 1 17 below 69 70 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size Scale D SQRT Sci2 Standard deviation at each level o Between level correlation p Effect size A C
48. Fixed Term Test New Interim Test 2 2013513108 4 3637891506 11 305 90 0004548763 Run E att columns Figure 4 13 15 Completed Probit Regression study design Melim Chapter 5 Cluster Randomized Trials 212 5 1 CRT Two Means Completely Randomized 5 1 1 Introduction Continuous data is found in nearly every area of research interest Procedures such as Z and t tests are used to evaluate the differences between two continuous means Clustered data is very common in a wide variety of academic social policy and economic studies This two sample test is used to test hypotheses about the difference between two means in a completely randomized cluster randomized trial This table facilitates the calculation of the power and sample size for hypothesis tests comparing means in a cluster randomized trial Power and sample size is computed using the method outlined by Donner and Klar 1996 5 1 2 Methodology This table provides sample size and power calculations for studies which will be comparing means which use a completely randomized cluster randomization study design A completely randomized design assigns clusters randomly to control and treatment groups This table assumes a balanced study design The sample size calculation for means is taken from Donner and Klar 1996 and is an extension of the t test for the comparison of two independent means The extension uses the intraclust
49. For this example it is known that the proportion of interest in Group 1 ranges from 0 45 to 0 55 and the proportion of interest in Group 2 ranges from 0 39 to 0 51 Therefore we want to see what the required samples sizes would be at the extremes of these ranges For example at the maximum proportion for Group 1 and the minimum proportion for Group 2 95 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size fest significance level a 1 or 2 sided test Number of levels M Between level correlation p Group 1 proportion p1 Group 2 proportion p2 Odds ratio p2 1 p1 p1 1 p2 Group 1 size ni Group 2 size n2 Ratio n2 ni 1 1 1 1 Power Cost per sample unit Total study cost Calculate required sample sizes for given power X Run All columns Figure 4 3 8 Repeated Measures for Two Proportions Test Table 5 By incorporating the Specify Multiple Factors table shown in Figure 4 3 9 the user can specify many designs columns by entering the desired parameter values and ranges in the provided boxes 6 We just want to define a two sided test design Enter 2 in the 1 or 2 sided test box In this study we want 3 levels so enter 3 in the Number of levels M box We also know that the between level correlation is 0 4 so enter 0 4 in the Between level correlation box Specify M
50. L Figure 2 4 3 Example of Fixed Term Design Interface As it can be seen from Figure 2 4 3 the Fixed term design window is split into three main sections i the test table ii Assistant Tables amp Output and iii Help Guide Cards The main table represents the test table In this example it is an ANCOVA table Values for various parameters can be entered by the user For some tests additional values need to be calculated This is provided for by using the Assistants tables found at the bottom half of the interface Additional calculations can be done and the appropriate values can be transferred from the Assistants tables to the main test table Once all the appropriate information has been entered in the test table the user must select the appropriate calculation to run i e whether you want to solve for power given a specified sample size or solve for sample size given a specified power The user can select the appropriate calculation to run from the drop down menu between the main test table and the Assistants table Once the appropriate test is selected the user must click on Run to run the analysis If multiple columns have been specified by the user there is an option to run the calculation for all the columns This is achieved by simply ticking the All columns box beside the Run button before clicking Run This will tell nTerim to concurrently run the calculations for all columns Then by
51. One Test D Interim Proportions Two Confidence Interval Survival gt Two Equivalence Agreement Regression Cluster Randomized CRT Two Means Completely Randomized CRT Two Proportions Inequality Completely Randomized CRT Two Proportions Equivalence CRT Two Proportions Non Inferiority CRT Two Proportions Superiority OK Cancel Figure 5 4 1 Study Goal and Design Window 2 In order to select the CRT Two Proportions Non Inferiority Completely Randomized design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Cluster Randomized as the Goal Two as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 5 4 2 Edit New Fixed Term Test New Interim Test W Plot Power vs Sample Size LUI Open Plot View Assistants Tools Window Help File Two Proportions Non Inferiorit 2 3 4 Likelihood score Likelihood score Likelihood score x Likelihood score Better z Better Test Type Higher Proportions Better Worse Better Better Control Group Proportion p2 Solve using p1 p2 p1 or p1 p2 Differences Non inferiority Test Statistic Actual Value of Test Statistic Intracluster Correlation ICC Clusters in Treatment Group K1 Clusters in Control Group K2 Cluster Sample Size in Treatment Group
52. Plot Tools Window Help i E New Fixed Term Test New Interim Test Plot Power vs Sample Size ON Open Negative Binomial 1 SS A 2 3 4 Test significance level a 0 05 Mean Rate of Event for Control r0 0 8 Rate Ratio r1 r0 0 85 Average Exposure Time pT 0 75 Dispersion Parameter k 0 7 Rates Variance Reference Group Rate r Reference Group Rate z Reference Group Rate z Reference Group Rate z Sample Size Ratio N1 N0 1 Control Group Sample Size N0 Treatment Group Sample Size N1 Cost per sample Total study cost All columns Figure 4 9 3 Values entered for Negative Binomial study design 163 164 Melim 5 Select Calculate required sample size for given power from the dropdown menu beside the Run button Then click Run Query nTeri File Edit View Assistants Plot Tools Window Help i E New Fixed Term Test gt New Interim Test Plot Power vs Sample Size Luo Negative Binomial 1 4 enna ened 2 3 7 Test significance level a 0 05 Mean Rate of Event for Control r0 0 8 Rate Ratio ri r0 0 85 Average Exposure Time pT 0 75 Dispersion Parameter k 0 7 Rates Variance Reference Group Rate z Reference Group Rate z Reference Group Rate z Reference Group Rate z Sample Size Ratio N1 N0 1 Control Group Sample Size NO 1433 Treatment Group Sample Size N1 1433 Power
53. Plot Tools Window Help 3 New Fixed Term Test New Interim Test a Plot Power vs Sample Size LU Open Manual Statistical Solutions Support Calculate power using Pillai Bartlett trace Run All columns Factor Level Table wi Factor Level Table Means Matrix Group Sizes W Covariance Matrix i Specify Multiple Factors Output Figure 4 6 2 Multivariate Analysis of Variance Design Window 5 The next step in this process is to specify the number of levels per factor This can be done using the Factor Level Assistant table illustrated in Figure 4 6 4 6 In this example we are going to specify 4 levels for Factor A and 3 levels for Factor B Seeing as we only highlighted two response variables in this example we can leave Factor C empty 7 We can also alter the default settings of 0 05 for the alpha value This represents the significance level for each factor In this example we will leave it at 0 05 8 Finally the as we are calculating attainable power the Power is where our output power values for each factor will appear thus we leave this column empty 129 File Edit View Assistants Plot Tools Window Help ew Fixed Term Test EB New Interim Test a Plot Power vs Sample Size LU Open Manual Statistical Solutions Support ce a I Calculate power using Pillai Bartlett trace Run All columns Factor Level Table o levels Alp
54. Response Rate 0 5 oe Response Rate 0 9 Response Rate 1 3 Response Rate 2 0 a a a I FS Ca SL AL SSD FSIS A SSL SPL 40 0 200 400 600 800 1000 1200 1400 Sample Size Figure 4 12 14 Power vs Sample Size plot 13 Finally by clicking on the desired study design column and going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 4 12 15 194 Output x OUTPUT STATEMENT A poisson regression of a dependent variable on a Normal distributed independent variable using a sample of 21 observations would achieve 95 792 power at the 0 05 significance level to detect a response rate ratio of at least 0 5 due to a unit change in the independent variable if the baseline rate was 1 and the mean exposure time was 1 Side Table value s Mean 0 Standard Deviation 1 g Normal Side Table jug Exponential Side Table jg Uniform Side Table jag Binomial Side Table E Specify Multiple Factors Jug Output Figure 4 12 15 Study design Output statement The next calculation is a sensitivity analysis for sample size where we change the distribution of the independent variable and see its effect on sample size These values are taken from Table 2 of Signorini 14 First delete the two Variance of b1 variables in the main table and the sample sizes for all four columns Then enter 0 5 for Response Rate Ratio and
55. Treatment Group Incidence Rate A2 0 8 Difference in Rates A1 A2 0 2 Sample Size per Group in Person Yea sos Cost per sample Total study cost 7m Erun All columns Figure 4 10 3 Values entered for Two Incidence Rates study design 5 Select Calculate required sample size for given power from the dropdown menu beside the Run button Then click Run This will give a result of 354 for the sample size as in Figure 4 10 4 The result in Smith and Morrow is 353 but this is due to two decimal place rounding in their calculations 171 IM File Edit View Assistants Plot Tools Window Help 3 New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Two Incidence Rates 1 Co m 2 3 4 Test significance level a r 1 or 2 sided test Control Group Incidence Rate A1 Treatment Group Incidence Rate A2 0 8 Difference in Rates A1 A2 Sample Size per Group in Person Yea Cost per sample Total study cost 4 m Calculate required sample size for given power C Run an columns Figure 4 10 4 Completed Two Incidence Rates study design 6 The next calculation is a sensitivity analysis for sample size where we change the treatment group incidence rate to investigate the impact this has on the sample size estimate To do this copy the same values across to columns 2 to 4 and delete the values for sampl
56. Two Proportions Repeated Measures Study 91 92 Melim 8 The cost per sample unit has been estimated as 120 in this particular study Therefore to calculate the overall cost associated with the sample size enter 120 in the Cost per sample unit row in order to calculate the total study cost associated with the sample size 9 Then select Calculate required sample size for given power from the drop down menu below the main table and click Run This is displayed in Figure 4 3 4 Plot File Assistants Tools Window Edit View Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Group 1 proportion p1 0 45 Group 2 proportion p2 0 55 Odds ratio W p2 1 p1 p1 1 p2 1 49383 Group 1 size n1 349 Group 2 size n2 349 Ratio n2 n1 1 1 1 1 Power 90 01 120 7 E All columns Figure 4 3 4 Completed Repeated Measures Design for Two Proportions Calculate required sample sizes for given power 10 Now we are going to repeat this study design example except we re going to explore how the sample size varies as we alter the proportion in both Group 1 and Group 2 Previously in Column 1 we had a Group 1 proportion of 0 45 and Group 2 proportion of 0 55 Next we are going to proportions 0 40 and 0 55 for Group 1 and Group 2 respectively 11 We want to see the effects of changing the group proportion levels ha
57. a normally distributed variable from Table 2 of Signorini 1991 3 First enter 0 05 for the Test Significance level row then select 1 for the 1 or 2 sided test variable dropdown menu enter 1 and 0 5 for the Baseline Response Rate and Response Rate Ratio variables respectively enter 1 for both the Mean Exposure Time and Overdispersion Parameter variables and enter O for R squared parameter 4 Finally enter 95 in the Power row the table will appear as per Figure 4 12 7 189 190 nQuery nTeri File Edit View Assistants Plot Tools Window Help I New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Wo Poisson Regression 1 F eS ee 2 3 4 Test significance level a 0 05 1or 2 sided test 1 r 2 x 2 x 2 z Baseline Response Rate eB0 1 Response Rate Ratio eB1 eB0 0 5 Mean Exposure Time pT 1 Overdispersion Parameter 9 1 Distribution of X1 sidetable required Variance of b1 Null Hypothesis Variance of bi Alternative Hypothesis R squared X1 and independent variables Sample Size N Power z Normal z Normal z Normal z Total study cost IKE gt Calculate attainable power with the given sample size Run all columns Figure 4 12 7 Values entered for Poisson Regression study design 5 Next we fill in the side table for a normally distributed variable by
58. a value for the common standard deviation This is a measure of the variability between subjects within a group and is assumed to be the same for all groups Given the common standard deviation and variance of means nTerim will automatically calculate the effect size using the formula A2 LA 4 4 3 o2 In order to calculate power a value for the total sample size N must be entered remember this can also be read in from the effect size assistant nTerim then calculates the power of the design by first determining the critical value For DF a Where DF G 1 is the numerator degrees of freedom and DF N G is the denominator degrees of freedom The non centrality parameter A is then calculated using the equation A NA 4 4 4 Using these two values nTerim calculates the power of this design as the probability of being greater than Fpr pe a ON a non central F distribution with non centrality parameter A In order to calculate sample size nTerim does not use a closed form equation Instead a search algorithm is used This search algorithm calculates power at various sample sizes until the desired power is reached 4 4 3 Examples Example 1 One way ANOVA with unequal n s in a Blood Pressure Study In this example we will compare the reduction in blood pressure resulting from the use of three potential treatments i Placebo ii current Standard Drug and iii New Drug According to similar previous studies on
59. assuming that the common standard deviation is 30 using a 2 sided z test with 0 05 significance level These results assume that 5 sequential tests are made and the Pocock spending function is used to determine the test boundaries Drift 3 55903 In the main table in Column 2 enter the same parameter values again except enter a value of 2 for the Ratio parameter Don t forget to change the spending function to Pocock Select Run and the sample size will be re calculated as shown in Figure 3 1 8 below 29 30 File Edit View Assistants Plot Tools __ New Fixed Term Test __ New Interim Test Window Help Z Plot Power vs Sample Size O x wW Open Manual Statistical Solutions Support GST Two Means 1 XxX Test significance level a 0 05 1 or 2 sided test 12 Group 1 mean p1 220 Group 2 mean p2 200 Difference in means pi p2 20 Group 1 standard deviation o1 J30 Group 2 standard deviation o2 30 Effect size 3 0 667 Group 1 size n1 57 Group 2 size n2 57 Ratio N2 N1 p Power 90 33 Cost per sample unit 250 Total study cost 28500 Number of looks 5 Information times Equal Max Times 1 Determine bounds Spending function Pocod Truncate bounds No Truncate at Futility boundaries Dont Spending function Phi p0 m 0 05 z 2 220 200 20 30 30 0 667 43 86 2 90 5 250 32250
60. bit or 64 bit processor Minimum of 450MHz processor Hard Disc 150MB for the nTerim software package RAM 512MB Additional Software Microsoft NET Framework Service Pack 3 5 Note Administrative privileges to the end users machine will be required for installation process only 1 2 Validation The calculations contained within this software package have been widely and exhaustively tested Various steps of each calculation along with the results have been verified using many text books and published journal articles Furthermore the calculations contained within this software package have been compared to and verified against various additional sources when possible 1 3 Support For issues pertaining to the methodology and calculations of each test in nTerim there is a brief outline of how each test is calculated in the Methodology section of each test chapter of the manual There are accompanying references for each test throughout the text and can be located in the References section of the manual If further clarification is required please contact our support statisticians by email at support statsols com If there are any issues with any aspect of the installation process there are three approaches you can take i you can check the system requirements outline in Section 1 1 of this manual ii look up the installation help and FAQ s on our website http www statsols com products nquery advisor nterim and iii you can
61. can be undertaken using the following steps 3 First enter 0 05 for the Test Significance level row Next select Likelihood Score from the Test Type option Better from the Higher Proportions Better Worse option and enter a control group proportion of 0 5 Next select Differences from the Solve Using dropdown option and enter 0 1 for the Superiority Test Statistic and 0 15 for the Actual Value of Test Statistic Enter 0 001 for the intracluster correlation Enter 100 for both cluster sample size variables and 80 for power Finally enter K1 in the Clusters in Control Group K2 row This will solve so that K1 and K2 must be equal Other ratios between K1 and K2 can be calculated by using arguments for K2 such as 2 K1 or K1 2 to have K2 be twice as large as K1 for example The table will appear as in Figure 5 5 3 255 256 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size Two Proportions Superiority 1 a 2 3 4 Test significance level a 0 05 Test Type Likelihood score Likelihood score Likelihood score Likelihood score Higher Proportions Better Worse Better J Better Better u Better g Control Group Proportion p2 0 5 Solve using p1 p2 p1 or p1 p2 Differences Differences z Differences z Differences z Superiority Test Statist
62. comparing proportions in a completely randomized cluster randomized trial Power and sample size is computed using the method outlined by Donner and Klar 2000 219 220 Im 5 2 2 Methodology This table provides sample size and power calculations for studies which will be comparing proportions in a trial which uses a completely randomized cluster randomization study design A completely randomized design assigns clusters randomly to control and treatment groups The sample size calculation for cluster randomized proportions is taken from Donner and Klar 2000 and is an extension of the methods used to compare two proportions in a fully randomized trial The extension uses the intracluster correlation ICC which is the ordinary product moment correlation between any two observations in the same cluster to adjust for the effect of within cluster correlation It is assumed the ICC is the same in both groups This table can be used to calculate the power the number of clusters in the treatment group the sample size per cluster in the treatment group the intracluster correlation and the smallest detectable difference given all other terms in the table are specified To calculate power the number of clusters in the treatment or the sample size per cluster in the treatment group the user must specify the test significance level whether to use a one or two sided test the control group proportion p2 which test type to be used th
63. covariance matrix they wish to use in their study design 14 The Specify Multi Factor assistant table is used to define a range values to be filled in across several columns in the test design table Once the user fills in this table with the range of values they require by clicking Run nTerim will fill out the required number of columns to satisfy the outlined range of parameters 2 7 Plotting A plotting menu has been introduced to nTerim 3 0 for all the additional graphing features that have been added Additional features have been added to the Power vs Sample Size and Boundary plots including multiple plotting capabilities highlighting various boundary functions of interest and scrolling features to enable users to pin point exact values The plotting menu bar is displayed in Figure 2 7 1 below File Edit View Assistants Tools Window Help A L New Fixed Term Test New Power vs Sample Size Plot iz LL Open Manual Statistical Solutions Support Spending Function Plot Inverse Boundaries Graph Figure 2 7 1 Plot Menu Options In relation to Interim designs a boundary plot is automatically displayed after running the calculations This is always displayed on the bottom right hand corner of the nTerim window An example of an O Brien Fleming boundary is given in Figure 2 7 2 below Boundaries Graph E a O Brien Fleming Boundaries with Alpha 0 05 1 2 3 4 Figure 2 7 2 Example of a Boun
64. degrees of freedom 4 1 3 The non centrality parameter A is then calculated using the equation A NA 4 1 4 Using these two values nTerim calculates the power of this design as the probability of being greater than FprF pr a ON a non central F distribution with non centrality parameter A In order to calculate sample size nTerim does not use a closed form equation Instead a search algorithm is used This search algorithm calculates power at various sample sizes until the desired power is reached 59 60 im 4 1 3 Examples Example 1 Examining the specific contrast between high and low doses of a new drug This test can be incorporated when examining different levels within a certain variable In this example we want to examine the contrast between high doses and low doses of a specific new drug The following steps outline the procedure for Example 1 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval Survival D gt Two Equivalence Agreement Regression One sample t test Paired t test for difference in Means i Univariate one way repeated measures analysis of variance
65. design column and going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 5 3 7 Output x OUTPUT STATEMENT In a cluster randomised trial comparing two binary variables a sample size of 5 clusters with 100 individuals per cluster in the treatment group and a sample size of 5 clusters per group with 100 individuals per cluster in the control group achieves 82 982 power to detect a difference between two proportions when the Differences under null hypothesis and alternative hypotheses are 0 1 and 0 1 respectively the intracluster correlation is 0 001 at the 0 05 significance level using the Pooled statistic ag Specify Multiple Factors u Output Figure 5 3 7 Study design Output statement This Output statement can then be easily transferred directly from the output window into a report 239 240 Im 5 4 CRT Two Proportions Non Inferiority Completely Randomized 5 4 1 Introduction Binary data is commonly studied in variety of different fields Non inferiority trials are commonly used to assess whether a new treatment is at least as effective as a pre existing treatment in the clinical setting e g comparing a generic drug to its competitor Clustered data is very common in a wide variety of academic social policy and economic studies This two sample test is used to test hypotheses about the non inferiority of a treatment i
66. entering O for the Mean and 1 for the Standard Deviation of the independent variable This is displayed in Figure 4 12 8 ui Specify Multiple Factors ui Normal Side Table Exponential Side Table a Uniform Side Table ui Binomial Side Table u Output Figure 4 12 8 Values entered for the Normal Distribution side table 6 Next click Compute on the side table and this will calculate and then transfer the Variance of b1 Null Hypothesis and Variance of b1 Alternative Hypothesis values into the main table as shown in Figure 4 12 9 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Z Plot Power vs Sample Size w Poisson Regression 1 A el 2 4 Test significance level a 0 05 1 or 2 sided test 1 y 2 z 2 z 2 z Baseline Response Rate e 0 1 Response Rate Ratio e 1 e 0 0 5 Mean Exposure Time pT 1 Overdispersion Parameter 1 Distribution of X1 sidetable required Normal x Normal x Normal x Normal z Variance of b1 Null Hypothesis 1 Variance of bi Alternative Hypothesis 0 786 R squared X1 and independent variables 0 Power 95 Cost per sample Total study cost PP m gt Calculate required sample size for given power x Run All columns Figure 4 12 9 Values entered for Poisson Regression study design 7 Select Calculate requir
67. expected difference the intracluster correlation ICC the number of clusters in the control group Kz and the sample size per cluster in the control group M3 The formula uses the normal approximation to calculate power The formulae use the difference between the proportions under the superiority margin A and the actual expected difference 6 regardless of the format of statistic used Proportions and ratios are converted to the relevant differences If higher values are considered better then the expected difference should be higher than the superiority margin and if the lower values are considered better then the expected difference should be lower than the superiority margin The formulae for the power are thus given by the following equations A o _ za rattan A lt OUnpool OUnpool A o o _ za Test meN A gt s OUnpool OUnpool Power 5 5 1 where Orest rype is the standard error defined by the test type being used and dynyoo1 is the unpooled standard error The formula for the three standard error statistics is as follows 251 Melim 1 252 Farrington and Manning Test Statistic Likelihood Score The Farrington and Manning test uses the constrained maximum likelihood estimator of the two proportions to calculate the standard error for the calculation of power and was proposed as method to test against a null hypothesis of a specified difference A The standard error used b
68. highlighting the desired columns and clicking on the Plot Power vs Sample Size button on the menu bar 18 To highlight the desired columns click on the column title for Column 1 and drag across to Column 3 19 Then click on the Plot Power vs Sample Size button on the menu bar The multiple column plot is displayed in Figure 4 3 7 below Power vs Sample Size paN K To con m eo hal Power vs Sample Size 110 1110 2110 3110 4110 100 ba i ey e o A C O r gt a we i oo f 7 a 90 i 80 80 gt 70 70 Column 1 e rn Column 2 4 yg Column 3 60 60 t y 504 4 50 Aea 110 610 1110 1610 2110 2610 3110 3610 4110 4610 Power 82 65 Sample Size N1 N2 Sample Size 2236 Figure 4 3 7 Power vs Sample Size Plot It can be seen from the legend on the left hand side legend can be altered manually that the blue line represents Column 1 the orange line represents Column 2 and the red line represents Column 3 The cross on the graph illustrates how the user can identify what the sample size is for a corresponding power value for each column In the bottom right corner of the plot indicated the exact values for Power and Sample Size for each identifier on the graph Example 2 Specifying and Comparing Multiple Designs In this example we use the Multiple Factor table to specify multiple designs and then compare the designs appropriately The fo
69. level These results assume that 5 sequential tests are made and the O Brien Fleming spending function is used to determine the test boundaries Drift 3 29983 27 Example 2 Pocock Spending Function and Unequal N s This example is taken from Reboussin et al 1992 using the Pocock spending function 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Interim Test from the menu bar at the top of the window A Study Goal and Design window will appear as shown below Select the options as mapped out in Figure 3 1 6 then Click OK No of Groups Analysis Method Test Group Sequential Test of Two Means Figure 3 1 6 Study Goal and Design Window 2 Setup the table as in the Example 1 3 We will again use 5 looks but this time change the Spending Function to Pocock in the dropdown box 28 nQuery nTerim 2 0 File Edit View New Fixed Term Test Assistants Plot Tools New Interim Test Window Z Plot Power vs Sample Size Help LL Open Manual b Statistical Solutions Support Xx GST Two Means 1 Test significance level a 1 or 2 sided test Group 1 mean p1 Group 2 mean p2 Difference in means pi p2 Group 1 standard deviation o1 Group 2 standard deviation 02 Effect size 3 Group 1 size n1 Group 2 size n2 Ratio N2 N1 Power Cost per sample unit Total study cost
70. opened the menu enables the user to toggle between the various tables and plots they may be working on during their session The Help menu gives access to the nTerim manual and supplies the nTerim version information and license agreement Below is a complete list of menu options from the menu bar File gt New Open Fresh Table Save Save As Close Test Exit Edit gt Fill Right Clear Table Clear Column Clear Selection View gt Option not available until a test window is opened Looks Specify Multiple Factor Table Covariance Matrix MANOVA design table only Boundaries Graph Power vs Sample Size Plot Boundaries Plot Spending Function Plot Output Help Notes Assistants gt Specify Multiple Factor Table Compute Effect Size Randomization Survival Parameter Converter Distribution Function Windows Calculator Plot gt Power vs Sample Size Plot Spending Function Plot Boundaries Plot Tools gt Print Main Table to Clipboard Print Looks Table to Clipboard Settings Windows gt Close All if no test window open Close All List of Open Windows Help gt Help About Manual 2 4 Opening a New Design The next aspect of the interface we will review is opening a new design both Fixed term and Interim There are two ways in which the user can open a new design in nTerim i by clicking on the File gt Open option or ii using the shortcut buttons highlighted in Figure 2 4 1 bel
71. or a proportion of the sample size enrolled The common element among most of the different spending functions is to use lower error values for the earlier looks By doing this it means that the results of any analysis will only be considered significant in an early stage if it gives an extreme result Boundaries The boundaries in nTerim 3 0 represent the critical values at each look These boundaries are constructed using the alpha and beta spending functions Users in nTerim 3 0 are given the option to generate boundaries for early rejection of the null hypothesis Hy using the alpha spending function or to generate boundaries for early rejection of either the null or alternative hypothesis Hj or H using a combination of both the alpha and beta spending functions The notion of using an alpha spending function approach to generate stopping boundaries for early rejection of Ho was first proposed by Lan and DeMets 1983 we refer to such boundaries in nTerim 3 0 as efficacy boundaries Building on the work of Lan and DeMets Pampallona Tsiatis and Kim 1995 2001 later put forward the concept of using a beta spending approach to construct boundaries for early rejection of H4 we refer to these boundaries in nTerim as futility boundaries Essentially if a test statistic crosses an efficacy boundary then it can be concluded that the experimental treatment shows a statistically significant effect the trial can be stopped with rejection of th
72. select the CRT Two Means Completely Randomized design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Cluster Randomized as the Goal Two as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 5 1 2 nQuery nTerit File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size Means Completely Randomised ee 2 3 g as 1 or 2 sided test 2 2 x2 x2 z Difference Between Means X1 X2 Standard Deviation 0 Intracluster Correlation ICC Number of Clusters per Group m Sample Size per Cluster N Power Cost per sample Total study cost a o p Calculate attainable power with the given sample size Run An columns Figure 5 1 2 CRT Two Means Inequality Completely Randomized Test Table The first calculation that will be performed is for power This example is can be undertaken using the following steps 3 First enter 0 05 for the Test Significance level row then select 2 for the 1 or 2 sided test variable dropdown menu enter 0 2 for the Difference Between Means enter 1 for the Standard Deviation variable and enter 0 01 for the Intracluster Correlation 4 Finally enter 5 for the number of clusters per group and 100 for the sample siz
73. simply clicking on a column the output statement will be presented Similarly to opening a Fixed Term test if the user clicks on the New Interim Test button below the menu bar the Study Goal and Design menu window will appear with the list of interim designs available in nTerim This Study Goal and Design window is presented below in Figure 2 4 4 Oe Design Goal No of Groups Analysis Method Fixed Term Means e Test Interim 5 Proportions Two Survival Group Sequential Test of Two Means OK Cancel Figure 2 4 4 Open New Interim Design The options for Interim term designs are presented in Figure 2 4 4 For example if you want to choose the Group Sequential Test of Two Means table you must first select Means as the Goal gt Two as the No of Groups and Test as the Analysis Method You can then select Group Sequential Test of Two Means from the list of tests Once you click OK the design table will be launched As it can be seen from Figure 2 4 5 the Interim term design window is split into four main sections i the test table ii Looks Table amp Output iii Boundary Graph and iv Help Guide Cards The main table represents the test table In this example it is a Group Sequential Test of Two Means table The top half of the main test table is for various parameters to be entered by the user The bottom half is for the user to
74. that must be specified is the smallest meaningful time averaged difference to be detected Given the above values in order to calculate the power for this design the user must enter the expected sample size for each group N and N nTerim then uses the total sample size N to calculate the power of the design using the following equation Za a MN n 1 m Power 1 4 2 1 o J1 pM 1 where is the standard normal density function and Ny 4 2 3 T 4 2 3 In order to calculate sample size for a given power the following formula is used Za Zg 1 M 1 p o y Cat ZC M 1 p 07 aa Md x 1 T where P is the probability of a type II error B 1 Power 4 2 5 73 74 Im 4 2 3 Examples Example 1 Comparing the Difference in Sample Size due to change in Significance Level In this example we are going to investigate how a difference in the level of significance for a study design can impact the sample size required to obtain a given power The following steps outline the procedure for Example 1 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear Study Goal And Design eS A b z l Goal No of Groups Analysis Method Means One Test D Proportions Two Confidence Interval Su
75. unpooled standard error The formula for the three standard error statistics is as follows 241 Melim 1 242 Farrington and Manning Test Statistic Likelihood Score The Farrington and Manning test uses the constrained maximum likelihood estimator of the two proportions to calculate the standard error for the calculation of power and was proposed as method to test against a null hypothesis of a specified difference A The standard error used by the Farrington and Manning test statistic is defined as follows 1 IF 1 IF eae Pri Prm1 1 Pem2 Prm2 IF2 5 4 2 K M KM2 where K4 is the number of clusters in the treatment group M4 is the sample size per cluster in the treatment group Dry is the maximum likelihood estimator for each group proportion and IF is the inflation factor for the effect of clustering in the treatment and control groups respectively IF is defined as follows IF 1 ICC M 1 i 1 2 5 4 3 The constrained maximum likelihood where p pz A estimator of the two proportions is calculated using the following calculations K M IF ___ 5 4 4 KM IF pa a 1 t 5 4 5 b 1 t p t tp ACE 2 5 4 6 c A A 2p t 1 p tpz 5 4 7 d p A 1 A 5 4 3 b gt be d v 35 ce2 Gee ee u sign v oj 5 4 10 g 3a 3a Vv a et oe Ge Baan 3 Prm1 2ucos w b 3a 5 4 12 Prmz2 Prm1 A 5 4 13 2 Unpooled Test S
76. well as early stopping for Futility Group Sequential Designs Group Sequential designs differ from Fixed Period designs in that the data from the trial is analyzed at one or more stages prior to the conclusion of the trial As a result the alpha value applied at each analysis or look must be adjusted to preserve the overall Type 1 error The alpha values used at each look are calculated based upon the spending function chosen the number of looks to be taken during the course of the study as well as the overall Type 1 error rate For a full introduction to group sequential methods see Jennison amp Turnbull 2000 and Chow et al 2008 Spending Function There are four alpha and beta spending functions available to the user in nTerim 3 0 as well as an option to manually input boundary values As standard all alpha spending functions have the properties that a 0 0 and a 1 a Similarly all beta spending functions have the properties that B 0 0 and 1 p Functionally the alpha and beta spending functions are the same In Table 3 2 1 we list the alpha spending functions available in nTerim 3 0 Table 3 2 1 Spending Function Equations O Brien Fleming a t 2 1 lt 2 Vt Pocock a t aln 1 e 1 t Power a t at gt 0 G e Hwang Shih DeCani a t a e 0 The parameter T represents the time elapsed in the trial This can either be as a proportion of the overall time elapsed
77. 0 199 200 Melim 4 13 2 Methodology Probit analysis is often used in the study of relative potency between test and control treatments The Probit Regression Model is given by the following equation Y F 1 P By Bi logy D 4 13 1 where F 1 P is the cumulative normal distribution for the lethality proportions and D is dose of the substance of interest For LDso trials we define LDso T as the dose which is lethal for 50 of the treatment group and LDs9 C as the dose which is lethal for 50 of the control group The relative potency p is defined as follows to give the following formulae LD5o T 4 13 2 LDso eee logio P logio LDso T logy9 LDso C 4 13 3 The mean and variance of log p are derived in Kodell et al 2010 and are used to generate the following statistic for hypothesis testing logio y Var 10g10 P The sample size calculation for this Probit analysis is taken from Kodell et al 2010 This table can be used to calculate the power the sample size and the relative potency 4 13 4 Calculations use a t distribution approximation To calculate power and sample size the user must specify the test significance level the number of dose levels the sum of the probit weights the slope of the Probit Regression Model f1 and the relative potency p The sum of the probit weights is calculated from the side table using the values of the target response proportions expecte
78. 0 for power 5 Finally enter K1 in the Clusters in Control Group K2 row This will solve so that K1 and K2 must be equal Other ratios between K1 and K2 can be calculated by using arguments for K2 such as 2 K1 or K1 2 to have K2 be twice as large as K1 for example The table will appear as in Figure 5 3 3 236 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test gt New Interim Test Plot Power vs Sample Size Two Proportions Equivalence 1 sao PO 2 3 4 Test significance level a 0 05 Test Type Pooled z Likelihood score z Likelihood score z Likelihood score z Control Group Proportion p2 0 5 Solve using p1 p2 p1 or p1 p2 Differences z Differences z Differences z Differences z Test Statistic for Upper Equivalence Margin 0 1 Test Statistic for Lower Equivalence Margin 0 1 Actual Value of Test Statistic 0 Intracluster Correlation ICC 0 001 Clusters in Treatment Group K1 Clusters in Control Group K2 Ki Cluster Sample Size in Treatment Group M1 100 Cluster Sample Size in Control Group M2 Cost per sample Total study cost pae m gt Calculate attainable power with the given sample size and number of clusters Run AII columns Figure 5 3 3 Values entered for CRT Two Proportions Equivalence study design Select Calculate required treatment group clusters K1 given power and sample size fro
79. 15 Select Calculate attainable power with the given sample size from the drop down menu below the main table and click Run This is displayed in Figure 4 5 6 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size Number of groups G Variance of means v Common standard deviation o Number of covariates c R Squared with covariates R2 Calculate attainable power with the given sample sizes X All columns Figure 4 5 6 Completed ANCOVA Test Table 116 It can be seen from Figure 4 5 6 that a sample size of 150 is required to obtain a power of 85 37 Due to the cost per sample unit of 100 the overall cost of sample size required has amounted to 15 000 By clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation When the total sample size across the 4 groups is 150 distributed across the groups as specified an analysis of covariance will have 85 37 power to detect at the 0 05 level a difference in means characterized by a Variance of means of 13 29 assuming that the common standard deviation is 25 and assuming the covariate s has an R squared of 0 75 Example 2 Investigating the effects of R squared on attainable Power In this example we will examine how the R squared with covariates value has an impact on the attainable power given a certain sa
80. 2 3 4 Test significance level a 0 05 1 or 2 sided test 1 2 2 2 m Standard Deviation of X1 a 0 3126 R squared of X1 and other X s 0 1837 Log Hazard Ratio B 1 Overall Event Rate P 0 738 Sample Size N 106 v 80 Cost per sample unit Total study cost EE o o o m gt Calculate required sample size for given power v All columns Figure 4 11 4 Completed Cox Regression study design The next calculation is a sensitivity analysis for sample size where we change the event rate to explore its effect on sample size These values are taken from Table 1 page 555 of Hsieh and Lavori 2000 6 In the first column enter 0 05 for Test Significance 1 for 1 or 2 sided test 1 for the standard deviation zero for the R value 0 5 for the log hazard ratio 0 1 for the Overall Event Rate and 80 for the power 7 Then copy these values across to columns 2 to 4 and change the value for overall Event Rate for columns 2 to 4 to 0 3 0 5 and 1 respectively This will give a table as per Figure 4 11 5 alee nO MZUETN File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size Cox Regression 1 Overall Event Rate P 0 1 0 3 0 5 ZEAL E 80 80 Cost per sample unit Total study cost Ce Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 1 BE z z Stand
81. 3 1 3 Completed Two Means Test Table The boundaries calculated are shown in Figure 3 1 4 Cumulative exit probability under HO 0 4 Lower bound 4 87688 3 35695 Upper bound 4 87688 3 35695 Nominal alpha 9 00000 0 00079 Incremental alpha 0 00000 0 00079 Cumulative alpha 0 00000 0 00079 Exit probability 0 03 10 17 Cumulative exit probability 10 03 10 21 Nominal beta Incremental beta Cumulative beta Exit probability under HO 0 6 1 0 8 2 68026 2 28979 2 03100 2 68026 2 28979 2 03100 0 00736 0 02203 0 04226 0 00683 0 01681 0 02558 0 00762 0 02442 0 05000 35 07 29 92 15 17 45 27 75 19 90 36 uJ Looks i Specify Multiple Factors ug Output Figure 3 1 4 Boundary Table for Two Means Test 10 Finally the boundaries calculated in the table in Figure 3 1 4 are automatically plotted as illustrated in Figure 3 1 5 Boundaries Greph E O Brien Fleming Boundaries with Alpha 0 05 Figure 3 1 5 Boundary Plot for Two Means Test By clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation Sample sizes of 49 in group 1 and 49 in group 2 are required to achieve 90 36 power to detect a difference in means of 20 the difference between group 1 mean 1 of 220 and group 2 mean u2 of 200 assuming that the common standard deviation is 30 using a 2 sided z test with 0 05 significance
82. 3 for the number of groups G as shown in Figure 4 4 4 103 104 Edit View Assistants Plot Tools Window Help 3 New Fixed Term Test New Interim Test Z Plot Power vs Sample Size ANOVA 1 Number of groups G Variance of means V Common standard deviation o Effect size A V o Power N as multiple of n1 Jri Sni ni Total sample size N Cost per sample unit Total study cost _ S S gt Calculate required sample sizes for given power X All columns Figure 4 4 2 One way Analysis of Variance Test Table Total sample size N N as multiple of n1 Sri Sni n1 4 Compute Effect Size Assistant u Specify Multiple Factors Output Figure 4 4 3 Compute Effect Size Assistant Window 5 Once you enter a value for the number of groups G the Compute Effect Size Assistant table automatically updates as shown in Figure 4 4 4 6 In order to calculate a value for Effect Size the Variance of Means V needs to be calculated first 7 The mean for each level and the corresponding sample size need to be entered in the Compute effect Size Assistant table 8 For the Mean values for each group enter 5 for group 1 12 for group 2 and 12 for group 3 9 For the group sample size n values for each group enter 20 for group 1 12 for group 2 and 18 for group 3 As a result the ratio 7 is calculated for e
83. 6 Test Statistic under H1 0 04 0 04 0 04 0 04 Intracluster Correlation ICC 0 01 0 01 0 01 0 01 Clusters in Treatment Group K1 38 27 21 57 Clusters in Control Group K2 38 54 84 30 Cluster Sample Size in Treatment Group M1 100 100 100 100 Cluster Sample Size in Control Group M2 100 100 100 100 81 018 81 452 80 937 80 111 Cost per sample Total study cost ME P Calculate required treatment group clusters K1 given power and sample size v Run v All columns Figure 5 2 6 Results from Sensitivity analysis The effect of changing these parameters on the relationship between Power and Sample size can be explored further using the Plot Power vs Sample Size button at the top of the screen 10 Select the first column by clicking the 1 at the top of column 1 Then hold down Ctrl and click the 3 at the top of column 3 and the 4 at the top of column 4 All three columns will now be highlighted 11 Click the Plot Power vs Sample Size button at the top of the screen This will give you a Figure 5 2 7 which will show the relationship between power and the number of clusters in the treatment group for each column Right click to add features such as a legend to the graph and double click elements for user options and editing 227 40 50 60 70 Power 82 84 Number of Clusters K1 Sample Size 43 80 90 Figure 5 2 7 Power vs Number of Clusters plot 12 By clicking on the desired stu
84. 95 for Power in all four columns Then select Exponential Uniform and Binomial from the Distribution of X1 variable for columns 2 to 4 respectively 15 For the normally distributed column 1 enter a mean of 0 and standard deviation of 1 in the Normal Side Table then click compute 16 For the exponentially distributed column 2 enter a lambda of 1 in the Exponential Side Table then click compute 17 For the uniform distributed column 3 enter a minimum of 1 732 and maximum of 1 732 in the Uniform Side Table then click compute 18 For the binomial distributed column 4 enter a proportion of 0 5 in the Binomial Side Table then click compute This will give you a table that is displayed in Figure 4 12 16 195 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test _ New Interim Test Z Plot Power vs Sample Size Lu Poisson Regression 2 A a 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 1 z z z z Baseline Response Rate eB0 1 1 1 1 Response Rate Ratio eB1 eB0 05 05 0 5 0 5 Mean Exposure Time pT 1 1 1 1 Overdispersion Parameter 1 1 1 1 Distribution of X1 sidetable required Normal z Exponential z Uniform z Binomial x Variance of b1 Null Hypothesis 1 1 1 4 Variance of bi Alternative Hypothesis 0 786 4 854 1 037 6 R squared X1 and indep
85. 996 4 10 2 Methodology This table provides generic sample size and power calculations for studies which will be comparing two incidence rates The tables for two Poisson means or two negative binomial means can be used instead if the researcher will know the analysis method and estimates of the additional associated parameters beforehand The sample size calculation for two incidence rates is taken from Smith and Morrow 1996 This table can be used to calculate the power the sample size and the treatment group incidence rate given all other terms in the table are specified To calculate power and sample size the user must specify the test significance level whether to use a one or two sided test the control group incidence rate A the treatment group incidence rate A and the difference in rates A Az The sample size formula uses the normal approximation to yield the following 2 _ Ca g 1 aj2 Ga 42 4 10 1 i a2 where y is the sample size per group expressed in person years or person units of time For the one sided test Z1_q 2 is replaced with Z _ Simple rearrangement of the above formula gives the following equation for the power VY 42 _ r 4 10 2 Ay Az Power A closed form equation is not used to calculate the treatment group incidence rate Instead a search algorithm is used The search algorithm calculates power at various values for the treatment group incidence ra
86. BT TE s min a p a q 1 df ap df s N r pts Hotelling Lawley Trace The test statistic for Pillai Bartlett trace is calculated using the formula HLT tr HE The transformation of this test statistic to an approximate F is given by nf af Mahe 1 0 dfy _ HLT s I T HLT s df ap 126 4 6 9 4 6 10 4 6 11 4 6 12 4 6 13 4 6 14 4 6 15 4 6 16 4 6 17 4 6 18 4 6 19 4 6 20 4 6 21 4 6 22 4 6 23 4 6 24 df s N r p 1 2 4 6 25 Depending on which of these three statistics is chosen nTerim then calculates the power of the design by first determining the critical value Faf af a and then the noncentrality parameter A Where A ndf 4 6 26 Using these two values nTerim will calculate the power of this design as the probability of being greater than Far af a ON a non central F distribution with non centrality parameter A In order to calculate sample size values for power must be specified in the Factor Level Table nTerim does not use a closed form equation Instead a search algorithm is used This search algorithm calculates power at various sample sizes until the desired power is reached 127 4 6 3 Examples Example 1 Pillai Bartlett Trace In this example we will calculate the attainable power given a specified sample size using the Pillai Bartlett trace method The following steps outline the procedu
87. Clusters in Control Group K2 back to K1 K1 2 4 K1 and 30 for columns 1 to 4 respectively Then set the power values back to 80 12 Finally select Calculate required treatment group clusters K1 given power and sample size from the dropdown menu beside the Run button Next tick the box to run All Columns Then click Run This will give Figure 5 4 8 nQuery nTerim 3 File Edit View Assistants Plot Tools Window New Fixed Term Test New Interim Test Z Plot Power vs Sample Size _ Two Proportions Non Inferiorit Test significance level a 0 05 Test Type Unpooled Higher Proportions Better Worse Worse Control Group Proportion p2 0 5 Solve using p1 p2 p1 or p1 p2 Differences Non inferiority Test Statistic 0 1 Actual Value of Test Statistic 0 05 Intracluster Correlation ICC 0 001 Clusters in Treatment Group K1 14 Clusters in Control Group K2 14 Cluster Sample Size in Treatment Group M1 100 Cluster Sample Size in Control Group M2 100 o 81 199 Cost per sample Total study cost 0 05 x Unpooled x Worse 0 5 Differences 0 1 0 05 0 001 11 22 100 100 82 817 4 0 05 Unpooled Worse 0 5 Differences 0 1 0 05 0 001 9 36 100 100 82 252 E LUI Open 4 0 05 Unpooled X Worse X 0 5 Differences i 0 1 0 05 0 001 9 30 100 100 80 914 gt Calcul
88. D o SQRT 1 p Power Group size N Cost per sample unit Total study cost Calculate attainable power with the given sample sizes Contrast C Scrpi Scale D SQRT Sci2 Figure 4 1 16 Automatically Updated Compute Effect Size Assistant Table 11 Once the table in Figure 4 1 17 is completed and values for Contrast C and Scale D are computed click on Transfer to automatically transfer these values to the main table a Compute Effect Size Assistant ua Specify Multiple Factors u Output Figure 4 1 17 Completed Compute Effect Size Assistant Table 12 Now that values for Contrast C and Scale D have been computed we can continue with filling in the main table For the Standard Deviation enter a value of 10 For the between level correlation enter a value of 0 7 13 We want to calculate the sample size required obtain a power of 90 Therefore enter 90 in the Power row 14 The cost per sample unit cannot be estimate yet in this study so we will leave this row blank for this calculation This value has no impact on the sample size or power calculation 15 Select Calculate required sample size for given power from the drop down menu below the main table and click Run This is displayed in Figure 4 1 18 F nQuery nTerim 24 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size RM Contrast 1
89. Fixed Term Test New Interim Test Plot Power vs Sample Size Poisson One Mean 1 A eed coe 2 3 4 Test significance level a 0 05 0 05 0 05 Null or Baseline Mean Poisson Rate AO 0 03 0 03 0 03 Alternative Mean Poisson Rate A1 0 1 0 05 0 08 Sample Size N 106 106 106 Power 28 287 74 168 Cost per sample Total study cost Ce ee ee gt Calculate attainable power with the given sample size X 7 All columns Figure 4 7 6 Results from Sensitivity analysis The effect of changing Alternative Mean Poisson Rate on the relationship between Power and Sample size can be explored further using the Plot Power vs Sample Size button at the top of the screen 7 Select all three columns by clicking the 1 at the top of column 1 Then hold down Shift and click the 3 above columns All three columns will now be highlighted 8 Click the Plot Power vs Sample Size button at the top of the screen This will give you a plot as displayed in Figure 4 7 7 which will show the relationship between power and sample size for each column Right click to add feature such as a legend to the graph and double click elements for user options and editing Power vs Sample Size 1020 Alt Mean Poisson Rate 0 1 r Alt Mean Poisson Rate 0 05 Alt Mean Poisson Rate 0 08 Figure 4 7 7 Power vs Sample Size plot 9 Finally by clicking on the desired study design column and
90. Group 2 size n2 Ratio n2 n1 a 1 1 1 Power 30 Cost per sample unit 65 5 C EE D Figure 4 2 9 Design Entry for Two Means Repeated Measures Study The cost per sample unit has been estimated as 65 in this particular study Therefore to calculate the overall cost associated with the sample size enter 65 in the Cost per sample unit row as shown in Figure 4 2 9 As we want to try several different parameter values for both Power and between level correlation we can use the Fill Right function to fill out multiple columns with the same information entered in Column 1 10 Once all the parameter information has been entered click on Edit and Fill Right as shown in Figure 4 2 10 File Edit View Assistants Plot Tools Window Help E Ne Fill Right Clear Table erim Test a Plot Power vs Sample Size Clear Column Clear Selection z 2 z 2 z 2 iz Number of levels M 4 Difference in means pi p2 15 Standard deviation at each level o 25 Between level correlation p 0 4 Group 1 size n1 Group 2 size n2 Ratio n2 n1 1 1 1 1 1 Power 30 Cost per sample unit Calculate required sample sizes for given power X Run AH columns Figure 4 2 10 Fill Right function 11 As shown in Figure 4 2 11 all columns have been filled in with the same parameter information contained in Column 1 We want to alter the othe
91. N cssssececcccccenssssseecececeaneususscecececeaeaussseececeseusuuaussesseceeeasanagssss 183 4 12 1 NTPOOMC OM ituctsnieeysrnestexecsoeatrerienesactonta detuaadeatdanmsodansaiea a ea aer Saaai kaia 183 4 12 2 Methodology seiorn EEEE EEEE 184 A123 EXAMP l ES e E Wega E T AR e 187 4 13 Probit REBPESSION x 2sav2vcecceseecetsdecyaedacsedvedvavevanswatasvedsaetha a r a a i 199 4 13 1 MAF OOMCTOM secus ineine anaia Ea a aaa Eoia 199 4 13 2 Methodol by inocu a aaa 200 4 13 3 EXAMP lE S cistithe anssen enii en aaiae tonai a eaaa aiana eaa ahai 202 Chapter Drine a e e a E E r a 211 Cluster Randomized Trials sseeoseesssesssessssesesesssersstsrstensesnseressresstersrestreneeenssersstenstensernsers 211 5 1 CRT Two Means Completely Randomized cccccccesssececssseeeeceesseeecesssaeeeesseneees 212 SALINE roOdUCTIOM cstivviresenssoizarsianiesiesastatsntat wines uedeeasceah di aasa Sinn EEEE aa EEE 212 5 1 2 M thodolo fVin na EAT 213 5 1 3 EXAMPle S i ccccacutd snaciansoeadcwiactestronseuntlonns Raa EA EEEa EER 214 5 2 CRT Two Proportions Inequality Completely Randomized c cccccccccssssssseeeeees 219 SA AAPG WCE Meesi eas aaa aaa a aa EEA T ERa 219 B22 IMME COTO OB sie tess schrcvucnatee tau nat eae aacezalinsataceuseadexituclstacuantesetasalstiag Medussahleuseattenden 220 5 2 3 EMAINIPIGS xis ascctiabsnsanxancasaddacadesstdnaeundiolosyyitanceaghiadaaasaceauehsadasharaadniadiannguaxeannnaaielacge 223 5 3 CRT Two P
92. Probit Regression Side Table for both columns 3 and 4 Enter 0 05 0 2 0 35 0 5 0 65 0 8 0 95 in the Target Response Proportions and then click Compute This will give the results as shown in Figure 4 13 10 Probit Regression Side Table x MINNIE oea Dose2 Dose Dose 4 Doses posee Dose7 l 0 2 0 5 0 35 0 65 0 8 0 95 0 2239360066 0 4898667305 0 603056802 0 6366197724 0 603056802 0 4898667305 0 2239360066 u8 Specify Multiple Factors ui Probit Regression Side Table u8 Output Figure 4 13 10 Completed Probit Regression Side Table for Sensitivity analysis 13 Select Calculate required sample size for given power from the dropdown menu beside the Run button Then tick the box to run All Columns Then click Run This will give the resultant sample sizes of 11 8 7 and 5 per group sequentially as per Kodell et al 2010 and this is displayed in Figure 4 13 11 207 Im nQuery nTerim File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Probit Regression 1 es 1 2 ee col 4 Test significance level a 0 05 0 05 0 05 0 05 Number of Dose Levels 5 5 7 7 Sum of Weights 2 2013513108 2 2013513108 3 2703388507 3 2703388507 Slope of Probit Regression B1 23 25 23 25 23 41 23 41 Relative Potency p 1 1 i i 11 Sample Size per Group N 11 8 7 5 90 5379299894 81 576000848 82
93. R i e with a single CPU at a single location THIS LICENSE SHALL NOT APPLY TO AND DOES NOT PERMIT THE ELECTRONIC TRANSFER OF THE SOFTWARE FROM ONE COMPUTER TO ANOTHER unless a Network Addendum to the Agreement is executed by Licensee and returned to LICENSOR Licensor reserves all rights not expressly granted to LICENSEE LICENSOR also agrees to provide free maintenance of the SOFTWARE for sixty 60 days 2 TRIAL PERIOD LICENSEE shall have sixty 60 days commencing on day of receipt by LICENSEE in which to return the SOFTWARE provided hereunder and shall be entitled to receive a full refund All refunds are contingent upon receipt of LICENSOR in undamaged condition of all materials provided hereunder 3 OWNERSHIP OF SOFTWARE LICENSOR retains title to and ownership of the SOFTWARE This LICENSE is not a sale of the original SOFTWARE or any copy 4 COPY RESTRICTIONS This SOFTWARE and the accompanying written materials are copyrighted Unauthorised copying of the SOFTWARE including SOFTWARE which has been modified merged or included with other software or of the written materials is expressly forbidden You may be held legally responsible for any copyright infringement that is caused or encouraged by your failure to abide by the terms of the LICENSE Subject to these restrictions you may make one 1 copy of the SOFTWARE solely for backup purposes You may reproduce and include the copyright notice on the backup copy 5 USE REST
94. RICTIONS As the LICENSEE you may physically transfer the SOFTWARE from one computer to another provided that the SOFTWARE is used on only one computer at a time You may not translate reverse engineer decompile or disassemble the software You may not distribute copies of the SOFTWARE or accompanying written materials to others 6 TRANSFER RESTRICTIONS This SOFTWARE is licensed only to you the LICENSEE and may not be transferred to anyone without the prior written consent of LICENSOR Any authorised transferee of the SOFTWARE shall be bound by the terms and conditions of this Agreement 7 TERMINATION This LICENSE is effective until terminated This LICENSE will terminate automatically without notice from LICENSOR if you fail to comply with any provision of this LICENSE Upon termination you shall destroy the written materials and all copies of the SOFTWARE including modified copies if any and shall notify LICENSOR of same 8 GOVERNING LAW MISCELLANEOUS This Agreement is governed by the laws of Ireland If any of the provisions or portions thereof of this License Agreement are invalid under any applicable statute or rule of law they are to that extent to be deemed omitted 9 DECISION OF ARBITRATORS At the option of the LICENSOR any dispute or controversy shall be finally resolved in accordance with the rules of the International Chamber of Commerce The Arbitration shall be conducted in Ireland with 3 Arbitrators unless Lic
95. Randomized design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Cluster Randomized as the Goal Two as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 5 2 2 223 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size LUJ Open DP ees ifi Loo O 1 or 2 sided test 2 x 2 x 2 x2 Test Type Likelihood score x Likelihood score x Likelihood score x Likelihood score Control Group Proportion p2 Solve using p1 p2 p1 or p1 p2 Differences z Differences x Differences d Differences X Test Statistic under HO Test Statistic under H1 Intracluster Correlation ICC Clusters in Treatment Group K1 Clusters in Control Group K2 Cluster Sample Size in Treatment Group M1 Cluster Sample Size in Control Group M2 Power Cost per sample Total study cost 4 Jal Calculate attainable power with the given sample size and number of clusters v Run 0 All columns Figure 5 2 2 CRT Two Proportions Inequality Completely Randomized Test Table The first calculation will be for the Clusters in Treatment Group K1 This example is can be undertaken using the following steps 3 First enter 0 05 for the Test Significance level row and select 2 for the 1 o
96. STATISTICAL Y SOLUTIONS Aletirr Version 3 0 User Manual 4500 Airport Business Park Cork lreland Web www statsols com Email sales statsols com Tel 353 21 4839100 Fax 353 21 4840026 Printed in the Republic of Ireland nTerim 3 0 User Manual Statistical Solutions Ltd One International Place 100 Oliver Street Suite 1400 Boston MA 02110 Web www statsols com Email sales statsols com Tel 617 535 7677 Fax 617 535 7717 No part of this manual may be reproduced stored in a retrieval system transmitted translated into any other language or distributed in any form by any means without prior permission of Statistical Solutions Ltd fietin Statistical Solutions Ltd nTerim License Agreement IMPORTANT READ BEFORE PROCEEDING WITH INSTALLATION THIS DOCUMENT SETS FORTH THE TERMS AND CONDITIONS OF THE LICENSE AND THE LIMITED WARRANTY FOR nTerim PROCEEDING WITH THIS INSTALLATION CONSTITUTES YOUR ACCEPTANCE OF THIS LICENSE AGREEMENT WITH RESPECT TO ALL ACCOMPANYING nTerim SOFTWARE RECEIVED BY YOU IF YOU DO NOT ACCEPT THIS AGREEMENT YOU MAY RETURN THIS SOFTWARE UNDAMAGED WITHIN 10 DAYS OF RECEIPT AND YOUR MONEY WILL BE REFUNDED 1 GRANT OF LICENSE In consideration of payment of the license fee which is part of the price you paid for this product Statistical Solutions Ltd as LICENSOR grants to you the LICENSEE a non exclusive right to use this copy of nTerim SOFTWARE on a single COMPUTE
97. W3 In MLE for column 3 W4 In CMLE for column 4 and W5 Variance Stabilizing for column 5 This will give a table as per Figure 4 8 5 155 File Edit View Assistants Plot Tools Window Help i E New Fixed Term Test New Interim Test Plot Power vs Sample Size Test significance level a Null Poisson Rate Ratio RO y0 y1 1 1 1 1 1 Alt Poisson Rate Ratio R1 y0 y1 4 _4 4 4 Test Statistic W1 MLE z W2 CMLE a W3 Ln MLE m W4 Ln CMLE z W5 Variance Stabilf wi 2 Observation Time for Group 1 t1 2 2 2 2 Observation Time for Group 2 t2 2 2 2 2 2 Mean Poisson Rate in Group 1 y1 0 0005 0 0005 0 0005 0 0005 0 0005 Sample Size Allocation Ratio N2 N1 0 5 o5 05 0 5 0 5 Sample Size in Group 1 N1 Sample Size in Group 2 N2 Mah Run i An columns Figure 4 8 5 Sensitivity analysis around the Test Statistic options 7 Select Calculate required Group 1 and 2 sample sizes for given power and sample size allocation from the dropdown menu beside the Run button Then tick the box to run All Columns Then click Run This will give the resultant Group 1 sample sizes of 8564 W1 6889 W2 6685 W3 6685 W4 and 8590 W5 for each statistic sequentially as in Figure 4 8 6 Similar to the W1 example above these answers differ due to rounding File Edit View
98. able difference given all other terms in the table are specified To calculate power the number of clusters in the treatment group or the sample size per cluster in the treatment group the user must specify the test significance level a the control group proportion p2 which test type is being used the format of the test statistic whether higher values for the proportion are better or worse from the researcher s perspective the non inferiority margin the actual expected difference the intracluster correlation ICC the number of clusters in the control group Kz and the sample size per cluster in the control group M3 The formula uses the normal approximation to calculate power The formulae use the difference between the proportions under the non inferiority margin A and the actual expected difference 6 regardless of the format of statistic used Proportions and ratios are converted to the relevant differences If higher values are considered better then the expected difference should be higher than the non inferiority margin and if the lower values are considered better then the expected difference should be lower than the non inferiority margin The formulae for the power are thus given by the following equations A O BGs a A lt 6 OUnpool OUnpool eee eater A gt 6 Ounpool Ounpool Power 5 4 1 where Orest Type is the standard error defined by the test type being used and dynyoo1 is the
99. able power with the given sample size and number of clusters v All columns Figure 5 2 9 Sensitivity analysis using different Test Statistics 16 Select Calculate attainable power with the given sample size and number of clusters from the dropdown menu beside the Run button Then tick the box for All Columns Then click Run This will give 30 751 as per Figure 5 2 9 229 230 File Edit View Assistants Plot Tools Window Help I New Fixed Term Test New Interim Test Plot Power vs Sample Size LLI Open Two Proportions Inequality Co PL___s o lt U Test significance level a 1 or 2 sided test Test Type Control Group Proportion p2 Solve using p1 p2 p1 or p1 p2 Test Statistic under HO Test Statistic under H1 Clusters in Treatment Group K1 Clusters in Control Group K2 Cluster Sample Size in Treatment Group M1 Cluster Sample Size in Control Group M2 Power Cost per sample Total study cost 0 05 2 Unpooled 0 06 Proportions 0 06 0 04 0 01 10 10 100 100 30 751 me x Unpooled 0 06 z Differences 0 0 02 0 01 10 10 100 100 30 751 0 05 x2 z Unpooled 0 06 z Ratios 1 0 666667 0 01 10 10 100 100 30 751 2 z Likelihood score z Differences k Calculate attainable power with the given
100. ach group as a proportion of group 1 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size ANOVA 1 Variance of means V Total sample size N N as multiple of n1 Sri Ini ni Figure 4 4 4 Automatically updated Compute effect size Assistant Table 10 Once the table in Figure 4 4 5 is completed and values for Variance of Means V and total Sample Size N are computed click on Transfer to automatically transfer these values to the main table Group 3 Variance of means V Total sample size N E Compute Effect Size Assistant 4 Specify Multiple Factors u Output Figure 4 4 5 Completed Compute Effect Size Assistants Table 105 106 11 Now that values for Variance of Means V and total Sample Size N are computed we can continue with filling in the main table For the Common Standard Deviation enter a value of 6 Now the Effect Size is automatically calculated 12 We want to calculate the attainable power given the sample size of 50 13 It has been estimated that it will cost 85 per sample unit in this study Therefore enter 85 in the Cost per sample unit row 14 Select Calculate attainable power with the given sample size from the drop down menu below the main table and click Run This is displayed in Figure 4 4 6 File Edit View Assistants Plot Tools Wi
101. al 1992 and Jennison amp Turnbull 2000 35 36 Calculate missing proportion given N Nz power and the other proportion Calculate p given pz In order to solve for p given p and all other information Equation 3 2 1 can be re expressed as a quadratic with respect to pz the roots of which give p4 Similarly if p is specified the roots give the values of p gt Calculate p4 given p2with Continuity Correction In order to solve for p given p and all other information Equation 3 2 2 can be re expressed as a quadratic with respect to pz the roots of which give p4 Similarly if p is specified the roots give the values of p gt 3 2 3 Examples Example 1 Pocock Spending Function This example is adopted from Reboussin et al 1992 using Pocock spending function 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Interim Test from the menu bar at the top of the window A Study Goal and Design window will appear as shown below Select the options as mapped out in Figure 3 2 1 then Click OK 7 7 Study Goal And Design A Design Goal No of Groups Analysis Method Fixed Term D Means Test Interim Proportions Two O Survival Group Sequential Test of Two Proportions Figure 3 2 1 Study Goal and Design Window 2 Now you have opened the test table as illustrated in Figure 3 2 2 you can begin enter
102. an M D 1984 An Improved Approximate Two sample Poisson Test Applied Statistics 33 2 pp 224 226 Hwang I K Shih W J and deCani J S 1990 Group Sequential Designs using a Family Type Error Probability Spending Functions Statistics in Medicine 9 pp 1439 1445 Jennison C and Turnbull B W 2000 Group Sequential Methods with Applications to Clinical Trials Chapman amp Hall Keppel G 1991 Design and Analysis A Researcher s Handbook Third Edition Prentice Hall Kodell R L Lensing S Y Landes R D Kumar K S amp Hauer Jensen M 2010 Determination of Sample Sizes for Demonstrating Efficacy of Radiation Countermeasures Biometrics 66 1 pp 239 248 261 262 im Liu H H Wu T T 2005 Sample Size Calculation and Power Analysis for Time Averaged Difference Journal of Modern Applied Statistical Methods 4 2 pp 434 445 Muller K E and Barton C N 1989 Approximate Power for Repeated Measures ANOVA Lacking Sphericity Journal of the American Statistical Association 84 pp 549 555 with correction in volume 86 1991 pp 255 256 Muller K E LaVange L M Ramey S L and Ramey C T 1992 Power Calculations for General Linear Multivariate Models Including Repeated Measures Applications Journal of the American Statistical Association 87 pp 1209 1226 O Brien P C and Fleming T R 1979 A Multipe Testing Procedure f
103. analyses Including the final analysis this requires Number of Looks to be 5 5 The looks will be equally spaced and the Power Family spending function is to be used Enter 3 for Phi 6 For this example we want to truncate the boundaries so as not to be over conservative Enter Yes for truncate bounds and then enter 3 for the value to truncate at 7 Select Calculate the attainable power with the given sample sizes from the drop down menu and then click Run 41 File Edit View Assistants I New Fixed Term Test Plot __ New Interim Test Tools Z Plot Power vs Sample Size Window Help oy x LL Open Manual Statistical Solutions Support GST Two Proportions 1 xX Test significance level a 0 05 1 or 2 sided test 2 Group 1 proportions n1 os Group 2 proportions n2 0 465 Odds ratio W n2 1 n1 ni 1 n2 1 25074 Group 1 size n1 1400 Group 2 size n2 1400 Ratio N2 N1 fa Continuity correction On Power 18117 Cost per sample unit 180 Total study cost 504000 Number of looks 5 Information Times Equally Spaced Max times pi Determine bounds Spending Function Spending function Power Family Phi J3 Truncate bounds Yes Truncate at 3 i Dont Calculate O Brien Fleming x2 x of 5 z Equally Spaced 1 z Spending Function O Brien Fleming z No
104. ard Deviation of X1 0 1 1 1 1 R squared of X1 and other X s 0 0 0 0 Log Hazard Ratio B 0 5 05 0 5 0 5 Cun L E an coimas Figure 4 11 5 Sensitivity analysis on the Overall Event Rate 8 Select Calculate required sample size for given power from the dropdown menu beside the Run button Then tick the box for All Columns and then click Run This will give the resultant sample sizes of 248 83 50 and 25 sequentially as in Figure 4 11 6 Similar to the example above the answers for column 1 and 2 differ from those presented in Hsieh and Lavori 2000 due to intermediate rounding nQuery nTerim File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size Cox Regression 1 CO 1m 2 4 Test significance level a 0 05 0 05 0 05 0 05 1or 2 sided test 1 z zli z z Standard Deviation of X1 a A 1 1 1 R squared of X1 and other X s 0 0 0 0 Log Hazard Ratio B 0 5 0 5 0 5 Overall Event Rate P 0 3 0 5 1 Sample Size N 83 50 25 80 80 80 Cost per sample unit Total study cost Calculate required sample size for given power X Run All columns Figure 4 11 6 Results from Sensitivity analysis 9 The effect of changing the test statistic on the relationship between Power and Sample size can be explored further using the Plot Power vs Sample Size button at the top of the screen
105. ate required sample sizes for given power from the drop down menu below the test table 15 As we want to run this calculation for multiple columns tick the All Columns box beside the Run button as shown in Figure 4 2 12 then click Run T nQuery nTerim 2 0 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size _ RM Two Means 1 2 2 8 4 a a Test significance level a 0 05 0 05 0 05 0 05 0 05 1 or 2 sided test 2 y 2 x 2 y 2 y 2 X Number of levels M 4 4 4 4 4 Difference in means pi p2 15 15 15 15 15 Standard deviation at each level o 25 25 25 25 25 Between level correlation p 0 4 0 4 0 4 0 7 0 2 Group 1 size n1 33 28 24 46 24 Group 2 size n2 33 28 24 46 24 Ratio n2 ni 1 1 1 1 1 Power Calculate required sample sizes for given power Figure 4 2 12 Completed multiple design Repeated Measures for Two Means Table As it can be seen in Figure 4 2 12 there is a drop in sample size of 5 units per group if you reduce the power to 85 and a further drop of 4 units per group when reducing power to 80 Depending on the different constraints on the study design 80 power may be acceptable and would reduce costs by approximately 25 when compared with the same study design with 90 power When we examined the volatility in relation to the between
106. ate required treatment group clusters K1 given power and sample size V All columns Figure 5 4 8 Results from second Sensitivity analysis This will give the same answers as for the above sensitivity calculation due to the control proportion being 0 5 in which case these values lower and higher than 0 5 are symmetric in terms of the calculation If the control group proportion were not 0 5 we would have expected different values for the two calculations 249 Im 5 5 CRT Two Proportions Superiority Completely Randomized 5 5 1 Introduction Binary data is commonly studied in variety of different fields Superiority trials are commonly used to assess whether a new treatment is better than a pre existing treatment in the clinical setting e g comparing a new drug to its competitor Clustered data is very common in a wide variety of academic social policy and economic studies This two sample test is used to test hypotheses about the superiority of a treatment in a completely randomized cluster randomized trial This table facilitates the calculation of the power and sample size for superiority hypothesis tests comparing proportions in a completely randomized cluster randomized trial Power and sample size is computed using the method outlined by Donner and Klar 2000 250 5 5 2 Methodology This table provides sample size and power calculations for studies which will be conducting a superiority trial betw
107. ate sample size calculations tick the All columns box beside the run button then select Calculate required sample sizes for given power from the drop down menu below the main table and click Run File Edit Plot View Assistants Tools Window Help New Fixed Term Test New Interim Test Ta Plot Power vs Sample Size _ RM Two Proportions 1 0 05 0 05 0 05 0 05 2 2 x2 mal 2 3 Number of levels M Between level correlation p 0 4 0 4 0 4 0 4 Group 1 proportion p1 0 45 0 55 0 45 0 55 Group 2 proportion p2 0 39 0 39 0 51 0 51 Odds ratio p2 1 p1 p1 1 p2 0 78142 0 5231 1 27211 0 85158 Group 1 size ni 852 121 873 1962 Group 2 size n2 852 121 873 1962 Ratio n2 ni 1 1 1 1 Power 90 89 9 90 30 Cost per sample unit 100 100 100 100 Total study cost 170400 24200 174600 392400 Calculate required sample sizes for given power x V All columns Figure 4 3 12 Comparison of four Repeated Measures Designs It can be seen in Figure 4 3 12 that all combinations of the minimum and maximum values for Group 1 and 2 proportions are created This allows us to evaluate how the sample size varies as the values of the group proportions change We can see from Columns 1 and 2 that if we fix the Group 2 proportion at the minimum value of 0 39 and increase the Group 1 proportion the required sample size decreases We can also see from C
108. ation in Column 1 to Column 2 To do this highlight Column 1 by clicking on the column title as shown in Figure 4 4 7 Then right click and select Copy File Edit View Assistants Plot Tools Window Help __ New Fixed Term Test New Interim Test a Plot Power vs Sample Size dhe ANOVA 1 2 3 4 Select All Copy Cut Paste Fill Right Clear Table Clear Column Clear Selection Print Table o Calculate attainable power with the given sample sizes X Run All columns Figure 4 4 7 Copy Column 1 2 Then right click on the first cell in Column 2 and select Paste as illustrated in Figure 4 4 8 below File Edit View Assistants Plot Tools Window Help i New Fixed Term Test New Interim Test Fa Plot Power vs Sample Size J ANOVA1 il 1 g z 2 4 EEEE E E a 10 05 Select All Number of groups G 3 Copy Variance of means V 11 76 Cu Common standard deviation o 6 mack Effect size A V o 0 32667 ser Power 94 82 Fill Right N as multiple of n1 Sri Ini ni 2 5 Clear Table Total sample size N 30 Clear Column Cost per sample unit Clear Selection Total study cost 4250 Z Print Table d om gt Calculate attainable power with the given sample sizes X Run All columns Figure 4 4 8 Paste contents of Column 1 into Column 2 107 108 Melim 3 Once the contents of Column 1 have been copied over to Column 2 you can change the value of the C
109. atrix Group Sizes u Covariance Matrix u Specify Multiple Factors wi Output Figure 4 6 13 Factor Level Table 136 11 As we have defined 3 response variables all with 3 levels each we will require a Means Matrix with 3 rows and 3x3x3 columns There is an extra row included to enable the user to specify the individual level sample size only needed if unequal sample sizes per level 12 The next step is to fill in all the values for each part of the Means Matrix In this example we will define the Means Matrix as below first column of matrix are row names 1 1 1 2 12342342 5 1 M 2 1 2 1 4 1 2 142 1141 3 68 74 5 644 4 4 5 4 6 1 2 12342342 5112 2 1412 142 1 1 4 122 1 8 745 644445 4 68 7 13 Enter this matrix in the Means Matrix Assistant table as illustrated in Figure 4 6 14 Calculate group size using Wilks lambda v Run All columns Means Matrix Group Sizes ag Factor Level Table ag Means Matrix Group Sizes jg Covariance Matrix a Specify Multiple Factors ui Output Figure 4 6 14 Completed Means Matrix Group Sizes Assistant Table 14 The bottom row is summed to give the total sample size required and automatically entered into the main design table In this case we are leaving the bottom row empty as we are going to specify the all groups have equal sample size In this event nTerim will automatically update this matrix once we have entered a value for Group Size in the main desi
110. been estimated that it will cost 80 per sample unit in this study Therefore enter 80 in the Cost per sample unit row 15 As we want to compare the effects that the R Squared value has on the Power of the study we will re run this design for several values of R Squared To do this right click on Column 1 as shown in Figure 4 5 12 and select Fill Right This will replicate the information in Column 1 across all the columns in this window File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size Select All Copy Cut Paste Fill Right Clear Table Clear Column 80 Clear Selection 9600 Print Table E All columns Calculate attainable power with the given sample sizes Figure 4 5 12 Fill Right Shortcut Feature 16 Now we want to change the R Squared values in Columns 2 3 and 4 to represent the remaining possible estimated R Squared values for our study design We would like to investigate R Squared ranging from 0 5 in Column 1 to 0 8 in Column 4 To do this enter 0 6 in the R Squared with covariates row in Column 2 0 7 in Column 3 and 0 8 in Column 4 as illustrated in Figure 4 5 13 below File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size Number of groups G 3 3 3 3 Varian
111. bles For observation i the regression model can be expressed as the following e Hiti uiti P Y yilki ti 4 12 3 F where u exp Bo B1X1i P2X2i BeXxi The sample size calculation for an independent variable in Poisson Regression is taken from Signorini 1991 with an extension adapted from Hsieh and Lavori 2000 to account for the multivariable case This table can be used to calculate the power the sample size the test significance level or the response rate ratio given all other terms in the table are specified Calculations use a standard normal approximation To calculate power and sample size the user must specify the test significance level whether to use a 1 or 2 sided test the baseline response rate e the response rate ratio e 1 ePo the mean exposure time ut the overdispersion parameter y the expected multiple correlation coefficient R between the independent variable of interest X4 and the other independent variables in the model and the distribution with the appropriate parameters specified of the independent variable of interest If there is only one independent variable in the model then R is set to zero The formula for the sample size in a 2 sided test for the null hypothesis of 4 0 versus the alternative hypothesis of 6 b for an independent variable is given by the following equation 2 2 zi y7 4 12 4 Ht 1 R eFob N 9 where V an
112. ce of means V 15 97222 15 97222 15 97222 15 9722 Common standard deviation o 30 30 30 30 Number of covariates c 1 1 1 1 R Squared with covariates R2 05 0 6 0 7 0 8 Power Total sample size N 120 120 120 120 0 05 Calculate attainable power with the given sample sizes X Run All columns 17 18 Figure 4 5 13 Altered columns for R Squared Comparison Now that all the information in each column has been entered we are ready to run the calculations In order to calculate the power for all the columns together tick the All columns box beside the Run button as shown in Figure 4 5 13 Now select Calculate attainable power given sample size from the drop down menu below the main table and click Run File New Fixed Term Test Edit View Assistants Plot Tools Window Help New Interim Test Z Plot Power vs Sample Size ANCOVA 1 SSS 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 Number of groups G l3 3 3 3 Variance of means V 15 97222 15 97222 15 97222 15 97222 Common standard deviation o 30 30 30 30 Number of covariates c i l 1 1 1 R Squared with covariates R los 0 6 0 7 0 8 Power 42 91 51 94 64 99 83 02 Total sample size N 120 120 120 120 Cost per sample unit 80 80 80 80 Total study cost 9600 9600 9600 9600 qm m D Calculate attainable power with the given sample sizes X 7 All columns Figure 4 5 14 C
113. ction Finally by clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation Sample sizes of at least 1400 in group 1 and 1400 in group 2 are required to achieve 81 17 power to detect an odds ratio of 1 25074 for proportions of 0 41 in group 1 and 0 465 in group 2 using a 2 sided continuity corrected y test with 0 05 significance level These results assume that 5 sequential tests are made and the Power Family spending function is used to determine the test boundaries 43 44 mM 3 3 Survival 3 3 1 Introduction nTerim 3 0 is designed for the calculation of Power and Sample Size for both Fixed Period and Group Sequential design In relation to Group Sequential designs calculations are performed using the Lan DeMets alpha spending function approach DeMets amp Lan 1984 DeMets amp Lan 1994 for estimating boundary values Using this approach boundary values can be estimated for O Brien Fleming O Brien amp Fleming 1979 Pocock Pocock 1977 Hwang Shih DeCani Hwang Shih amp DeCani 1990 and the Power family of spending functions Calculations follow the approach of Reboussin et al 1992 and Jennison amp Turnbull 2000 Calculations can be performed for studies that involve comparisons of means comparisons of proportions and survival studies as well as early stopping for Futility Group Sequential Designs Group Sequential designs differ from Fixed P
114. d Klar 2000 and is an extension of the methods used for equivalence trials in a fully randomized trial as outlined in Chow et al 2008 The extension uses the intracluster correlation ICC which is the ordinary product moment correlation between any two observations in the same cluster to adjust for the effect of within cluster correlation It is assumed the ICC is the same in both groups This table can be used to calculate the power the number of clusters in the treatment group the sample size per cluster in the treatment group the intracluster correlation and the smallest detectable difference given all other terms in the table are specified To calculate power the number of clusters in the treatment group or the sample size per cluster in the treatment group the user must specify the test significance level a the control group proportion p2 which test type is being used the format of the test statistic the upper equivalence margin the lower equivalence margin the actual expected difference the intracluster correlation ICC the number of clusters in the control group K2 and the sample size per cluster in the control group M3 The formula uses the normal approximation to calculate power The formulae use the difference between the proportions under the equivalence margins A and the actual expected difference 6 regardless of the format of statistic used Proportions and ratios are converted to the relevant di
115. d for each dose level The slope of the probit regression model can also be optionally calculated from the side table if the user provides the number of doses for the control group associated with each target response proportion The formula for the sample size using the appropriate simplifications and approximations outlined in Kodell et al 2010 for the null hypothesis of p 1 versus the alternative hypothesis of p gt 1 alternatively Hg log 9 p 0 vs H logi9 p gt 0 is as follows _ 2Gpa a trae 4 13 5 logio Pi i Wi where ty is the inverse cumulative t distribution function with f degrees of freedom evaluated at probability p and Wi is the sum of the Probit weights The values of f and w are evaluated as follows f 29 3 4 13 6 B o i eS L siaa 4 13 7 Ra ce where g is the number of dose levels x is the density function of the normal distribution 7t is the cumulative distribution of the normal distribution and P is the target response proportion for each dose level The power is calculated by re arrangement of the above formula to give the following equation Clogio p 61 D2 wi N Power t f 2 Stria 4 13 8 where te x is the cumulative t distribution function evaluated at f degrees of freedom The relative potency is given by the following formulation p 10 4 13 9 2 me trae t tyap 4 13 10 N f j Wi where Im 201 Im 4 13 3 Examples Examp
116. d test 2 y 2 Number of levels M Difference in means pi p2 Standard deviation at each level o Between level correlation p Group 1size ni Ta ad Group 2 size n2 Ratio n2 n1 1 1 1 Power est sig fiC e level Cost per sample unit J C Calculate required sample sizes for given power Figure 4 2 2 Repeated Measures for Two Means Test Table _ All columns 6 We also know that the between level correlation is 0 5 so enter 0 5 into the Between level correlation row File Edit View Assistants Plot Tools Window Help x Plot Power vs Sample Size _ New Fixed Term Test New Interim Test RM Two Means 1 n i 2 Test significance level a 0 05 1 or 2 sided test 2 y 2 2 Number of levels M 4 Difference in means p1 p2 10 Standard deviation at each level a 20 Between level correlation p los Group 1 size ni 53 Group 2 size n2 53 Ratio n2 ni j 1 power so 90 9540 Calculate required sample sizes for given power X All columns Figure 4 2 3 Completed Repeated Measures Design for Two Means 75 7 We want to calculate the required sample size for each group in order to obtain 90 power To do this enter 90 in the Power row 8 It has also been estimated that the associated cost per unit in this study will amount to 90 Theref
117. dV are the variances of the estimated regression coefficient for the independent variable under the null and alternative hypotheses respectively These are defined below For the one sided test Z _ would be used in place of z _ 2 The power is calculated by re arrangement of the above formula to give the following equation Power 4 12 5 u 1 reeFob w JP 24 vi with a similar re arrangement yielding the following equation for the test significance level a 2h 0 In the above equations V and V are defined by the choice of distribution for the independent variable X1 These are outlined below m 1 R eob N Jde 7 nav J 4 12 6 1 Normal Distribution for X4 The calculation requires the mean u and standard deviation o of X4 These are used to calculate Vo and V as follows 1 v 4 12 7 z riz y r nu 2 4 12 8 oO 2 Exponential Distribution for X4 The calculation requires the exponential mean A These are used to calculate Vo and V as follows 1 4 12 9 3 V L 4 12 10 3 Uniform Distribution for X4 The calculation requires the minimum A and maximum B of the uniform distribution interval These are used to calculate Vp and V as follows 185 186 12 ee ee 4 12 11 Vo B A w v 4 12 12 where E e 1 B e14 4 12 13 D C b E y EOB D e l A 1 4 12 14 a B A b B e 1P 2 2b B b B
118. dary Plot 15 16 im In relation to Power vs Sample Size plots there is also a shortcut button provided in the tool bar just below the menu bar as highlighted in Figure 2 7 3 below In order to use this function the user must highlight the columns which they would like to compare and then click on the Plot Power vs Sample Size button nQuery nTerim 2 0 lexan el x m File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size LL Open Manual Statistical Solutions Support Figure 2 7 3 Power vs Sample Size Plot Shortcut Tab An example of the new Power vs Sample Size plot is displayed in Figure 2 7 4 below This plot shows three columns being compared The legend on the right side of the window can be altered to label each line appropriately Power vs Sample Size re ee me oo 20 30 40 50 60 70 80 90 100 Power 82 62 Total sample size N Sample Size 59 Figure 2 7 4 Power vs Sample Size Plot A crosshair is provided to enable the user to pin point exact values for power and sample size at various points on each line These exact values are given in the box in the bottom right hand corner of the plot window In order to save a plot in nTerim simply right click anywhere on the plot window and a list of options will be presented as illustrated in Figure 2 7 5 The options include Save Image Print Print P
119. define parameters relating to the interim design such as number of looks spending function to be used and so on Once all the appropriate information has been entered in the test table the user must select the appropriate calculation to run i e whether you want to solve for power given a specified sample size or solve for sample size given a specified power The user can select the appropriate calculation to run from the drop down menu between the main test table and the Looks table Once the appropriate test is selected the user must click on Run to run the analysis 11 12 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size LJ Open Manual Statistical Solutions Support GST Two Means 1 ll x m zj Help x Aoa Two group z test for the J _ Lor 2 sided test difference between Group 1 mean 1 independent means Group 2 mean p2 Enter a value for alpha a the DRA significance level for the test and sce z select a one or two sided test Specify Group 1 standard deviation o1 two of effect size power and sample F Group 2 standard deviation o2 size and nTerim will compute the third Effect size pnp pA Test significance level a 2size n2 Alpha is the probability of rejecting the esse bore null hypothesis of equal means when it Rati
120. der HO H1 variables 14 Test Statistic under H1 15 For column 2 Differences enter zero for Test Statistic under HO and 0 02 for For column 3 Ratios enter 1 for Test Statistic under HO and 0 666667 for Test Statistic under H1 The values for column 2 and 3 are the values for these two formats which correspond to the values given for the first column using Proportions This will give Figure 5 2 9 nQuen erim 3 File Edit View Assistants Plot Tools New Fixed Term Test New Interim Test _ Two Proportions Inequality Co Window Z Plot Power vs Sample Size LLI Open Test significance level a 1 or 2 sided test Test Type Control Group Proportion p2 Solve using p1 p2 p1 or p1 p2 Test Statistic under HO Test Statistic under H1 Custers in Treatment Group K1 Clusters in Control Group K2 Cluster Sample Size in Control Group M2 Power Cost per sample Total study cost 0 05 2 Unpooled 0 06 Proportions 0 06 0 04 0 01 10 10 Cluster Sample Size in Treatment Group M1 100 100 4 0 05 2 Unpooled 0 06 Differences 0 0 02 0 01 10 10 100 100 4 4 0 05 2 Unpooled 0 06 Ratios 1 0 6667 0 01 10 10 100 100 4 4 2 Y Likelihood score v Differences v Calculate attain
121. der to select the Negative Binomial design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Means as the Goal Two as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 4 9 2 162 File Edit View Assistants Plot Tools Window Help EN x n Test E NewInterimTest Plot Power vs Sample Size LL Open Negative Binomial 1 A ee 2 3 4 Mean Rate of Event for Control r0 Rate Ratio r1 r0 Average Exposure Time pT Dispersion Parameter k Rates Variance Reference Group Rate z Reference Group Rate z Reference Group Rate z Reference Group Rate z Sample Size Ratio N1 N0 Control Group Sample Size N0 Treatment Group Sample Size N1 Power Cost per sample Total study cost prere a e e e e A ee E a i Figure 4 9 2 Two Negative Binomial Rates Test Table The first calculation will be for Sample Size to begin we enter the values as follows 3 First enter 0 05 for the Test Significance level row then enter 0 8 and 0 85 for the Mean Rate of the Event for the Control and Rate Ratio variable rows respectively 4 Next enter 0 75 for the Average Exposure Time 0 7 for the Dispersion Parameter and 1 for the Sample Size Ratio Finally enter 80 in the Power row File Edit View Assistants
122. dsdiags va E E RE AE A EEEE 4 Getting Started Guide soci crssencsiadssassdosnessdouessalaceubigds seeasodndelanged adeosadadwcbaytededeilandoocassbaucenainensuantes 4 2 1 Starting NCNM snc wits ities aicisin e neien ieee Ai 5 2 2 Home Vi OW ceseroun peran ts hentd eda duedeanan ane ena T e aai nasa E aT 5 2 3 Men Balissccieeccchueusedcevscaccasyeliectscstegienbusssulsavedgecessisueteaperdaceasslcecdeaweleudea ecuesesuengedeavecectvaseheds 6 2 4 Opening a New DeSIQN cceeeseeee eter et eneneneneneneneneneneneneneneneneserenereserenerenereserenenenenenenenes 8 2 5 Selecting an nQuery Advisor Design Table through nTerim ccssecccessereeeeeeneeeeees 13 2 6 Using the Assistant Tables ccccccccsssessssecececeesesessaeseeeeeescesseseeaeseeeesesseesesaeaeeeeeeseneees 14 ZF POULIN E ye stecces gees seasued caves nes oaSeysedanvavecvessesuudaasanantey evendsasthaudesds ds dbvesuicdeatevesdonetedesecsededdeavexs 15 2 8 Helpand GU OR cease nesnese rniii nades rais aa a aaae EEN Tra E aneian 18 Chapters comer a vcnseuneesesveratesetieactesectesenucin Seseeveseaaebuncesecteavesuenss 19 Group Sequential Interim Design ccccescessceeseeeseecsseceeeseeeeeseecusceseeseeeeessecsaeeseeseeeeeeneees 19 3L TWO MEANS soinera a detesedisdacueld vececs tds chadeticeleey aa a 20 Bid Ae Introductio oa E R A E E 20 3 1 2 Methodology ctiecacncdeiadudesanngeiidaexsqunedinioed ded dtuanansdiedenmesauiedceetavsdaeaadeddconnsieddeunsaunctiedge
123. dy cost DKE C Bj Calculate attainable power with the given sample sizes All columns Figure 4 1 12 Completed One way Repeated Measures Contrast Table It can be seen from Figure 4 1 12 that a sample size of 30 per group for each of the three groups thus a total sample size N of 90 is required to obtain a power of 94 82 By clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation When the group sample size n is 30 the test of a single contrast at the 0 05 level in a one way repeated measures analysis of variance with 3 levels will have 94 82 power to detect a contrast C dci pi of 6 with a scale D SQRT Sci of 2 44949 assuming a standard deviation at each level of 3 677 and a between level correlation of 0 67 68 im Example 3 Investigating Self Esteem Scores over time In this example we will be examining self esteem scores over time For the researchers involved they expect the self esteem scores to increase monotonically over time Therefore the researchers would wish to test the linear contrast following the repeated measures ANOVA to assess what sample size is requires for the contrast to have 90 power The following steps outline the procedure for Example 3 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A
124. dy design column and going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 5 2 8 Output OUTPUT STATEMENT In a cluster randomised trial comparing two binary variables a sample size of 38 clusters with 100 individuals per cluster in the treatment group and a sample size of 38 clusters per group with 100 individuals per cluster in the control group achieves 81 018 power to detect a difference between two proportions when the Proportions under null hypothesis and alternative hypotheses are 0 06 and 0 04 respectively the control group proportion is 0 06 the intracluster correlation is 0 01 and when using a 2 Sided test at the 0 05 significance level using the Unpooled statistic pa Specify Multiple Factors a Output Figure 5 2 8 Study design Output statement 228 Example 2 Validation example calculating power using different Test Statistic formats A calculation is conducted to show the equivalence of the test statistic approaches The following steps outline the procedure for this example 13 Delete columns 2 to 4 and delete the Power value in column 1 Then replace K1 and K2 with 10 in column 1 Next in column 2 change the Solve Using switch to Differences and change it to Ratios in column 3 Copy and paste across the column 1 values to column 2 and 3 with the exception of the two Test Statistic Un
125. e Ratio of N4 to N Drift Parameter KR Za Number of Time points Looks Calculate Sample Size for a given Power Using the number of time points K number of sides type of spending function the hypothesis to be rejected the type 1 error and the power 1 the drift parameter can be obtained using algorithms by Reboussin et al 1992 and Jennison amp Turnbull 2000 S2 HR log 2 1 For the Exponential Survival Curve this is defined by the expression below This can be solved for d the required number of events using the equation below Then to calculate the Proportional Hazards Curve Equation 3 3 4 is employed This can be solved for d the required number of events using Equation 3 3 5 log HR dy gy 8 log HR o LTHRV ade 1 HR 1 HR e 1 HR a 3 3 1 3 3 2 3 3 3 3 3 4 3 3 5 To calculate the sample size N the following formula is used 2d N 2 S1 S Calculate Attainable Power with the given Sample Size 3 3 6 Given a N group survival proportions s1 S2 number of time points K number of sides type of spending function the hypothesis to be rejected the requirement is to obtain the power For the Exponential Survival Curve Equation 3 3 7 is used N 2 s s2 log HR 8 N 2 s s2 1 HR 2 1 HR 3 3 7 3 3 8
126. e format of the test statistic the value of the test statistic under the null and alternative hypotheses the intracluster correlation ICC the number of clusters in the control group K2 and the sample size per cluster in the control group M3 The formulae use the normal approximation to calculate power The formulae use the difference between the proportions under the null A and alternative 6 hypotheses regardless of the format of statistic used Proportions and ratios are converted to the relevant differences The formula for the power is given by the following equation A Or T Power Z4 2 E OUnpool OUnpool 7 A OTest Type e ee OUnpool OUnpool where Orest Type is the standard error defined by the test type being used and Oynpoo1 is the unpooled standard error The formula for the three standard error statistics is as follows 5 2 1 1 Farrington and Manning Test Statistic Likelihood Score The Farrington and Manning test uses the constrained maximum likelihood estimator of the two proportions to calculate the standard error for the calculation of power and was proposed as method to test against a null hypothesis of a specified difference A The standard error used by the Farrington and Manning test statistic is defined as follows z Prm 1 Pemi IF Prm2 1 Prmz IFz Ory 1AA A 5 2 2 K M KM where K4 is the number of clusters in the treatment group M4 is the sample
127. e 5 5 The Pocock spending function is to be used however the looks will not be evenly spaced 6 For Information Times select User Input Then in the Times row in the lower table enter the values 0 1 0 2 0 3 0 6 and 1 7 It is estimated that the cost per unit is roughly 100 so enter 100 in the Cost per sample unit row File Edit View Assistants New Fixed Term Test Plot E New Interim Test Tools Window Z Plot Power vs Sample Size Help UJ Open Manual Statistical Solutions Support 1 or 2 sided test SS Eee Seen Z ape ate Hazard ratio h In n1 In n2 0 5 0 4 0 756 Survival time assumption 1000 85 32 550 100 100000 5 User Input ji Spending Function Pocock No Calculate attainable power with the given sample sizes X Figure 3 3 10 Complete Survival Table with Pocock Spending Function 5 Equally Spaced r Spending Function jro x Don t Calculate x O Brien Fleming O Brien Fleming 5 Equally Spaced 1 x Spending Function O Brien Fleming no z Don t Calculate x O Brien Fleming 5 z Equally Spaced r Spending Function Ejro B O Brien Fleming Don t Calculate O Brien Fleming Proportional Hazard Exponential Survival Exponential Survival Exponential Survival L Al columns 8 Once all the values have bee
128. e bottom right corner of the plot indicated the exact values for Power and Sample Size for each identifier on the graph 4 6 Multivariate Analysis of Variance MANOVA 4 6 1 Introduction This table facilitates the calculation of power and sample size for multivariate analysis of variance MANOVA designs In multivariate models there are several test statistics that can be used In nTerim we provide the option for power and sample size calculations using three common test statistics Wilks likelihood ratio statistic Pillai Bartlett trace and Hotelling Lawley trace Calculations are performed using the approximations outlined by Muller and Barton 1989 and Muller LaVange Ramey and Ramey 1992 Multivariate analysis of variance MANOVA analysis is very similar to its univariate counterpart analysis of variance ANOVA MANOVA can be described simply as an ANOVA with several response variables In ANOVA differences in means between two or more groups are tested on a single response variable In MANOVA the number of response variables is increased to two or more The purpose of MANOVA is to test for the difference in the vectors of means for two or more groups To give an example we may be conducting a study where we are comparing two different treatments a new treatment and a standard treatment and we are interested in improvements in subjects scores for depression life satisfaction and physical health In this example improveme
129. e null hypothesis If the test statistic crosses a futility boundary then this indicates with high probability that an effect will not be found that the trial can be terminated by rejecting the alternative hypothesis In the case where the user wishes to generate boundaries for early rejection of either the null or alternative hypothesis H or H they are given two options either to have the boundaries binding or non binding With binding boundaries if the test statistic crosses the futility boundary the test must be stopped otherwise the type 1 error may become inflated The reason for this is that there is an interaction between the efficacy and futility boundaries in their calculation that could cause the efficacy boundary to shift In the case of non binding boundaries the efficacy boundaries are calculated as normal that is as if the futility boundaries did not exist This eliminates the danger of inflating the type 1 error when the futility boundary is overruled The downside of the non binding case is that it may increase the required sample size relative to the binding case The boundaries calculated in nTerim 3 0 follow the procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 33 Metin 34 3 2 2 Methodology The variables are defined as Symbol Description a Probability of Type error B Probability of Type II error 1 Power of the Test Pi P2 Gr
130. e per cluster The table will appear as per Figure 5 1 3 nQuery nTeri File Edit View Assistants Plot Tools Window Help 3 lt New Fixed Term Test E New Interim Test a Plot Power vs Sample Size Means Completely Randomised a 2 3 4 Test significance level a 0 05 1or 2 sided test 2 x2 x2 x2 z Difference Between Means X1 X2 0 2 Standard Deviation 0 1 Intracluster Correlation ICC 0 01 Number of Clusters per Group m 3 Sample Size per Cluster N 100 oo O Cost per sample Total study cost a E prr m L gt Run E An columns Figure 5 1 3 Values entered for CRT Two Means Completely Randomized study design 215 im 5 Select Calculate attainable power with given sample size from the dropdown menu beside the Run button Then click Run This will give a result of 26 8 for the power as in Figure 5 1 4 File Edit View Assistants Plot Tools Window Help 3 New Fixed Term Test New Interim Test Es Plot Power vs Sample Size Means Completely Randomised ee Test significance level a 0 05 1 or 2 sided test 2 2 x2 x2 z Difference Between Means X1 X2 0 2 Standard Deviation 0 1 Intracluster Correlation ICC 0 01 Number of Clusters per Group m 3 See atta niches el Cost per sample Total study cost M C Run E an columns Figure 5 1 4 Completed CRT Two M
131. e size Then change the values for columns 2 to 4 to 0 9 1 1 and 1 2 respectively This will give a table as per Figure 4 10 5 File Edit View Assistants Plot Tools Window Help I New Fixed Term Test New Interim Test Plot Power vs Sample Size Two Incidence Rates 1 ee ey ee ee ee 4 Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 2 x2 2 x2 x Control Group Incidence Rate A1 1 1 1 1 Treatment Group Incidence Rate A2 0 8 0 9 11 1 2 Difference in Rates A1 A2 0 2 0 1 0 1 0 2 Sample Size per Group in Person Yea 80 80 80 80 Cost per sample Total study cost Calculate required sample size for given power a Run All columns Figure 4 10 5 Sensitivity analysis around the Treatment Group Incidence Rate 7 Select Calculate required sample size for given power from the dropdown menu beside the Run button Then tick the box to run All Columns Then click Run 172 This will give the resultant sample sizes of 354 1492 1649 and 432 sequentially as in Figure 4 10 6 Similar to the example above these answers differ from Smith and Morrow 1996 due to rounding File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size Two Incidence Rates 1 h 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 2 2 x 2 2 x Control Group Incidence Rate A1
132. eans In fact the one way ANOVA can be viewed as being an extension of a two group t test To give an example of a one way ANOVA design consider a study on cholesterol Suppose we wanted to compare the reduction in cholesterol resulting from the use of a placebo the current standard drug and a new drug The one way ANOVA tests the null hypothesis that the mean reductions in cholesterol in all three groups are equal The alternative hypothesis is that the mean reductions in cholesterol in the three groups are not all equal 101 102 Im 4 4 2 Methodology Power and sample size are calculated using central and non central F distributions and follow the procedures outlined by O Brien and Muller 1993 To calculate power and sample size the user must specify the test significance level and the number of groups G The user must then enter a value for the variance of means V Alternatively the user can enter the expected means in each group using the compute effect size assistant nTerim will then calculate the variance of means using the formula 2a ri Ui ft y 4 4 1 i 11i where G m Nili 4 4 2 i gt 4 4 2 i 1 The compute effect size assistant also allows the user to enter the expected sample sizes in each group or the expected ratio to group 1 for each group 7 This is particularly useful when you expect unequal sample sizes per group Once the variance in means is calculated the user must input
133. eans as the Goal Two as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 4 10 2 170 ery n View Assistants Plot File Tools Edit Window Help a _ New Fixed Term Test New Interim Test Va Plot Power vs Sample Size Two Incidence Rates 1 Pi 2 1 or 2 sided test 2 x2 x2 Control Group Incidence Rate A1 Treatment Group Incidence Rate A2 Difference in Rates A1 A2 Sample Size per Group in Person Years Power Cost per sample Total study cost calculate attainable power with the given sample size Run All columns Figure 4 10 2 Two Incidence Rates Test Table The first calculation will be for Sample Size to begin we enter the values as follows 3 First enter 0 05 for the Test Significance level row then enter 1 and 0 8 for the Control Group Incidence Rate and Treatment Group Incidence Rate variable rows respectively The Difference in Rates variable will update automatically to 0 2 4 Then enter 80 in the Power row this table will appear as per Figure 4 10 3 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Two Incidence Rates 1 a 2 Test significance level a 0 05 1or 2 sided test 2 z 2 w 2 Control Group Incidence Rate A1 1
134. eans Completely Randomized study design The next calculation is a sensitivity analysis for power where we change the difference between means and the sample size per cluster and see their effect on power These values are taken from Table 1 of Donner and Klar 1996 6 Copy the first column above into columns 2 to 4 Then change the value for Difference between Means to 0 5 in columns 2 and 4 and Sample Size per Cluster to 300 in columns 3 and 4 This will give a table as per Figure 5 1 5 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test gt New Interim Test Z Plot Power vs Sample Size z Means Completely Randomised A a 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 1or 2 sided test 2 z 2 z 2 x2 x Difference Between Means X1 X2 0 2 0 5 0 2 0 5 Standard Deviation 0 1 1 1 1 Intracluster Correlation ICC 0 01 0 01 0 01 0 01 Number of Clusters per Group m 3 3 3 3 Sample Size per Cluster N 100 100 300 300 Cost per sample Total study cost a gt Calculate attainable power with the given sample size X Run E All columns Figure 5 1 5 Sensitivity analysis around Difference Between Means and Sample Size 216 7 Select Calculate attainable power with given sample size from the dropdown menu beside the Run button Then tick the box to run All Columns Then click Run This will give the resultant powe
135. east 2237 in group 1 and 2237 in group 2 are required to achieve 80 09 power to detect an odds ratio of 0 72752 for proportions of 0 11 in group 1 and 0 083 in group 2 using a 2 sided continuity corrected y test with 0 05 significance level These results assume that 4 sequential tests are made and the Pocock spending function is used to determine the test boundaries Example 2 Power Family spending function with truncated bounds This example is an adaptation from Reboussin et al 1992 using Power Family spending function with truncated bounds 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Interim Test from the menu bar at the top of the window A Study Goal and Design window will appear as shown below Select the options as mapped out in Figure 3 2 6 then Click OK E Teres OOOO Design Goal No of Groups Analysis Method 5 Fixed Term Means Test Interim Proportions Two D Survival Group Sequential Test of Two Proportions Cancel Figure 3 2 6 Study Goal and Design Window 2 Enter 0 05 for alpha 2 sided 0 41 for Group 1 proportion 0 465 for Group 2 proportion The odds ratio is calculated as 1 25074 3 Select On for the Continuity Correction We are interested in solving for power given a sample size of 1400 per group so enter 1400 in the Group 1 size row 4 This study planned for 4 interim
136. ed In this example the Analysis of Covariance ANCOVA table was selected A screen shot of this design table is given in Figure 2 4 3 10 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size LLI Open Manual Statistical Solutions Support ANCOVA 1 x Help x Analysis of covariance ANCOVA Variance of means V Enter a value for alpha a the Common standard deviation o significance level for the analysis of covariance Input the number of groups that are to be studied the variance of Power the means and the common Total sample size N standard deviation within 2 roups Specify the number ost per sampi ant of pastes a the R Total study cost squared with covariates then specify for power or sample size and nTerim will compute the other Number of covariates c R Squared with covariates R Test significance level a Calculate required sample sizes for given power gt Run C All columns Alpha is the probability of rejecting the null hypothesis Compute Effect Size Assistant Xx of equal means when it is true the probability of a Type error Total sample size N Input Advice N as multiple of n1 Sri Sni n1 Enter 0 05 a frequent standard Entry Options 0 001 to 0 20 Compute Transfer References aana ny z ui Compute Effect Size Assistant E Specify Multiple Factors ui Output a Help pa Notes
137. ed sample size for given power from the dropdown menu beside the Run button Then click Run This will give a result of 21 for the sample size as in Figure 4 12 10 The result in Signorini 1991 is 20 however this is due to the two decimal place rounding used in the paper Quen im 3 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test ea Plot Power vs Sample Size Ww Poisson Regression 1 T nnal 2 4 Test significance level a 0 05 1or 2 sided test 1 x2 x2 x2 z Baseline Response Rate e 0 1 Response Rate Ratio e 1 e 0 0 5 Mean Exposure Time pT 1 Overdispersion Parameter 1 Distribution of X1 sidetable required Normal z Normal z Normal Normal z Variance of b1 Null Hypothesis 1 Variance of bi Alternative Hypothesis 0 786 R squared X1 and independent variables 0 21 Power 95 792 Cost per sample Total study cost gt Calculate required sample size for given power Run E All columns Figure 4 12 10 Completed Poisson Regression study design 191 192 im The next calculation is a sensitivity analysis for sample size where we change the Response Rate Ratio and see its effect on sample size These values are taken from Table 2 of Signorini 1991 8 First delete the two Variance of b1 variables in the main table and re enter 95 f
138. een proportions in trials which use a completely randomized cluster randomization study design Superiority trials are those in which the researcher is testing that a treatment is better than the pre existing control treatment A completely randomized design assigns clusters randomly to control and treatment groups The sample size calculation for cluster randomized proportions is taken from Donner and Klar 2000 and is an extension of the methods used for superiority trials in a fully randomized trial as outlined in Chow et al 2007 The extension uses the intracluster correlation ICC which is the ordinary product moment correlation between any two observations in the same cluster to adjust for the effect of within cluster correlation It is assumed the ICC is the same in both groups This table can be used to calculate the power the number of clusters in the treatment group the sample size per cluster in the treatment group the intracluster correlation and the smallest detectable difference given all other terms in the table are specified To calculate power the number of clusters in the treatment group or the sample size per cluster in the treatment group the user must specify the test significance level a the control group proportion p2 which test type is being used the format of the test statistic whether higher values for the proportion are better or worse from the researcher s perspective the superiority margin the actual
139. email us for technical help at support statsols com In order to help us address your questions in the best way possible the more information you can provide us with the better If it is a technical question about one of our test tables screen shots of the completed tables of issues you are having are very helpful In order to address any installation issues or technical questions relating to the users machines the more information provided about the type of machine in question can speed up the process by a great deal Screen shots of installation issues are very helpful to us in solving any issue you may have Chapter 2 Getting Started Guide This chapter is a guide to help users get acquainted with the layout and various aspects of the interface of nQuery Advisor nTerim 3 0 This chapter aims at getting the user a firm understanding of how to approach study design using nTerim in a quick and easy way Every aspect of the nTerim interface will be presented in this chapter from the home window to the various plotting menus and side tables 2 1 Starting nTerim There are two main ways to open nTerim on your desktop By double clicking on the desktop icon nTerim will be automatically launched Alternatively if you chose not to have a desktop shortcut to nTerim you can find it by clicking on the Windows Start button and then select All Programs A list of all the programs on the user s machine will be listed in alphabetica
140. endent variables 0 0 0 0 Sample Size N 95 95 95 95 Cost per sample Total study cost DEE E O Run Figure 4 12 16 Sensitivity analysis around the Distribution of X1 19 Select Calculate required sample size for given power from the dropdown menu beside the Run button Then tick the box to run All Columns Then click Run This will give the resultant sample sizes of 21 58 325 and 112 sequentially as in Figure 4 12 17 Similar to the example above the answers for column 2 to 4 differ from Signorini due to rounding nQuen File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size Wo Poisson Regression 2 Cg 7 Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 1 z z z z Baseline Response Rate e 0 1 1 1 1 Response Rate Ratio e 1 e 0 0 5 0 5 0 5 0 5 Mean Exposure Time pT 1 1 1 1 Overdispersion Parameter 9 1 1 1 1 Distribution of X1 sidetable required Normal x Exponential jx Uniform x Binomial z Variance of b1 Null Hypothesis 1 1 1 4 Variance of bi Alternative Hypothesis 0 786 4 854 1 037 R squared X1 and independent variables 0 0 0 0 Sample Size N 21 58 23 112 95 792 95 047 95 043 95 07 Cost per sample Total study cost DEE D Calculate required sample size for given power Run all columns
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142. ent is to obtain N and Nz Using that N R x N the result from Equation 3 2 2 obtained is _ RPG P PA P Sou 1 R py p2 The steps involved are e Obtain e Solve Equation 3 2 4 for N and N R x N User supplies R only and selects Continuity Correction If the user has selected to use the continuity correction then apply the formula from Fleiss et al 1980 fie 3 2 5 R N p1 p2 to obtain Nice It follows that Ncc is then R X Nice If the user has NOT selected to use continuity correction then N N and Na R x N4 User specifies N4 only or N only When the user specifies N4 then Equation 3 2 1 can be re expressed as a quadratic in N from which two roots are obtained one less than and one greater than N4 Similarly if N is specified the roots gives the values of N4 Calculate Attainable Power with the given Sample Sizes Given a N4 proportions p14 p2 R or Nz time points and type of spending function the requirement is to obtain the power If the user has NOT selected to use continuity correction The steps are e Obtain by solving Equation 3 2 1 given that N1 R p4 p2 p are known e Obtain power using the algorithm by Reboussin et al 1992 and Jennison amp Turnbull 2000 If the user has selected to use continuity correction The steps are e Obtain by solving Equation 3 2 2 given that N1 R p4 p2 p are known e Obtain power using the algorithm by Reboussin et
143. epeated measures design is an experimental design in which multiple measurements are taken on one or more groups of subjects over time or under different conditions This type of design leads to a more precise estimate of an endpoint and can avoid the bias from a single measure For example an individual s blood pressure is known to be sensitive to many temporary factors such as amount of sleep had the night before mood excitement level exercise etc If there is just a single measurement taken from each patient then comparing the mean blood pressure between two groups could be invalid as there could be a large degree of variation in the single measures of blood pressure levels among patients However by obtaining multiple measurements from each individual and comparing the time averaged difference between the two groups the precision of the experiment is increased 4 3 2 Methodology Power and sample size are calculated using standard normal distributions following procedures outlined in Liu and Wu 2005 To calculate power and sample size the user must first specify the test significance level a and choose between a one or a two sided test The user must then enter a value for the number of levels M This value corresponds to the number of measurements that will be taken on each subject Values must then be provided for the between level correlation p and any two of group 1 proportions p4 group 2 proportions pz and odds ratio Y G
144. er To give an example of an ANCOVA design consider a study where we are examining test scores among students In this study it is found that boys and girls test scores for a particular subject differ However it is known that girls take more classes in the subject than boys We can use ANCOVA to adjust the test scores based on the relationship between the number of classes taken and the test score Thus enabling us to determine whether boys and girls have different test scores while adjusting for the number of classes taken 4 5 2 Methodology Power and sample size are calculated using central and non central F distributions and follow the procedures outlined by Keppel 1991 To calculate power and sample size the user must specify the test significance level and the number of groups G The user must then enter a value for the variance of means V Alternatively the user can enter the expected means in each group using the compute effect size assistant nTerim will then calculate the variance of means using the formula ei es 2 V a 4 5 1 i 1 1i where G _ Nili Z 4 5 2 7 2N i 1 The compute effect size assistant also allows the user to enter the expected sample sizes in each group or the expected ratio to group 1 for each group 7 This is particularly useful when you expect unequal sample sizes per group Once the variance in means is calculated the user must input a value for the common standard deviation
145. er x Control Group Proportion p2 0 5 0 5 0 5 0 5 Solve using p1 p2 p1 or p1 p2 Differences x Differences Differences x Differences lal Superiority Test Statistic 0 1 0 1 0 1 0 1 Actual Value of Test Statistic 0 15 0 15 0 15 0 15 Intracluster Correlation ICC 0 001 0 001 0 001 0 001 Clusters in Treatment Group K1 Clusters in Control Group K2 Ki K1 2 4 K1 30 Cluster Sample Size in Treatment Group M1 100 100 100 100 Cluster Sample Size in Control Group M2 100 100 100 100 80 80 80 80 Cost per sample Total study cost Mal o m gt Calculate required treatment group clusters K1 given power and sample size v Run m Figure 5 5 5 Sensitivity analysis around the Control Group Number of Clusters 9 Select Calculate required treatment group clusters K1 given power and sample size from the dropdown menu beside the Run button Then tick the box for All Columns Then click Run This will give the resultant values of K1 of 14 10 9 and 9 sequentially with the values of K2 updating automatically to reflect the desired ratio between K1 and K2 as in Figure 5 5 6 257 258 View Assistants Plot Tools Window Help E New Fixed Term Test _ New Interim Test ae Plot Power vs Sample Size 3 Test significance level a 0 05 0 05 a 0 05 Test Type Likelihood score x Likelihood score z Likelihood
146. er correlation ICC which is the ordinary product moment correlation between any two observations in the same cluster to adjust for the effect of within cluster correlation For means it can be calculated as the proportion of variability explained by the between cluster variation It is assumed the ICC is the same in both groups This table can be used to calculate the power the test significance level the number of clusters per group the sample size per cluster the intracluster correlation the standard deviation and the smallest detectable difference given all other terms in the table are specified To calculate power the number of clusters or the sample size per cluster the user must specify the test significance level whether to use a one or two sided test the difference in means 4 2 the standard deviation a and the intracluster correlation ICC The formulae use the non central t distribution to calculate power This distribution requires the appropriate degrees of freedom and a non centrality parameter NC The formula for the power is given by the following equation 1 a 1 a Power 1 t2 m 1 NC to m 1 1 5 t2 m 1 NC to m 1 1 5 5 1 1 where le H2l o NC 2 2 1 ICC N D nn 5 1 2 and m is the number of clusters per group and N is the sample size per cluster For the one sided test Z1 q 2 is replaced with Z _ A closed form equation is not used t
147. eriod designs in that the data from the trial is analyzed at one or more stages prior to the conclusion of the trial As a result the alpha value applied at each analysis or look must be adjusted to preserve the overall Type 1 error The alpha values used at each look are calculated based upon the spending function chosen the number of looks to be taken during the course of the study as well as the overall Type 1 error rate For a full introduction to group sequential methods see Jennison amp Turnbull 2000 and Chow et al 2008 Spending Function There are four alpha and beta spending functions available to the user in nTerim 3 0 as well as an option to manually input boundary values As standard all alpha spending functions have the properties that a 0 0 and a 1 a Similarly all beta spending functions have the properties that B 0 0 and 1 p Functionally the alpha and beta spending functions are the same In Table 3 3 1 we list the alpha spending functions available in nTerim 3 0 Table 3 3 1 Spending Function Equations O Brien Fleming a t 2 1 lt 2 Vt Pocock a t aln 1 e 1 t Power a t at gt 0 G e Hwang Shih DeCani a t a e 0 The parameter T represents the time elapsed in the trial This can either be as a proportion of the overall time elapsed or a proportion of the sample size enrolled The common element among most of the different spending functio
148. f covariates c R Squared with covariates R Power Total sample size N Cost per sample unit Total study cost a Calculate attainable power with the given sample sizes X All columns Compute Effect Size Assistant r x Variance of means V Total sample size N N as multiple of n1 Sri Eni ni Figure 4 5 10 Automatically updated Compute effect size Assistant Window 10 Once the table in Figure 4 5 11 has been completed the values for Variance of Means V and Total sample size N are computed click on Transfer to automatically transfer these values to the main ANCOVA test table 119 120 Im Compute Effect Size Assistant x Caer compute Transter ja Compute Effect Size Assistant Specify Multiple Factors Output Figure 4 5 11 Completed Compute Effect size Assistant Window 11 Now that values for Variance of Means V and Total sample size N are computed we can continue with filling in the main table For the Common Standard Deviation enter a value of 30 12 The number of covariates to be used in this study is set at 1 so enter the value 1 in the Number of covariates row Also the R Squared value has been estimated as 0 5 for this study design so enter 0 5 in the R Squared with covariates row 13 We want to calculate the attainable power give the sample size of 120 14 It has
149. f1 1 1 14 0 k asymptotically with a mean of log gt and a variance equal to gt al To To No Lut To Ory 0 calculate power and sample size the user must specify the test significance level the mean rate of the event for the control group 79 the ratio between the control group and treatment group mean event rates gt the average exposure time for each subject u 0 the common dispersion parameter of the two groups k the specification of how the null hypothesis variance is calculated Rates Variance in table and the sample size allocation ratio 0 ma It is assumed that the dispersion parameter is the same for both groups and 0 guidance on how to estimate the dispersion parameter is outlined in Zhu and Lakkis 2014 The formula for the sample size for the null hypothesis of gt 1 equivalent to f4 0 0 versus the alternative hypothesis of gt 1 equivalent to 6 0 is given by the following 0 equation 2 217o 21 pV 1 Ng 2 s m On 4 9 2 os G where ngo and n are the sample sizes of group O and 1 respectively Vo and V are the estimated variances of the normal approximation of 6 under the null and alternative hypotheses respectively These are defined below 159 Melim The power is calculated by re arrangement of the above formula to give the following equation Jno log 7 44 8 Vo Power 4 9 3 WW In the ab
150. fferences The equivalence trial is assumed to be using the two one sided test TOST methodology for analysis Thus the formula for the power is given by the following equation A OTest Type Power Zi a e E OUnpool Ounpool 6 Au ae es nue Ounpool Ounpool 5 3 1 where Orest Type is the standard error defined by the test type being used and dynyoo1 is the unpooled standard error The formula for the three standard error statistics is as follows 1 Farrington and Manning Test Statistic Likelihood Score The Farrington and Manning test uses the constrained maximum likelihood estimator of the two proportions to calculate the standard error for the calculation of power and was proposed as method to test against a null hypothesis of a specified difference A The standard error used by the Farrington and Manning test statistic is defined as follows 1 IF 1 IF by Prmi PremidlFy Prm2 Prm2 lF2 5 3 2 K M KM2 where K4 is the number of clusters in the treatment group M4 is the sample size per cluster in the treatment group Dry is the maximum likelihood estimator for each group proportion and IF is the inflation factor for the effect of clustering in the treatment and control groups respectively IF is defined as follows IF 1 ICC M 1 i 1 2 5 3 3 The constrained maximum likelihood where p pz A estimator of the two proportions is calculated using the f
151. given power The following example will look at a number of clusters calculation problem is conducted and then a sensitivity analysis is conducted to show the effect of changing control group number of clusters The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 5 3 1 below f Study Goal And Design T Design Goal No of Groups Analysis Method Fixed Term O Means One Test D Interim Proportions Two Confidence Interval Survival gt Two Equivalence Agreement Regression Cluster Randomized CRT Two Means Completely Randomized CRT Two Proportions Inequality Completely Randomized CRT Two Proportions Non Inferiority CRT Two Proportions Superiority OK Cancel Figure 5 3 1 Study Goal and Design Window 2 In order to select the CRT Two Proportions Equivalence Completely Randomized design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Cluster Randomized as the Goal Two as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 5 3 2 235 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size
152. gn Repeated measures for two means Negative Binomial Two Incidence Rates Two Poisson Means Cancel Figure 4 8 1 Study Goal and Design Window 2 In order to select the Two Poisson Means design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Means as the Goal Two as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 4 8 2 153 154 File Edit View Assistants Plot gt _ New Fixed Term Test lt New Interim Test Tools Window Help Z Plot Power vs Sample Size Two Poisson Means 1 Test significance level a Null Poisson Rate Ratio RO y0 y1 Alt Poisson Rate Ratio R1 y0 y1 Test Statistic Observation Time for Group 2 t2 Mean Poisson Rate in Group 1 y1 Sample Size Allocation Ratio N2 N1 Sample Size in Group 1 N1 Sample Size in Group 2 N2 Power Cost per sample Total study cost W1 MLE W1 MLE z W1 MLE x W1 MLE w2 MLE z Calculate attainable power with the given sample size All columns _ New Fixed Term Test Figure 4 8 2 Two Poisson Means Test Table in the Power row The table will appear as per Figure 4 8 3 File Edit View Assistants Plot New Interim Test Tools Window Help Plot Power vs Sa
153. gn table 15 The next step is to create the Covariance Matrix There are two ways of doing this in nTerim one is to enter specify the matrix manually in the Covariance Matrix Assistant table and the other way is to enter values for common standard deviation and correlation so nTerim can create the matrix automatically 16 In this example we are going to enter values for common standard deviation and correlation 137 Melim 17 In the Common standard deviation row enter a value of 2 In the Between level correlation row enter a value of 0 6 This is shown in Figure 4 6 15 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test E New Interim Test ex Plot Power vs Sample Size Ww Open Manual Statistical Solutions Support A a 2 Number of response variables p 3 Factor level table Means matrix Common standard deviation o Between level correlation Total sample size N Cost per sample unit Total study cost a D Run E AN columns Means Matrix Group Sizes 138 uJ Factor Level Table ui Means Matrix Group Sizes ui Covariance Matrix u Specify Multiple Factors ui Output Figure 4 6 15 Completed MANOVA Table 18 The next step is to enter the Group Size and as the groups will have equal sizes in this example of 4 enter 4 in the Group size n row Notice that the Means Matrix in Figure 4 6 15 has now been updated with the sample size pe
154. going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 4 7 8 Output x OUTPUT STATEMENT In a study investigating one poisson mean a sample size of 106 achieves 90 338 power to detect a difference between a null hypothesis mean of 0 03 and an alternative hypothesis mean of 0 1 with a significance level of 0 05 using a one sided one sample Poisson test ug Specify Multiple Factors u Output Figure 4 7 8 Study design Output statement This Output statement can then be easily transferred directly from the output window into a report 147 148 4 8 Two Poisson Means 4 8 1 Introduction Count data is often obtained in a variety of clinical and commercial activities such as the number of accidents at a junction or number of occurrences of a disease in a year The most common distribution used to model count data is the Poisson distribution The two sample test is used to test hypotheses about the difference between two Poisson distributed samples This table facilitates the calculation of the power and sample size for hypothesis tests of the mean of two Poisson distributed samples Power and sample size is computed using the method outlined by Gu et al 2008 Metin For this study design the fixed time intervals t4 t2 the sample sizes N N2 the number of events that occurred X X 2 and the mean Poisson even
155. h and Lavori 2000 175 176 4 11 2 Methodology The Cox Proportional Hazards model assumes that the hazard function A t for the survival time T given the discrete or continuous predictors X X3 Xx is described by the following regression equation be log mG X h ube 4 11 1 where o t is the baseline hazard The sample size calculation for an independent variable in the Cox Proportional Hazards model is taken from Hsieh and Lavori 2000 This table can be used to calculate the power the sample size the test significance level or the log hazard ratio given all other terms in the table are specified Calculations use a standard normal approximation and assume the statistical hypothesis is being tested using the Wald statistic To calculate power and sample size the user must specify the test significance level whether to use a 1 or 2 sided test the overall event rate P the expected log hazard ratio for the independent variable given the other terms in the model f the standard deviation of the independent variables a and the expected multiple correlation coefficient R between the independent variable and the other independent variables in the model If there is only one independent variable in the model then R is set to zero The formula for the sample size in a 2 sided test for the null hypothesis of 4 0 versus the alternative hypothesis of 4 B for an independent variable is gi
156. ha Power 0 05 wij Factor Level Table Means Matrix Group Sizes ui Covariance Matrix ui Specify Multiple Factors ui Output Figure 4 6 3 Enter Number of Response variables 9 Once the number of levels for each factor has been specified the next step is to populate the Means Matrix The Means Matrix is displayed in Figure 4 6 5 10 Depending on the values entered into the Factor Level table the size of the means matrix will be created wij Factor Level Table a Means Matrix Group Sizes a Covariance Matrix a Specify Multiple Factors u Output Figure 4 6 4 Factor Level Table 130 i Factor Level Table ij Means Matrix Group Sizes ui Covariance Matrix wilj Specify Multiple Factors u Output Figure 4 6 5 Means Matrix Group Sizes Assistants Table 11 As we have defined 2 response variables one with 4 levels and one with 3 levels we will require a Means Matrix with 2 rows and 3x4 columns There is an extra row included to enable the user to specify the individual level sample size only needed if unequal sample sizes per level 12 The next step is to fill in all the values for each part of the Means Matrix In this example we will define the Means Matrix as below first column of matrix are row names 1 1 1212342 3425 M 2 1 2 1412142114 n 6 87 456444454 13 Enter this matrix in the Means Matrix Assistant table as illustrated in Figure 4 6 6 wi Factor Level Table ui Mea
157. he top of column 5 All four columns will now be highlighted 9 Click the Plot Power vs Sample Size button at the top of the screen This will give you a plot as displayed in Figure 4 8 7 which will show the relationship between power and sample size for each column Right click to add feature such as a legend to the graph and double click elements for user options and editing 2480 7480 12480 17480 22480 I a a A W1 MLE W2 CMLE W3 Ln MLE W5 Variance Stabilizing Power 99 Sample Size 23657 Figure 4 8 7 Power vs Sample Size plot 10 Finally by clicking on the desired study design column and going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 4 8 8 Output x OUTPUT STATEMENT In a study investigating the poisson rate ratio of two groups a sample of 8590 in group 1 observed for 2 time periods and a sample of 4295 subjects in group 2 observed for 2 time periods achieves 90 0014651088 power to detect a change in the mean response ratio y2 Y1 from 1 to 4 when the mean rate for group 1 is 0 0005 at the 0 05 significance level using a one sided test uJ Specify Multiple Factors Jud Output Figure 4 8 8 Study design Output statement 157 158 4 9 Two Negative Binomial Rates 4 9 1 Introduction The Negative Binomial Model has been increasingly used t
158. his enables the user to seamlessly transition between nTerim and nQuery By opening the Study Goal and Design window using the options outlined in the previous section Section 2 4 the user has the full range of design tables available in both nTerim and nQuery at their disposal Study Goal And i Goal Analysis Method Means Test Proportions Confidence Interval Equivalence One sample t test Finite Population Paired t test for difference in Means Paired t test for difference in Means Finite Population Univariate one way repeated measures analysis of variance One way repeated measures contrast Univariate one way repeated measures analysis of variance Greenhouse Geisser Poisson One Mean Selected test is only available in nQuery Advisor Clicking OK will open the test in nQuery Advisor Figure 2 5 1 Study Goal and Design Window As shown in the Study Goal and Design window in Figure 2 5 1 above the user has selected a One sample t test This test is available in nQuery therefore a message has appeared at the bottom of the Study Goal and Design window stating Selected test is only available in nQuery Advisor Clicking OK will open the test in nQuery Advisor This message is highlighted in the red box in Figure 2 5 1 Once the user clicks OK this will prompt nQuery to open the specified test 13 IM 2 6 Using the Assistant Tables
159. iance of means V Common standard deviation o Number of covariates c R Squared with covariates R Power Total sample size N Cost per sample unit Total study cost pD Calculate required sample sizes for given power X All columns Figure 4 5 8 Analysis of Covariance Test Table Compute Effect Size Assistant het Compute Effect Size Assistant Specify Multiple Factors ui Output Figure 4 5 9 Compute Effect size Assistant Window 5 Once you enter a value for the number of groups G the Compute Effect Size Assistant table updates automatically as shown in Figure 4 5 10 6 In order to calculate a value for Effect Size the Variance of Means V needs to be calculated first 7 The mean for each level and the corresponding sample size need to be entered in the Compute Effect Size Assistant table 118 8 For the Mean values for each group enter 31 for group 1 41 for group 2 and 45 for group 3 9 For the group sample size n values for each group enter 40 for group 1 45 for group 2 and 35 for group 3 As a result the ratio 7 is calculated for each group as a proportion of group 1 File Edit View Assistants Plot Tools Window Help New Fixed Term Test lt New Interim Test W Plot Power vs Sample Size ANCOVA 1 i Pei Test significance level a 0 05 Number of groups G 3 Common standard deviation o Number o
160. iation at each level is 25 Therefore enter a value of 10 in the Difference in Means row and a value of 20 in the Standard deviation at each level row as shown in Figure 4 2 9 79 80 Im File Window Help Edit View Assistants Plot Tools New Fixed Term Test New Interim Test Z Plot Power vs Sample Size RM Two Means 1 1 or 2 sided test Number of levels M Difference in means p1 p2 Standard deviation at each level 0 Between level correlation p Group 1 size n1 j Group 2 size n2 Ratio n2 n1 Power Cost per sample unit Total study cost m m m DEN Calculate required sample sizes for given power X All columns Figure 4 2 8 Repeated Measures for Two Means Test Table 6 The between level correlation is estimated as 0 4 so enter 0 4 in the Between level correlation row 7 We want to calculate the required sample size to obtain a power of 90 so enter 90 on the Power row File Edit View Assistants Plot Tools Window Help gt New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Two Means 1 cues eel af milans Test significance level a 0 05 Jor 2sided test i 2 Number of levels M 4 Difference in means pi p2 15 Standard deviation at each level o 25 Between level correlation p 0 4 Group 1 size n1
161. ic 0 1 Actual Value of Test Statistic 0 15 Intracluster Correlation ICC 0 001 Clusters in Treatment Group K1 Clusters in Control Group K2 K1 Cluster Sample Size in Treatment Group M1 100 Cluster Sample Size in Control Group M2 100 Power 80 Cost per sample Total study cost MaL o m r D Calculate attainable power with the given sample size and number of clusters X Run F All columns Figure 5 5 3 Values entered for CRT Two Proportions Superiority study design Select Calculate required treatment group clusters K1 given power and sample size from the dropdown menu beside the Run button Then click Run This will give a result of 14 for K1 and K2 as in Figure 5 5 4 Q han nd L erim E File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size Two Proportions Superiority 1 m 2 3 4 Test significance level a 0 05 Test Type Likelihood score z Likelihood score Likelihood score z Likelihood score z Higher Proportions Better Worse Better z Better z Better z Better z Control Group Proportion p2 0 5 Solve using p1 p2 p1 or p1 p2 Differences z Differences z Differences z Differences z Superiority Test Statistic oa Actual Value of Test Statistic 0 15 Intracluster Correlation ICC 0 001 Custers in Treatment Group K1 14 Clusters
162. ide Table to define a normally distributed independent variable This will appear as in Figure 4 12 3 Normal Side Table x Standard Deviation Variance of b1 Null Hypothesis Variance of bi Alternative Hypothesis uJ Specify Multiple Factors ui Normal Side Table Exponential Side Table a Uniform Side Table a Binomial Side Table ui Output Figure 4 12 3 Normal Distribution Assistant table There are three other side tables one for an exponentially distributed variable one for a uniform distributed variable and one for a binomial distributed variable These can be selected in two manners First using the dropdown menu for the Distribution of X1 variable in the main table or secondly using the relevant tabs at the bottom of the side table The other three sides tables will look as follows Variance of b1 Alternative Hypothesis Variance of b1 Variance of b1 Alternative Hypothesis Hypothesis ui Specify Multiple Factors ui Normal Side Table u Exponential Side Table ui Uniform Side Table Binomial Side Table wi Output Figure 4 12 5 Uniform Distribution Assistant table Variance of bi Variance of bi Alternative Hypothesis Hypothesis uj Specify Multiple Factors ui Normal Side Table wi Exponential Side Table ui Uniform Side Table ui Binomial Side Table Output Figure 4 12 6 Binomial Distribution Assistant table The first calculation will be for Sample Size for using
163. ignificance level a Number of Dose Levels Sum of Weights Slope of Probit Regression B1 Relative Potency p Sample Size per Group N Total Number of Subjects Cost per sample Total study cost Io Calculate required sample size for given power X Run All columns 0 05 5 2 2013513108 23 25 1 1 90 0 05 5 2 2013513108 23 25 11 80 0 05 5 2 2013513108 23 25 1 16 90 0 05 5 2 2013513108 23 25 1 16 80 Figure 4 13 7 Sensitivity analysis around Relative Potency and Power 205 im 9 Select Calculate required sample size for given power from the dropdown menu beside the Run button Then tick the box to run All Columns Then click Run This will give the resultant sample sizes of 11 8 5 and 4 per group sequentially as per Kodell et al 2010 as in Figure 4 13 8 below File Edit View Assistants Plot Tools Window Help E New Fixed Term Test gt New Interim Test Z Plot Power vs Sample Size Probit Regression 1 C Test significance level a 0 05 0 05 0 05 0 05 Number of Dose Levels 5 5 5 5 Sum of Weights 2 2013513108 2 2013513108 2 2013513108 2 2013513108 Slope of Probit Regression B1 23 25 23 25 23 25 23 25 Relative Potency p 1 1 1 1 1 16 1 16 Sample Size per Group N 11 8 5 4 r l 90 5379299894 81 576000848 92 5490996322 87 4262231093 110 80 50 40 Total Nu
164. ignificance level row Next enter 0 03 and 0 1 for the Null or Baseline Mean Poisson Rate and Alternative Mean Poisson Rate variable rows respectively Finally enter 90 in the Power row The table will appear as per Figure 4 7 3 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size Poisson One Mean 1 o l 3 4 Test significance level a 0 05 Null or Baseline Mean Poisson Rate A0 0 03 Alternative Mean Poisson Rate A1 0 1 Sample Size N 30 Cost per sample Total study cost Wap o o gt Calculate attainable power with the given sample size Run F all columns Figure 4 7 3 Values entered for Poisson One Mean study design 4 Select Calculate required sample size for the given power from the dropdown menu beside the Run button Then click Run File Edit View Assistants Plot Tools Window Help E New Fixed Term Test NewInterimTest Plot Power vs Sample Size Poisson One Mean 1 _ 2 3 4 Test significance level a 0 05 Null or Baseline Mean Poisson Rate A0 0 03 Alternative Mean Poisson Rate A1 Sample Size N Cost per sample Total study cost ICEN Caiana required sample size for given power ne n E All columns Figure 4 7 4 Completed Poisson One Mean study design This will give a result of 106 as displayed
165. in Control Group K2 14 Cluster Sample Size in Treatment Group M1 100 Cluster Sample Size in Control Group M2 100 82 313 Cost per sample Total study cost A allaa SEE ial o DD Calculate required treatment group clusters K1 given power and sample size v Run An columns Figure 5 5 4 Completed CRT Two Proportions Superiority study design The next calculation is a sensitivity analysis for the treatment group number of clusters when the control group number of clusters is changed 7 Delete the values for K1 and K2 in the first column then replace the updated power with 80 and enter K1 in the control group clusters row Then copy the first column into columns 2 to 4 Other ratios other than K1 and K2 being same between K1 and K2 can be calculated by using arguments for K2 such as 2 K1 or K1 2 to have K2 be twice as large as K1 for example 8 Enter K1 2 in column 2 for Clusters in Control Group K2 4 K1 in column 3 and 30 in column four This will give a table as per Figure 5 5 5 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Two Proportions Superiority 1 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 Test Type Likelihood score x Likelihood score x Likelihood score x Likelihood score v Higher Proportions Better Worse Better x Better y Better x Bett
166. in Figure 4 7 4 above Example 2 Validation example calculating required power for a given sample size The next calculation is a sensitivity analysis for Power where the Alternative Mean Poisson Rate is varied while the sample size is fixed at 106 5 To do this copy the same values across to column 2 and 3 Then change the value for Alternative Mean Poisson Rate to 0 05 and 0 08 for column 2 and 3 respectively This will give a table as per Figure 4 7 5 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size LU Open m Poisson One Mean 1 d a 2 3 4 Test significance level a 0 05 0 05 0 05 Null or Baseline Mean Poisson Rate A0 0 03 0 03 0 03 Alternative Mean Poisson Rate A1 0 1 0 05 0 08 Sample Size N 106 106 106 Cost per sample Total study cost Calculate attainable power with the given sample size Run V All columns Figure 4 7 5 Sensitivity analysis around the Alternative Mean Poisson rate 6 Select Calculate attainable power for the given sample size from the dropdown menu beside the Run button Then tick the box to run All Columns Then click Run This will give the resultant powers of 90 338 28 287 and 74 168 for columns 1 2 and 3 respectively as in Figure 4 7 6 145 146 Im nQuery nTeri Query nTerim 3 File Edit View Assistants Plot Tools Window Help E New
167. ing values 3 Enter 0 05 for alpha 2 sided 0 4 for Group 1 proportion 0 6 for Group 2 proportion The odds ratio is calculated as 2 25 4 Select Off for the Continuity Correction We are interested in solving for sample size given 90 power so enter 90 in the Power row 5 This study planned for 4 interim analyses Including the final analysis this requires Number of Looks to be 5 37 6 The looks will be equally spaced and the Pocock spending function is to be used There will be no truncation of bounds 7 It is estimated that the cost per unit is roughly 180 so enter 180 in the Cost per sample unit row nQue File Edit View Assistants Plot Tools New Fixed Term Test New Interim Test Window Z Plot Power vs Sample Size Help Lo x LJ Open Manual Statistical Solutions Support GST Two Proportions 1 Odds ratio W n2 1 n1 n1 1 n2 Group 1 size ni i Group 2 size n2 Ratio N2 N1 f Continuity correction off hast Cost per sample unit Total study cost Number of looks Information Times Max times Determine bounds Equally Spaced 1 Spending Function O Brien Fleming Phi No Truncate at Dont Calculate O Brien Fleming 5 z Equally Spaced 1 Spending Function O Brien Fleming ne z Don t Calculate x O Brien Fleming z g 5 5 Equally Spaced 1 1 O Brien Fleming
168. ion one normal covariate adjusted for others Linear Regression one covariate Linear Regression multiple covariates Linear Regression multiple covariates adjusted for others Linear Regression test of coefficient Poisson Regression Probit Regression ok cance Figure 4 11 1 Study Goal and Design Window 2 In order to select the Cox Regression design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Regression as the Goal One as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 4 11 2 178 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size Cox Regression 1 P 2 1or 2 sided test 2 x2 z 2 Standard Deviation of X1 0 R squared of X1 and other X s Log Hazard Ratio B Overall Event Rate P Sample Size N Power Cost per sample unit Total study cost Calculate attainable power with the given sample size Run An columns lz z O Figure 4 11 2 Cox Regression Test Table The first calculation will be for Sample Size for the multiple myeloma data set example 3 First enter 0 05 for the Test Significance level then select 1 for the 1 or 2 sided test variable dropdown menu 4 Enter 0 3126 for the Standa
169. iven two of p1 p2 or Y nTerim will compute the other using the following equation _ p21 pr 4 3 1 pi 1 p2 Given the above values in order to calculate the power for this design the user must enter the expected sample size for each group N and N2 nTerim then uses the total sample size N to calculate the power of the design using the following equation The formula used to calculate power is Ni pi Nop2 Niqi N2q2 P 1 O Z ee K N N p q N2p2q2 4 3 2 d MN 1 r n 1 p M 1 pig 1 T p2q2 where is the standard normal density function and N N N 4 3 3 Ny 4 3 4 w 4 3 4 q 1 p 4 3 5 q2 1 p2 4 3 6 d p pz 4 3 7 In order to calculate sample size a value for power must be specified nTerim does not use a closed form equation to calculate sample size Instead a search algorithm is used This search algorithm calculates power at various sample sizes until the desired power is reached 89 im 4 3 3 Examples Example 1 Investigate how Group Proportion affects Sample size for a given Power In this example we examine how the group proportion affects sample size values for a given power The following steps outline the procedure for Example 1 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A S
170. ixnesiltnaaraccaaneeaiean sia narnncauaaaaiatadaabiogaanaeuiatoinaianaalscalaaseomenmeahonieh 88 4 3 2 ICCNGACIO BY ccits ipusdieteatacesengteateyanscbduigsdaanosdivedsnag taai a iiaiai 89 AD SEMAN NGS seis seecaninseeraiassasdateatarn EEEE EEEE REEE AEAEE NEKE E ESENE 90 4 4 One Way Analysis of Variance ANOVA ccsssscssssecsscesssssecssseeesseceeseeseseneessanees 101 4 41 MOGI CLIO MN sasitvcaisannseiesacutunivisaawina tiknan sa eea EKES KNARANTE E RVE a VEKAS eananbans 101 44 2 Methodology as ieniscioisasrorai inen aE AE aS aaaea e aSa EAE ENa ah 102 4 4 3 Exa Mple Sceni a a A a 103 4 5 Analysis of Covariance ANCOVA ccccsssccceesssseceeseseeeceeseseceesssaeecesesaeeeeeesasseeeeneues 110 451e troduction ssassn naaa apa aaa Ea aariaa KEKE aa aeaiia 110 4 5 2 MethodOlogy cccsessccccececscsesssseaeeeceeecsesesseaeseceeeesseesesaeaeeeeeeseseseeeaeseeeeeesssessaaeas 111 A5 Dx EXAIMMOIGS cosecccestessesschsecatevectesoecuiasicaieetassuensntuns teste nousevarsuactnce ie wena se ceteneeaustens tested 113 4 6 Multivariate Analysis of Variance MANOVA ccccsscccceesssceceesssceeeesssaeeecesseeeeeensaes 123 4 6 1 UU FADING IAs des asda este e E E nde angalalls TE 123 4 6 2 MethOdOlOgy cccsecscccceccescsssessaecececscsesesseaeseceeeesseeseaeaeseeeesessseseeaeeeeeesesesesenaeas 124 4 6 3 EXAM Ple S eirinen iae i ease a o eai r EE a aa a aeiaai 128 4 7 0NE Poisson Mediese cede cotadebeeceadpeeest ld chee pes dads e
171. l order You can locate nTerim under the title nQuery Advisor nTerim 3 0 Click on this folder and then select nQuery Advisor nTerim 3 0 to launch the program 2 2 Home Window Once the user has launched nTerim the home window will appear as illustrated below in Figure 2 2 1 From the home window there are several options open to the user depending on what they want to do The user can open a new fixed term or interim design table open a previous design that was saved before access the manual or access the Statistical Solutions support website for help or guidance File Edit View Assistants Plot Tools Window Help E New Fixed Term Test I New Interim Test a Plot Power vs Sample Size LU Open Manual Statistical Solutions Support Figure 2 2 1 Home Window 2 3 Menu Bar The first aspect of the interface we will review is the menu bar and all the options available There are eight options on the menu bar File Edit View Assistants Plot Tools Window and Help These are highlighted in Figure 2 3 1 below File Edit View Assistants Plot Tools Window Help Figure 2 3 1 Menu Bar The File menu allows the user to open a new or previously saved design table as well as enabling the user to save a design and allowing the user to exit nTerim whenever they wish Design tables can be saved as nia format which is the Statistical Solutions file format for nTerim The Edit menu enables to user to fill a desig
172. le 1 Validation example calculating required sample size for a given power The following examples are taken from Kodell et al 2010 where a sample size calculation problem is conducted for a five dose level study followed by two sensitivity analyses one for the effect of power and relative potency on sample size and one for studies with a different number of dose levels Finally there will be a sample size calculation where the slope is calculated using the side table The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 4 13 1 below study Goal And Design i mesa Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval 5 Survival gt Two Equivalence Agreement Regression Cluster Randomized Logistic Regression one normal covariate Logistic Regression one normal covariate adjusted for others Linear Regression one covariate Linear Regression multiple covariates Linear Regression multiple covariates adjusted for others Linear Regression test of coefficient Cox Regression Poisson Regression Probit Regression ox cancet Figure 4 13 1 Study Goal and Design Window 2 In order to select the Probit Reg
173. level correlation and keeping the power fixed at 90 we can see that as the between level correlation increases so does the sample size required With a lower between level correlation a lower sample size is required Example 3 Differences in Group Size Ratios In this example we investigate how the sample size ratio between Group 1 and Group 2 affects the overall sample size required to obtain a given power The following steps outline the procedure for Example 3 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear 3 Study Goal And Design x Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval Survival gt Two Equivalence O Agreement Regression Two sample t test Student s t test equal variances Satterwaithe s t test unequal variances Two group t test for fold change assuming log normal distribution Two group t test of equal fold change with fold change threshold Wilcoxon Mann Whitney rank sum test continuous outcome Wilcoxon Mann Whitney rank sum test ordered categories Two group univariate repeated measures ANOVA Greenhouse Geisser correction 2x2 Crossover Design i Repeated Measures for two means
174. lity that an effect will not be found that the trial can be terminated by rejecting the alternative hypothesis In the case where the user wishes to generate boundaries for early rejection of either the null or alternative hypothesis H or H they are given two options either to have the boundaries binding or non binding With binding boundaries if the test statistic crosses the futility boundary the test must be stopped otherwise the type 1 error may become inflated The reason for this is that there is an interaction between the efficacy and futility boundaries in their calculation that could cause the efficacy boundary to shift In the case of non binding boundaries the efficacy boundaries are calculated as normal that is as if the futility boundaries did not exist This eliminates the danger of inflating the type 1 error when the futility boundary is overruled The downside of the non binding case is that it may increase the required sample size relative to the binding case The boundaries calculated in nTerim 3 0 follow the procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 21 22 Im 3 1 2 Methodology The variables are defined as Symbol Description a Probability of Type error B Probability of Type II error 1 Power of the Test Ly Le Group Means 01 02 Group Standard Deviations Ni N3 Group Sample Sizes R Ratio of N4 to N 0 Drift Parameter K Number
175. llowing steps outline the procedure for Example 2 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear i apa Study Goal And Design a Design No of Groups Analysis Method Fixed Term P One Test Interim Proportions Two Confidence Interval 5 Survival gt Two Equivalence D Agreement D Regression Chi squared test to compare two proportions Compute power or sample size Compute one or two proportions Chi squared test continuity corrected Compute power or sample size Compute one or two proportions Fisher s exact test Two group Chi square test comparing proportions in C categories Mantel Haenszel Cochran test i Mantel Haenszel Cochran test of OR 1 in S strata Mantel Haenszel Cochran test of OR 1 in S strata continuity corrected Repeated Measures for two proportions Cancel Figure 4 3 8 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear This test table is illustrated in Figure 4 3 8 3 An additional table that will be used in this example is the Specify Multiple Factors table displayed in Figure 4 3 9 This is used to generate multiple columns and designs by entering a range of values for particular parameters 4
176. lumn 1 values across to columns 2 and 3 12 For Rates Variance select True Rates and Maximum Likelihood for column 2 and 3 respectively This will give a table as per Figure 4 9 9 166 nQuery nTe File Edit View Assistants Plot Tools Window Help i E New Fixed Term Test New Interim Test Z Plot Power vs Sample Size LL Of Negative Binomial 1 A eens nee 2 3 4 Test significance level a 0 05 0 05 0 05 Mean Rate of Event for Control r0 0 8 0 8 0 8 Rate Ratio r1 r0 0 85 0 85 0 85 Average Exposure Time pT 0 75 0 75 0 75 Dispersion Parameter k 0 7 0 7 0 7 Rates Variance Reference Group Rate z True Rates z Maximum Likelihood z Reference Group Rate z Sample Size Ratio N1 N0 1 1 1 Control Group Sample Size N0 Treatment Sample Size N1 C 2 80 Cost per sample Total study cost qa o o E gt Run YI An columns li Figure 4 9 9 Comparing Rates Variance options 13 Select Calculate required sample size for given power from the dropdown menu beside the Run button Then tick the box to run All Columns Then click Run nQuery nTerir File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Ga Plot Power vs Sample Size LU Of Negative Binomial 1 F ee heel 2 3 4 Test significance level a 0 05 0 05 0 05 Mean Rate of Event for Control r0
177. m Fixed Period designs in that the data from the trial is analyzed at one or more stages prior to the conclusion of the trial As a result the alpha value applied at each analysis or look must be adjusted to preserve the overall Type 1 error The alpha values used at each look are calculated based upon the spending function chosen the number of looks to be taken during the course of the study as well as the overall Type 1 error rate For a full introduction to group sequential methods see Jennison amp Turnbull 2000 and Chow et al 2008 Spending Function There are four alpha and beta spending functions available to the user in nTerim 3 0 as well as an option to manually input boundary values As standard all alpha spending functions have the properties that a 0 0 and a 1 a Similarly all beta spending functions have the properties that B 0 0 and 1 p Functionally the alpha and beta spending functions are the same In Table 3 1 1 we list the alpha spending functions available in nTerim 3 0 Table 3 1 1 Spending Function Equations O Brien Fleming a t 2 1 Vt Pocock a t aln 1 e 1 t Power a t at gt 0 G e Hwang Shih DeCani a t a e 0 The parameter T represents the time elapsed in the trial This can either be as a proportion of the overall time elapsed or a proportion of the sample size enrolled The common element among most of the different spending fu
178. m the dropdown menu beside the Run button Then click Run This will give a result of 5 for K1 and K2 as in Figure 5 3 4 File Edit View Assistants Plot Tools Window Help i E New Fixed Term Test New Interim Test Plot Power vs Sample Size Two Proportions Equivalence 1 P P 2 3 a Test significance level a 0 05 Test Type Pooled z Likelihood score z Likelihood score z Likelihood score z Control Group Proportion p2 0 5 Solve using p1 p2 p1 or p1 p2 Differences z Differences z Differences z Differences Test Statistic for Upper Equivalence Margin 0 1 Test Statistic for Lower Equivalence Margin 0 1 Actual Value of Test Statistic 0 Intracluster Correlation ICC 0 001 Clusters in Treatment Group K1 5 Clusters in Control Group K2 5 Cluster Sample Size in Treatment Group M1 100 Cluster Sample Size in Control Group M2 Cost per sample Total study cost pre gt Calculate required treatment group clusters K1 given power and sample size Run All columns Figure 5 3 4 Completed CRT Two Proportions Equivalence study design 237 im The next calculation is a sensitivity analysis for the treatment group number of clusters when the control group number of clusters is changed 7 Delete the values for K1 and K2 in the first column then replace the updated power with 80 and enter K1 in the control group clusters ro
179. mber of Subjects Cost per sample Total study cost Run V All columns Calculate required sample size for given power Figure 4 13 8 Results from Sensitivity analysis The next calculation is a sensitivity analysis for sample size where we change Number of Dose Levels where these values are taken from Table 2 of Kodell et al 2010 10 For columns 3 and 4 delete the sum of weights and sample size values Then for the Slope of Probit Regression replace 23 25 with 23 41 For Relative Potency replace 1 16 with 1 1 and return the power values to 90 and 80 11 Finally set Number of Dose Levels to 7 for both This will give a table as displayed in Figure 4 13 9 206 File Window Help Edit View Assistants Plot Tools E New Fixed Term Test l New Interim Test Plot Power vs Sample Size Probit Regression 1 Pioo Laaa aaa Za 4 Test significance level a 0 05 0 05 0 05 0 05 Number of Dose Levels 5 5 7 7 Sum of Weights 2 2013513108 2 2013513108 Slope of Probit Regression B1 23 25 23 25 23 41 23 41 Relative Potency p 11 1 1 1 1 11 Sample Size per Group N 11 8 90 5379299894 81 576000848 90 80 Total Number of Subjects 110 80 Cost per sample Total study cost ce ee Pp Run att columns Figure 4 13 9 Sensitivity analysis around the Number of Dose Levels 12 Next we need to complete the
180. mith P G amp Morrow R H 1996 Field Trials of Health Interventions in Developing Countries A Toolbox Second Edition Macmillan Malaysia Zhu H amp Lakkis H 2014 Sample size calculation for comparing two negative binomial rates Statistics in Medicine 33 3 pp 376 387 263 North Central South America amp Canada Statistical Solutions One International Place 100 Oliver Street Suite 1400 Boston MA 02110 Tel 617 535 7677 Fax 617 535 7717 Email sales statsols com STATISTICAL SOLUTIONS www statistical solutions software com Europe Middle East Africa amp Asia Statistical Solutions 4500 Airport Business Park Cork Rep of Ireland Tel 353 21 4839100 Fax 353 21 4840026 Email sales statsols com
181. mple Size The first calculation will be for Sample Size using the W1 MLE test statistic to begin we enter the values as follows 3 First enter 0 05 for the Test Significance level row then enter 1 and 4 for the Null Poisson Rate Ratio and Alternative Poisson Rate Ratio variable rows respectively 4 Next enter two for both Observation time variables t1 t2 0 0005 for the Mean Poisson Rate in Group 1 and a Sample size allocation ratio of 0 5 and finally enter 90 Two Poisson Means 1 a O Test significance level a Null Poisson Rate Ratio RO y0 y1 Alt Poisson Rate Ratio R1 y0 y1 Test Statistic Observation Time for Group 1 t1 Observation Time for Group 2 t2 Mean Poisson Rate in Group 1 y1 Sample Size Allocation Ratio N2 N1 Sample Size in Group 1 N1 Sse GE 0 05 1 4 W1 MLE 2 2 0 0005 0 5 x w2 MLE _ Cost per sample Total study cost x w2 MLE i w2 MLE w2 MLE z Calculate attainable power with the given sample size a gt All columns Figure 4 8 3 Values entered for Two Poisson Mean study design 5 Select Calculate required Group 1 and 2 sample sizes for given power and sample size allocation from the dropdown menu beside the Run button Then click Run This will give a result of 8564 and 4282 for the Group 1 and 2 sample sizes respecti
182. mple size The following steps outline the procedure for Example 2 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear I meer Study Goal And Design mesm Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval Survival gt Two D Equivalence Agreement Regression One way analysis of variance One way analysis of variance Unequal n s Single one way contrast Single one way contrast Unequal n s Two way analysis of variance Multivariate analysis of variance MANOVA Analysis of Covariance ANCOVA OK Cancel Figure 4 5 7 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear 117 3 There are two main tables required for this test the main test table illustrated in Figure 4 5 8 and the effect size assistant table shown in Figure 4 5 9 4 Enter 0 05 for alpha the desired significance level and enter 3 for the number of groups G as shown in Figure 4 5 10 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size ANCOVA 1 a el 2 Number of groups G Var
183. mpleted and values for Contrast C and Scale D are computed click on Transfer to automatically transfer these values to the main table uJ Compute Effect Size Assistant ja Specify Multiple Factors ui Output Figure 4 1 11 Completed Compute Effect Size Assistant Table 12 Now that values for Contrast C and Scale D have been computed we can continue with filling in the main table For the Standard Deviation enter a value of 3 677 For the between level correlation enter a value of 0 13 We want to calculate the attainable power given the sample size therefore enter 30 in the Group size n row 14 The cost per sample unit cannot be estimate yet in this study so we will leave this row blank for this calculation This value has no impact on the sample size or power calculation 15 Select Calculate attainable power with the given sample sizes from the drop down menu below the main table and click Run This is displayed in Figure 4 1 12 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size RM Contrast 1 SS S 2 3 4 Test significance level a 0 05 Number of levels M 3 Contrast C Jci pi 6 Scale D SQRT ici 2 44949 Standard deviation at each level o 3 677 Between level correlation p 0 Effect size A C D o SQRT 1 p 0 66617 Power 94 82 Group size N eg 30 Cost per sample un Total stu
184. mpute Effect Size Assistant table automatically updates as shown in Figure 4 1 4 7 In order to calculate a value for Effect Size two parameters need to be calculated first the Contrast C and Scale D 8 The mean for each level and the corresponding coefficient value need to be entered in the Compute effect Size Assistant table 9 For the Mean values for each level enter 12 for level 1 12 for level 2 and 14 for level 3 10 For the Coefficient values for each level enter O for level 1 1 for level 2 and 1 for level 3 The sum of these values must always equate to zero This is illustrated in Figure 4 1 5 below 61 Im File Edit View Assistants Plot Tools Window Help i New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Contrast 1 ma 0 05 3 Scale D SQRT Sci2 Standard deviation at each level o Between level correlation p Effect size A C D o SQRT 1 p Power a eee ill ae Compute Effect Size Assistant Scale D SQRT ici Figure 4 1 4 Automatically Updated Compute Effect Size Assistant Table 11 Once the table in Figure 4 1 5 is completed and values for Contrast C and Scale D are computed click on Transfer to automatically transfer these values to the main table S Compute Effect Size Assistant u Specify Multiple Factors Output Figure 4 1 5 Completed Compute Effect Size As
185. mula the value of d is increased until the inequality above is fulfilled and the interval between the left hand side expression and right hand side expression contains at least one integer 141 im The power is calculated by finding X as outlined above for the specified value of Ag This value of X is used to calculate power as follows x X crit mea n nd Power 1 e aa Ho Ao 2 24 Ha o lt A4 faA ye 4 7 3 na MALY Hy Ag lt Au Hat Ay gt A Power e can o Ag S A1 Ha Ag gt Ay A closed form equation is not used to calculate the rate ratio Instead a search algorithm is used Firstly X is calculated as outlined above for the specified value of Ay The search algorithm then calculates power at various values of A until the desired power is reached 142 4 7 3 Examples Example 1 Validation example calculating required sample size for a given power The following example is taken from Question 1 31 from page 29 of Guenther 1977 where a sample size calculation problem is followed by a sensitivity problem for the Alternative Hypothesis Rate variable The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 4 7 1 below Ce Goal Analysis Method Means Test Proportions
186. n Then tick the box to run All Columns Then click Run This will give the resultant values of K1 of 5 4 3 and 3 sequentially with the values of K2 4 is updating automatically to reflect the desired ratio between K1 and K2 as in Figure 5 3 6 238 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Two Proportions Equivalence 1 C OS i Test significance level a 0 05 0 05 0 05 0 05 Test Type Pooled x Pooled x Pooled x Pooled ina Control Group Proportion p2 0 5 0 5 05 0 5 Solve using p1 p2 p1 or p1 p2 Differences z Differences z Differences z Differences z Test Statistic for Upper Equivalence Margin 0 1 0 1 0 1 0 1 Test Statistic for Lower Equivalence Margin 0 1 0 1 0 1 0 1 Actual Value of Test Statistic 0 0 0 0 Intracluster Correlation ICC 0 001 0 001 0 001 0 001 Clusters in Treatment Group K1 5 4 3 3 Clusters in Control Group K2 5 8 12 30 Cluster Sample Size in Treatment Group M1 100 100 100 100 Cluster Sample Size in Control Group M2 100 100 100 r l 85 859 81 004 86 787 Cost per sample Total study cost IKE Calculate required treatment group clusters K1 given power and sample size v Run Figure 5 3 6 Results from Sensitivity analysis lt All columns 10 By clicking on the desired study
187. n a completely randomized cluster randomized trial This table facilitates the calculation of the power and sample size for non inferiority hypothesis tests comparing proportions in a completely randomized cluster randomized trial Power and sample size is computed using the method outlined by Donner and Klar 2000 5 4 2 Methodology This table provides sample size and power calculations for studies which will be conducting a non inferiority trial between proportions in trials which use a completely randomized cluster randomization study design Non inferiority trials are those in which the researcher is testing that a treatment is no worse than the pre existing control treatment A completely randomized design assigns clusters randomly to control and treatment groups The sample size calculation for cluster randomized proportions is taken from Donner and Klar 2000 and is an extension of the methods used for non inferiority trials in a fully randomized trial as outlined in Chow et al 2008 The extension uses the intracluster correlation ICC which is the ordinary product moment correlation between any two observations in the same cluster to adjust for the effect of within cluster correlation It is assumed the ICC is the same in both groups This table can be used to calculate the power the number of clusters in the treatment group the sample size per cluster in the treatment group the intracluster correlation and the smallest detect
188. n click Run This will give the resultant values of K1 of 14 11 9 and 9 sequentially with the values of K2 updating automatically to reflect the desired ratio between K1 and K2 as in Figure 5 4 6 247 Toone File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size Ww Open Se een eee 2 3 4 0 05 0 05 0 05 0 05 Unpooled z Unpooled z Unpooled z Unpooled z Better x Better x Better x Better z Control Group Proportion p2 0 5 0 5 0 5 0 5 Solve using p1 p2 p1 or p1 p2 Differences z Differences z Differences iz Differences z Non inferiority Test Statistic 0 1 0 1 0 1 0 1 Actual Value of Test Statistic 0 05 0 05 0 05 0 05 Intracluster Correlation ICC 0 001 0 001 0 001 0 001 Clusters in Treatment Group K1 14 11 9 9 Clusters in Control Group K2 14 22 36 30 Cluster Sample Size in Treatment Group M1 100 100 100 100 Cluster Sample Size in Control Group M2 1100 100 100 100 Power 81 199 82 817 82 252 80 914 Cost per sample Total study cost ME m gt Calculate required treatment group clusters K1 given power and sample size v Run V All columns Figure 5 4 6 Results from Sensitivity analysis 10 By clicking on the desired study design column and going to the Output tab at the bottom of the screen you can get an output statemen
189. n each group will be different then the expected sample size in each group must be specified in the Means Matrix nTerim gives the option of calculating power using one of three commonly used test statistics Wilks lambda Pillai Bartlett Trace or Hotelling Lawley trace In order to perform calculations using either of these three statistics nTerim first calculates the matrices H E and T using the following formulas CM 4 6 3 where C is a matrix of contrasts that nTerim automatically generates This is an orthogonal matrix that is unique to each factor and factor interaction M is the means matrix which has been inputted by the user H 09 C X X C T t Oo 4 6 4 where is the matrix of hypothesised means which is zero for this test and X is the design matrix E X N 7r 4 6 5 where amp is the covariance matrix T H E 4 6 6 Wilks Lambda Using these matrices the test statistic for Wilks lambda is calculated using the formula W ET 4 6 7 The transformation of this test statistic to an approximate F is given by n af Fafa fo 1 n dfy n dfs 4 6 8 where 125 i n 1 W9 a q 1 df ap df g N r p a 1 2 ap 2 2 Pillai Bartlett Trace The test statistic for Pillai Bartlett trace is calculated using the formula PBT tr HT The transformation of this test statistic to an approximate F is given by __n af mee a df P
190. n entered select Calculate the attainable power with the given sample sizes from the drop down menu and click Run 9 The boundaries calculated are shown in Figure 3 3 11 0 2 0 6 2 65511 2 62320 2 58958 2 34880 2 65511 2 62320 2 58958 2 34880 0 00793 0 00871 0 00961 0 01883 0 00793 0 00684 0 00602 0 01464 0 00793 0 01477 0 02079 0 03543 5 20 8 79 10 68 34 58 Cumulative exit probability 5 20 13 99 2468 59 26 Nominalbeta E Incremental beta Cumulative beta Exit probability under HO Cumulative exit probability under HO 1 2 27923 2 27923 0 02265 0 01457 0 05000 26 06 85 32 4 Looks u Specify Multiple Factors wi Output Figure 3 3 11 Boundary Table for Pocock Spending Function 55 Melim 56 10 Finally the boundaries calculated in the table displayed in Figure 3 3 11 are automatically plotted as illustrated in Figure 3 3 12 ces ph _ a Pocock Boundaries with Alpha 0 05 Figure 3 3 12 Boundary Plot for Proportional Hazard Survival Test By clicking on the output tab at the bottom of the screen you can see a statement giving details of the calculation A total sample size of at least 1000 550 events is required to achieve 85 32 power to detect a hazard ratio of 0 756 for survival rates of 0 5 in group 1 and 0 4 in group 2 using a 2 sided log rank test with 0 05 significance level assuming that the hazards are pr
191. n table using the Fill Right option This is where the user when defining multiple columns enters certain information into a column and can copy this information across the remaining empty columns The View menu is initially unavailable until the user opens a design table Once a table has been opened several options appear enabling the user to view various plots and toggle between various assistant tables help guides cards and notes The Assistants menu is initially unavailable until the user opens a design table Once a table has been opened the menu enables the user to open and toggle between various side tables depending on the design table Another side table located under the Assistants menu is the Specify Multiple Factor table This table enables the user to specify a range of designs or columns in a table The Plot menu is initially unavailable until the user opens a design table Once a table has been opened the user can use this menu to create certain plots such as Power vs Sample Size plots Boundaries Plots and Spending Function Plots The Tools menu allows the user to define certain settings before running any analysis such as defining the minimum cell count and outlining various assumptions in relation to group proportions and means This also enables the user to save design tables and Looks tables as images for transporting The Window menu is initially unavailable until the user opens a design table Once a table has been
192. nctions is to use lower error values for the earlier looks By doing this it means that the results of any analysis will only be considered significant in an early stage if it gives an extreme result Boundaries The boundaries in nTerim 3 0 represent the critical values at each look These boundaries are constructed using the alpha and beta spending functions Users in nTerim 3 0 are given the option to generate boundaries for early rejection of the null hypothesis Hy using the alpha spending function or to generate boundaries for early rejection of either the null or alternative hypothesis Hj or H4 using a combination of both the alpha and beta spending functions The notion of using an alpha spending function approach to generate stopping boundaries for early rejection of Hg was first proposed by Lan and DeMets 1983 we refer to such boundaries in nTerim 3 0 as efficacy boundaries Building on the work of Lan and DeMets Pampallona Tsiatis and Kim 1995 2001 later put forward the concept of using a beta spending approach to construct boundaries for early rejection of H we refer to these boundaries in nTerim as futility boundaries Essentially if a test statistic crosses an efficacy boundary then it can be concluded that the experimental treatment shows a statistically significant effect the trial can be stopped with rejection of the null hypothesis If the test statistic crosses a futility boundary then this indicates with high probabi
193. ndow Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size ANOVA 1 2 3 4 Test significance level a 0 05 Number of groups G 3 Variance of means V 11 76 Common standard deviation o 6 0 32667 94 82 2 5 Effect size A v o2 2 s Ene N as multiple of n1 Sri Zni ni Total sample size N 50 85 x All columns Figure 4 4 6 Completed One Way Analysis of Variance Test Table Calculate attainable power with the given sample sizes It can be seen from Figure 4 4 6 that a sample size of 50 is required to obtain a power of 94 82 Due to the cost per sample unit of 85 the overall cost of sample size required has amounted to 4 250 By clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation When the total sample size across the 3 groups is 50 distributed across the groups as specified a one way analysis of variance will have 94 82 power to detect at the 0 05 level a difference in means characterized by a Variance of means V Sri ui um Sri of 11 76 assuming that the common standard deviation is 6 In this example we can also perform sensitivity analysis to see how volatile this study is to slight changes in a particular parameter For example let us examine how the attainable power alters under slight changes in Standard Deviation 1 Firstly we much copy the inform
194. ng of SOFTWARE to achieve the LICENSEE S intended results or for particular applications 13 DISCLAIMER IN NO EVENT SHALL LICENSOR OR ITS SUPPLIERS BE LIABLE TO LICENSEE FOR ANY SPECIAL INDIRECT INCIDENTAL OR CONSEQUENTIAL DAMAGES IN ANY WAY RELATING TO THE USE OR ARISING OUT OT THE USE OF SOFTWARE EVEN IF LICENSOR HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES LICENSOR S LIABILITY SHALL IN NO EVENT EXCEED THE TOTAL AMOUNT OF THE PURCHASE PRICE LICENSEE FEE ACTUALLY PAID BY THE LICENSEE FOR THE USE OF SOFTWARE Melim Acknowledgements We would like to sincerely thank all those who made the production of Statistical Solutions software package nQuery nTerim 3 0 possible The Statistical Solutions Team Andrew Grannell Ronan Fitzpatrick Denis Moore Steven Keady Brendan Nyhan Diana Scriven Mark Donnelly Caroline Costello Helen Murphy ine Dunleavy Shane Thornhill Kevin Sievewright Mary Byrne Special Thanks to Eoghan Murphy Brian Sullivan and Niall Fitzgerald Contents Chapter DL gsc scevrcodcctvesesipesveteceoty csvayesestdevaneensssciaateseanauceasuegsGaunancecs suey E AET E Er Aaaa Ea 1 Systems GUJE rasse nanesi ER A E A ERE E E AE A steeds 1 1 1 System Requirements cccccccccssssssseececececeaneessesceceeeceseaaessseeeeeesessasesseseeeesessaueaaeseeeess 2 1 2 ValidatiO Mienie nesir AEE T E AE Ei 2 13 0 0 811 p a E A E E E E E ee eve tendo ys 2 Chapter 2 os ssszissienecssss cevvessdendexesscssvesiioree
195. ns Matrix Group Sizes ij Covariance Matrix ui Specify Multiple Factors ui Output Figure 4 6 6 Completed Means Matrix and Group Sizes Assistants Table 14 The bottom row is summed to give the total sample size required and automatically entered into the main design table 131 132 Melim 15 The next step in this MANOVA process is to specify the Covariance Matrix 16 The user has the option to enter the standard deviation and correlation thus nTerim will automatically calculate the Covariance Matrix but in this example we do not have this information so we will enter the Covariance Matrix directly 17 Enter the Covariance Matrix as given below 3 2 z 3 18 The Covariance Matrix entered in nTerim is shown in Figure 4 6 7 below Covariance Matrix x u Factor Level Table i Means Matrix Group Sizes u Covariance Matrix u Specify Multiple Factors a Output Figure 4 6 7 Covariance Matrix Assistant Table 19 Now we can entered all the information required to calculate attainable the Power given a specified sample size 20 The final step is to select which method we want to use In this case we want to use the Pillai Bartlett Trace approach 21 In order to do this simply select the Calculate power using Pillai Bartlett trace and the click on Run as shown in Figure 4 6 8 below Calculate group size using Wilks lambda Calculate group size using Pillai Bartlett trace
196. ns is to use lower error values for the earlier looks By doing this it means that the results of any analysis will only be considered significant in an early stage if it gives an extreme result Boundaries The boundaries in nTerim 3 0 represent the critical values at each look These boundaries are constructed using the alpha and beta spending functions Users in nTerim 3 0 are given the option to generate boundaries for early rejection of the null hypothesis Hy using the alpha spending function or to generate boundaries for early rejection of either the null or alternative hypothesis Hj or H using a combination of both the alpha and beta spending functions The notion of using an alpha spending function approach to generate stopping boundaries for early rejection of Ho was first proposed by Lan and DeMets 1983 we refer to such boundaries in nTerim 3 0 as efficacy boundaries Building on the work of Lan and DeMets Pampallona Tsiatis and Kim 1995 2001 later put forward the concept of using a beta spending approach to construct boundaries for early rejection of H4 we refer to these boundaries in nTerim as futility boundaries Essentially if a test statistic crosses an efficacy boundary then it can be concluded that the experimental treatment shows a statistically significant effect the trial can be stopped with rejection of the null hypothesis If the test statistic crosses a futility boundary then this indicates with high probability
197. nts in depression life satisfaction and physical health are the response variables and our null hypothesis is that a subject s treatment has not effect on any of the three different ratings As there are three response variables MANOVA is used to this hypothesis 123 124 Im 4 6 2 Methodology Power and sample size is calculated using central and non central F distributions and follows the procedures outlined by Muller and Barton 1989 and Muller LaVange Ramey and Ramey 1992 To calculate power and sample size the user must first enter the number of response variables p The user must then specify the number of levels categories per factor in their design using the Factor Level Table assistant Note if you wish to not use a factor in your design then you can simply leave the number of levels blank for that factor Using this same table the alpha value and desired power per factor and per factor interaction must also be specified Note if you are solving for power then you must leave the power fields blank Having specified the number of response variables and the number of levels per factor the Means Matrix M becomes populated with empty cells that must be filled in by the user The numbered rows of this matrix represent the response variables p and the columns represent the factors or to be more specific the number of groups that a subject can be classified into q Where q Levels Levels Levelsc For example if
198. o n2 n1 1 1 is true the probability of a Type error Power m Cost per sample unit Input Advice T ne cas Enter 0 05 a frequent standard eet 5 a 5 Entry Options Information times Equally Spaced z Equally Spaced z Equally Spaced 0 001 to 0 20 Max Times a 1 1 x Determine bounds Spending Function z Spending Function x Spending Function i Help uf Notes Spending function O Brien Fleming r O Brien Fleming ix O Brien Fleming Boundaries Graph x Calculate required sample sizes for given power X A Looks 0 2 0 4 0 6 0 8 1 Lower bound Upper bound Nominal alpha Incremental alpha Cumulative alpha a Looks a Specify Multiple Factors u Output Figure 2 4 5 Example of an Interim Design Window If multiple columns have been specified by the user there is an option to run the calculation for all the columns This is achieved by simply ticking the All columns box beside the Run button before clicking Run This will tell nTerim to concurrently run the calculations for all columns Then by simply clicking on a column the output statement will be presented as well as the boundary graph for each column in the bottom right hand corner of the interface 2 5 Selecting an nQuery Advisor Design Table through nTerim A new feature added to nTerim 3 0 is the ability to open an nQuery design table through nTerim T
199. o calculate the number of clusters the sample size per cluster the test significance levels the standard deviation or the smallest detectable difference Instead a search algorithm is used The search algorithm calculates power at various values for the relevant parameter until the desired power is reached 213 214 Im 5 1 3 Examples Example 1 Validation example calculating required sample size for a given power The following examples are taken from Donner and Klar 1996 where a power calculation problem is conducted and then a sensitivity analysis is conducted to show the effect of changing the sample size per cluster and difference between means The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 5 1 1 below E moo oen E Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval Survival gt Two Equivalence Agreement Regression Cluster Randomized CRT Two Means Completely Randomized CRT Two Proportions Inequality Completely Randomized CRT Two Proportions Equivalence CRT Two Proportions Non Inferiority CRT Two Proportions Superiority Figure 5 1 1 Study Goal and Design Window 2 In order to
200. o model count data in clinical trials This technique allows for the modelling of heterogeneity in count data has greater flexibility in modelling the mean variance relationship over the Poisson model and is expected to be less biased by non missing at random data This flexibility and robustness has contributed to its increased use in the literature This table facilitates the calculation of the power and sample size for the difference between two negative binomial rates Power and sample size is computed using the method outlined by Zhu and Lakkis 2014 4 9 2 Methodology Let yj equal the number of events in time for subject i in group j treatment and control placebo groups Assuming y follows a negative binomial distribution with mean Lij and dispersion parameter k the mean for a negative binomial regression can be modelled as log uij log tij Bo Bi Xi 4 9 1 where x is O for subjects in Group 0 control placebo and equal to 1 for subjects in Group 1 treatment Let ro and r be the mean rate of events per unit time in Group O and 1 respectively Then r ef r ef0tF and thus efi 0 The sample size calculation for two negative binomial rates is taken from Zhu and Lakkis 2014 This table can be used to calculate the power the sample size and the rate ratio given all other terms in the table are specified Calculations use a standard normal approximation as 4 follows the normal distribution 1
201. od The following steps outline the procedure for Example 2 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear fer PAI Ve ve x Study Goal And Design S Design Goal No of Groups Analysis Method Fixed Term Means D One Test 5 Interim Proportions Two Confidence Interval 5 Survival 0 Equivalence O Agreement Regression One way analysis of variance One way analysis of variance Unequal n s H Single one way contrast Single one way contrast Unequal n s Two way analysis of variance Multivariate analysis of variance MANOVA Analysis of Covariance ANCOVA Cancel Figure 4 6 10 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear This window is illustrated in Figure 4 6 11 3 There are several tables required for this test including the main test table shown in Figure 4 6 11 the Factor Level table illustrated in Figure 4 6 4 the Means Matrix assistant table presented in Figure 4 6 5 and the Covariance Matrix assistant table shown in Figure 4 6 7 4 To begin we first need to specify the number of response variables to be used in the study In this example we are using 3 so enter 3 in the Number of response variables p
202. od outlined by Guenther 1977 4 7 2 Methodology The test for hypotheses regarding 2 the Poisson mean are taken from Ostle 1988 It is conducted in two steps 1 Calculate the critical value X such that the probability of rejecting Hp is equal to alpha by finding the minimum value of X which fulfils the relevant inequality x X crit n o a lt 1 gt emna hoy Ho Ao A4 Ha rg lt A4 f a ie 4 7 1 A x a gt gt eo aa Ho Ao lt Ay Ha Ao gt Ai 2 Select n items and sum the total number of events If the total number of events is greater than Xerit for the first equation or less than Xerit for the second equation the null hypothesis is rejected The sample size calculation is taken from Guenther 1977 This table can be used to calculate the power the sample size or the minimum detectable Poisson mean given all other terms in the table are specified To calculate power and sample size the user must specify the test significance level the Poisson mean under the null hypothesis Ag and the Poisson mean under the alternative hypothesis A The sample size is calculated by using one of the following equations Tane lt n lt Paa Haz Ap Heda lt Ay f Jag Learn USL 472 1 a ay ZN ZG Moi Ao S Aa Hai Ao gt As where Y2q p is the inverse of the cumulative distribution of the chi squared distribution evaluated at 2d degrees of freedom and probability p In this for
203. of Time points Looks Calculate Sample Sizes for a given Power Using the number of time points K number of sides type of spending function the hypothesis to be rejected the type 1 error a and the power 1 the drift parameter can be obtained using the algorithms by Reboussin et al 1992 and Jennison amp Turnbull 2000 The test statistic is defined as Hy H2 of E ox 3 1 1 N Nz The user supplies the means 4 U2 and either R N or Nz Since R Ne it follows that a 1 value of R 1 indicates equal sample sizes The approach to solving this problem is dependent on what information the user supplies Given any two of R N or Nz the unknown is obtained by solving Equation 3 1 1 Calculate Attainable Power with the given Sample Sizes Given a N4 group means 44 U2 group standard deviations 64 02 R or N3 time points and type of spending function The requirement is to obtain the power The steps are e Obtain 0 by solving Equation 3 1 1 given that N R My M2 0 and oz are known e Obtain power using the algorithm by Reboussin et al 1992 and Jennison amp Turnbull 2000 Calculate Means given all other information Given a N4 group standard deviations 01 02 R or N2 power 1 time points and type of spending function The requirement is to obtain either 44 or u given the other The steps are e Obtain by solving Equation 3 1 1 given that N R 4 H2 0
204. ollowing calculations K M IF eee ae Aaa 5 3 4 K M IF a a 1 t 5 3 5 b 1 t p tpz A e 2 5 3 6 c A A 2p t 1 p tpz 5 3 7 d p A 1 A 5 3 8 b bc d Sf ho Ea 5 3 9 a r z l 2 u sign v gt 5 3 10 v poe La eae 3 Prm1 2ucos w b 3a 5 3 12 Prmz Prmi A 5 3 13 233 234 For equivalence trials it is necessary to calculate the constrained maximum likelihood estimators for both equivalence margins separately if their absolute values are different 2 Unpooled Test Statistic This test statistic uses the estimated group proportions to calculate the standard error Its formula is as follows Pil p IF p2 1 p2 IFz 5 3 14 K M KM3 OUnpool where p and p are the estimated mean proportions for the two groups 3 Pooled Test Statistic The pooled test statistic uses the weighted average of the two proportions to calculate the standard error Its formula is as follows n 1 DIF n 1 p IF a pU piF PA pF 5 3 15 K M KM2 where K M KoM j 1M1P 2M2P2 5 3 16 K M K M A closed form equation is not used to calculate the other parameters Instead a search algorithm is used The search algorithm calculates power at various values for the relevant parameter until the desired power is reached 5 3 3 Examples Example 1 Validation example calculating required number of clusters for a
205. olumns 3 and 4 that if we fix the Group 2 proportion at the maximum value of 0 51 and increase the Group 1 proportion the sample size also increases With this approach we are able to quantify how the sample size is affected by changes in both Group 1 and 2 proportions 14 Another feature that enables us to compare designs side by side is by using the Power vs Sample Size plot Multiple columns can be plotted together by simply highlighting the desired columns and clicking on the Plot Power vs Sample Size button on the menu bar 15 To highlight the desired columns click on the column title for Column 1 and drag across to Column 4 16 Then click on the Plot Power vs Sample Size button on the menu bar The multiple column plot is displayed in Figure 4 3 13 Ponc vs Sepie Size a Power vs Sample Size 90 2090 4090 6090 Column 1 Column 2 Column 3 Column 4 Power 4 90 1090 2090 3090 4090 5090 6090 Power 82 68 Sample Size N1 N2 Sample Size 3144 Figure 4 3 13 Power vs Sample Size Plot It can be seen from the legend on the left hand side legend can be altered manually that the blue line represents Column 1 the orange line represents Column 2 the red line represents Column 3 and the navy line represents Column 4 The cross on the graph illustrates how the user can identify what the sample size is for a corresponding power value for each column In the bot
206. ommon Standard Deviation to 4 and click Run This will update Column 2 to its new attainable value for power as seen in Figure 4 4 9 Plot Tools Window Help View Assistants Edit File New Fixed Term Test New Interim Test Plot Power vs Sample Size 10 05 Number of groups G 3 3 Variance of means V 111 76 11 76 Common standard deviation o ais 4 Effect size A V o2 l 0 32667 0 735 Power 194 82 99 98 N as multiple of n1 Jri Sni ni las 25 Calculate attainable power with the given sample sizes X Run E All columns Figure 4 4 9 Re run calculations to update Column 2 4 Repeat Steps 2 amp 3 except paste the contents of Column 1 into Column 3 change the Common Standard Deviation to 8 and click Run This is displayed in Figure 4 4 10 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size 0 05 10 05 Number of groups G 3 3 3 Variance of means V 11 76 11 76 11 76 Common standard deviation o l6 4 8 Effect size A V o2 0 32667 0 735 0 18375 Power 194 82 99 98 75 13 N as multiple of n1 Sri Jni n1 las 2 5 25 Total sample size N 50 50 50 Cost per sample unit 85 85 85 Total study Calculate attainable power with the given sample sizes X Run E All columns Figure 4 4 10 Re run calculations for Column 3
207. ompleted multiple design ANCOVA Table 121 122 im As the results show in Figure 4 5 14 as the R Squared value is increase from 0 5 up to 0 8 the corresponding power also increase dramatically almost doubling from 42 91 to 83 02 It can be seen from this approach that we would want an R Squared value approximately equal to 0 8 to obtain a credible value for power 19 Another feature that enables us to compare designs side by side is by using the Power vs Sample Size plot Multiple columns can be plotted together by simply highlighting the desired columns and clicking on the Plot Power vs Sample Size button on the menu bar 20 To highlight the desired columns click on the column title for Column 1 and drag across to Column 4 Then click on the Plot Power vs Sample Size button on the menu bar The multiple column plot is displayed in Figure 4 5 15 Power vs Sample Size leal Power vs Sample Size Column 1 Column 2 Column 3 Column 4 100 150 200 250 300 350 400 450 500 550 600 Total sample size N Figure 4 5 15 Power vs Sample Size Plot It can be seen from the legend on the left hand side legend can be altered manually that the blue line represents Column 1 the orange line represents Column 2 and the red line represents Column 3 The cross on the graph illustrates how the user can identify what the sample size is for a corresponding power value for each column In th
208. on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear 3 Study Goal And Design a Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval Survival gt Two Equivalence O Agreement Regression Two sample t test Student s t test equal variances Satterwaithe s t test unequal variances Two group t test for fold change assuming log normal distribution Two group t test of equal fold change with fold change threshold Wilcoxon Mann Whitney rank sum test continuous outcome Wilcoxon Mann Whitney rank sum test ordered categories Two group univariate repeated measures ANOVA Greenhouse Geisser correction 2x2 Crossover Design i Repeated Measures for two means cancer Figure 4 2 7 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear This test table is illustrated in Figure 4 2 8 3 Enter 0 05 for alpha the desired significance level and enter 4 for the number of levels M as shown in Figure 4 2 9 4 Two sided test is the default setting in nTerim as well as a Ratio value of 1 for the group sizes 5 In this example we will examine a study where the difference in means is 15 and the standard dev
209. oportional These results assume that 5 sequential tests are made and the Pocock spending function is used to determine the test boundaries Metin Chapter 4 Fixed Term Design 58 im 4 1 One Way Repeated Measures Contrast Constant Correlation 4 1 1 Introduction This table facilitates the calculation of power and sample size for a one way repeated measures contrast design Calculations are performed using the methods outlined by Overall and Doyle 1994 A one way repeated measures contrast is used to analyse specific planned contrasts in a repeated measures one way analysis of variance ANOVA design This is an experimental design in which multiple measurements are taken on a group of subjects over time or under different conditions This design is the same as the one way ANOVA but for related not independent groups It can be viewed as an extension of the dependent t test To give an example of such a design consider a study of a three month intervention aimed at raising self esteem in children Self esteem will be measured before after one month after two months and after three months of the intervention It is assumed that self esteem will increase monotonically over time Thus for this study it may be of interest to test for a linear trend in self esteem The contrasts 3 1 1 3 would be appropriate for such a study Such planned contrasts are useful because they provide a more sharply focused analysis compared
210. or Power in column 1 Then copy the values in column 1 across to columns 2 3 and 4 Then change the value for Response Rate Ratio in columns 2 to 4 from 0 5 to 0 9 1 3 and 2 respectively This will give a table as per Figure 4 12 11 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test x Plot Power vs Sample Size Ww 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 1 BE z z gt Baseline Response Rate eB0 1 1 1 7 Response Rate Ratio eB1 e 0 0 5 0 9 13 2 Mean Exposure Time pT 1 1 1 1 Overdispersion Parameter 1 1 1 1 Distribution of X1 sidetable required Normal Normal Normal Normal z Variance of b1 Null Hypothesis Variance of b1 Alternative Hypothesis R squared X1 and independent variables 0 0 0 0 Sample Size N C 95 95 Cost per sample Total study cost OE e gt Calculate required sample size for given power X Run 7 All columns Figure 4 12 11 Sensitivity analysis around the Response Rate Ratio 9 Using the Normal Distribution side table as before recalculate the Variance of b1 Null Hypothesis and Variance of b1 Alternative Hypothesis variables for each column with a mean of zero and a standard deviation of one This will give a table as per Figure 4 12 12 nQuer nTe 3 File Edit View Assistants Plot Tools Window Help Ne
211. or 2 sided test 1 zli z 1 z z Baseline Response Rate e 0 1 1 i 1 Response Rate Ratio eB1 eB0 0 5 0 9 13 2 Mean Exposure Time pT 1 1 1 1 Overdispersion Parameter 1 1 1 1 Distribution of X1 sidetable required Normal z Normal z Normal z Normal B Variance of b1 Null Hypothesis 1 1 1 1 Variance of b1 Alternative Hypothesis 0 786 0 994 0 966 0 786 R squared X1 and independent variables 0 0 0 0 Sample Size N 21 973 155 21 Power 95 792 95 014 95 05 95 792 Total study cost i gt Calculate required sample size for given power Run V All columns Figure 4 12 13 Results from Sensitivity analysis 193 im The effect of changing the response rate ratio on the relationship between Power and Sample size can be explored further using the Plot Power vs Sample Size button at the top of the screen 11 Select the first column by clicking the 1 at the top of column 1 Then hold down the Shift key and click the 4 at the top of column 4 All four columns will now be highlighted 12 Click the Plot Power vs Sample Size button at the top of the screen This will give you a plot as displayed in Figure 4 12 14 which will show the relationship between power and sample size for each column Right click to add features such as a legend to the graph and double click elements for user options and editing Power vs Sample S i Power vs Sample Size 0 200 400 600 109 eee eee
212. or Clinical Trials Biometrika 35 pp 549 556 O Brien R G Muller K E 1993 Unified Power Analysis for t tests through Multivariate Hypotheses Edwards L K Ed Applied Analysis of Variance in Behavioral Science Marcel Dekker pp 297 344 Ostle B Malone L 1988 Statistics in Research Basic Concepts and Techniques for Research Workers Fourth Edition lowa State Press Ames lowa Overall J E Doyle S R 1994 Estimating Sample Sizes for Repeated Measures Designs Controlled Clinical Trials 15 pp 100 123 Pampallona S Tsiatis A A and Kim K 1995 Spending functions for type and type II error probabilities of group sequential trials Technical report Dept of Biostatistics Harvard School of Public Health Boston Pampallona S Tsiatis A A and Kim K 2001 Interim monitoring of group sequential trials using spending functions for the type and type II error probabilities Drug Information Journal 35 pp 1113 1121 Pocock S J 1977 Group Sequential Methods in the Design and Analysis of Clinical Trials Biometrika 64 pp 191 199 Reboussin D M DeMets D L Kim K and Lan K K G 1992 Programs for Computing Group Sequential Boundaries using the Lan DeMets Method Technical Report 60 Department of Biostatistics University of Winconsin Madison Signorini D F 1991 Sample size for Poisson regression Biometrika 78 2 pp 446 450 S
213. or the results given by the current calculation as per Figure 4 10 8 below Output OUTPUT STATEMENT A comparison of two incidence rates in terms of person years in a fully randomised trial with a sample of 354 person years per group would achieve 80 power at the 0 05 significance level to detect a difference in incidence rates of at least 0 2 if the control group incidence rate was 1 ag Specify Multiple Factors te Output Figure 4 10 8 Study design Output statement This Output statement can then be easily transferred directly from the output window into a report 174 4 11 Cox Regression 4 11 1 Introduction The Cox Proportional Hazards Regression model is a semi parametric method used to measure the effect on the hazard ratio of independent variable s on a dependent variable in survival analysis For example taking a new therapy may halve the hazard rate for a cancer occurring This model can be used to calculate the effect of a variable on the hazard rate without needing to specify the underlying hazard function while also allowing censoring This flexibility has made the Cox Proportional Hazards model a widely used tool in survival analysis This table facilitates the calculation of the power and sample size for an independent variable both in a single variable model and given its relationship with other variables in a multivariable model Power and sample size is computed using the method outlined by Hsie
214. ore enter 90 in the Cost per sample unit row in order to calculate the Total study cost associated with the sample size 9 Then select Calculate required sample size for given power from the drop down menu below the main table and click Run This is displayed in Figure 4 2 3 above By clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation When the sample size is 53 in group 1 and 53 in group 2 a test for the time averaged difference between two means in a repeated measures design with a 0 05 significance level will have 90 power to detect a difference in means of 10 in a design with 4 repeated measurements when the standard deviation is 20 and the between level correlation is 0 5 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test ue Plot Power vs Sample Size _ RM Two Means 1 i a a 2 Test significance level a 0 05 0 025 1or2sided test 2 x 2 x 2 x 2 z Number of levels M 4 4 Difference in means pi p2 10 10 i Standard deviation at each level o 20 20 Between level correlation p 0 5 0 5 Group 1 size ni 53 63 Group 2 size n2 53 63 Ratio n2 ni 1 1 1 1 Power 90 30 Cost per sample unit 90 30 Total study cost 9540 11340 Plah gt Calculate required sample sizes for given power E All columns Figure 4 2 4 Re run calculations to update Column 2 10 Now we are going
215. ore information you can provide us with the better If it is a technical question about one of our test tables screen shots of the completed tables of issues you are having are very helpful In order to address any installation issues or technical questions relating to the users machines the more information provided about the type of machine in question can speed up the process by a great deal Screen shots of installation issues are very helpful to us in solving any issue you may have Metin Chapter 3 Group Sequential Interim Design 20 mM 3 1 Two Means 3 1 1 Introduction nTerim 3 0 is designed for the calculation of Power and Sample Size for both Fixed Period and Group Sequential design In relation to Group Sequential designs calculations are performed using the Lan DeMets alpha spending function approach DeMets amp Lan 1984 DeMets amp Lan 1994 for estimating boundary values Using this approach boundary values can be estimated for O Brien Fleming O Brien amp Fleming 1979 Pocock Pocock 1977 Hwang Shih DeCani Hwang Shih amp DeCani 1990 and the Power family of spending functions Calculations follow the approach of Reboussin et al 1992 and Jennison amp Turnbull 2000 Calculations can be performed for studies that involve comparisons of means comparisons of proportions and survival studies as well as early stopping for Futility Group Sequential Designs Group Sequential designs differ fro
216. ortion row 90 File Edit View Assistants Plot Tools Window Help lt New Fixed Term Test New Interim Test Z Plot Power vs Sample Size RM Two Proportions 1 1 or 2 sided test Number of levels M Between level correlation p Group 1 proportion p1 Group 2 proportion p2 Odds ratio W p2 1 p1 p1 1 p2 Calculate required sample sizes for given power X All columns Figure 4 3 2 Repeated Measures for Two Proportions Test Table 6 We also know that the between level correlation is 0 5 so enter 0 5 into the Between level correlation row 7 We want to calculate the required sample size for each group in order to obtain 90 power To do this enter 90 in the Power row nQuery nTerim 2 File Edit View Assistants Plot Tools Window Help c New Fixed Term Test New Interim Test W Plot Power vs Sample Size RM Two Proportions 1 i P 2 x Test significance level a 0 05 2 el el Sh E Number of levels M 3 Between level correlation p 10 5 Group 1 proportion p1 10 45 Group 2 proportion p2 0 55 Odds ratio W p2 1 p1 p1 1 p2 1 49383 Group 1 size ni 1 Group 2 size n2 7 Ratio n2 ni 1 i 7 Power 90 Cost per sample unit 120 wt z Calculate required sample sizes for given power Run 0 AN columns Figure 4 3 3 Design Entry for
217. ost per sample unit 100 FillTable Clear Table wi Specify Multiple Factors Output Figure 4 3 10 Completed Specify Multiple Factors Table 10 Once all the parameter values and ranges have been entered correctly click on Fill Table at the bottom right side of the Specify Multiple Factors table 11 This will automatically fill in the required amount of columns in the test table as illustrated in Figure 4 3 11 In this example we require four columns File Edit Plot View Assistants Tools Window Help _ New Fixed Term Test New Interim Test Z Plot Power vs Sample Size RM Two Proportions 1 0 05 0 05 0 05 0 05 lor 2 sided test 2 x2 x 2 x2 z Number of levels M 3 3 3 3 Between level correlation p 0 4 0 4 0 4 04 Group 1 proportion p1 0 45 0 55 0 45 0 55 Group 2 proportion p2 10 39 0 39 0 51 0 51 Odds ratio p2 1 p1 p1 1 p2 0 78142 0 5231 1 27211 0 85158 Group 1 size ni Group 2 size n2 Ratio n2 n1 li 1 1 1 Power g0 90 90 90 Cost per sample unit 100 100 100 100 Total study cost Calculate required sample sizes for given power x Run All columns Figure 4 3 11 Design Entry for Multiple columns 97 98 Melim 12 It can be seen from Figure 4 3 11 that different designs have been created for each combination of the proportions for both groups 13 In order to calculate appropri
218. otal st COS Calculate required sample sizes for given power v E All columns 0 05 40 80 0 5 85 75 e Figure 4 2 17 Altered columns for comparison 14 As we want to run this calculation for multiple columns tick the All Columns box beside the Run button as shown in Figure 4 2 17 then click Run File Edit View Assistants Plot Tools Window Help gt New Fixed Term Test New Interim Test Z Plot Power vs Sample Size RM Two Means 1 eee z Test significance level a 0 05 0 05 0 05 0 05 lor2sidedtest mm 2 2 2 ix Number of levels M 5 5 5 5 Difference in means pi p2 40 40 40 40 _ Standard deviation at each level o 80 80 80 80 Between level correlation p 0 5 0 5 0 5 0 5 Group 1 size n1 44 33 29 27 Group 2 size n2 4 66 87 108 Ratio N2 N1 1 2 3 4 Power 85 85 85 85 Cost per sample unit 75 75 75 75 Total study cost 6600 7425 8700 10125 a D Calculate required sample sizes for given power v V All columns Figure 4 2 18 Completed multiple design Repeated Measures for Two Means Table 86 15 Another feature that enables us to compare designs side by side is by using the Power vs Sample Size plot Multiple columns can be plotted together by simply highlighting the desired columns and clicking on the Plot Power vs Sample Size button on the menu bar 16 To highlight the desired columns click on the column ti
219. oup Means 01 02 Group Standard Deviations Ni N3 Group Sample Sizes R Ratio of N4 to N 0 Drift Parameter K Number of Time points Looks q Spending Function O Brien Fleming Pocock etc Calculate Sample Sizes for a given Power Using the number of time points K number of sides type of spending function the hypothesis to be rejected the type 1 error a and power 1 the drift parameter can be obtained using algorithms by Reboussin et al 1992 and Jennison amp Turnbull 2000 The test statistic is defined as lpi Pol pam PD 3 2 1 Ny N NypitN where p me The user supplies the proportions p4 p2 and either R N or N3 1 2 N2 ary f Since R it follows that a value of R 1 indicates equal sample sizes and that 1 R HEP The approach to solving this problem is dependent on what information the 1 R user supplies For the case of continuity correction the formula can be written as m ar e P P2 ZN R 8 3 2 2 pil p R 1 NR as per Fleiss 1981 The validity of this formula relies on the assumption of minimum expected cell count being above a pre specified threshold As a rule of thumb the normal approximation to the binomial will hold if the following conditions are met PiN gt T 1 py N gt T P2N2 gt T 1 p2 N2 gt T where T is a predefined threshold 3 2 3 User supplies R only The requirem
220. ove equations Vp and V are defined as follows v 00 4 9 4 Oe fe Fy OF 8 gt E 4 9 5 1 m n Or 8 E where and 7 are the rates under the null hypothesis and r and r are the assumed true rates specified in the table i e their values under the alternative hypothesis The values of fo and and by extension Vo can be specified in three ways These are selected using Rates Variance option in the table These are detailed below 1 Reference Rate Under Ho r Thus can set 7 7 ro which gives the following 1 6 1 0 k Vocrr Er z 4 9 6 2 True Rates The values of 7 and fare set to their true values under the alternative hypothesis This gives the following 1 1 r 1 0 k 4 9 7 V OCR Ut Or 0 3 Maximum Likelihood Maximising the log likelihood function under the null hypothesis restriction yields the following maximum likelihood estimate of the overall event rate Nouto MMM _ To Ory i _ _ 4 9 8 NoMe Nikt 1 6 Setting 7 7 F gives the following variance formulation 1 6 1 0 k Vom gas t gt 4 9 9 Our To 0T 0 160 In Zhu and Lakkis 2014 simulation indicated the true rates and maximum likelihood methods gave the best estimates of the sample size for Wald and likelihood ratio tests for p4 A closed form equation is not used to calculate the rate ratio Instead a search algorithm is used This search algo
221. ow File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size LL Open Manual Statistical Solutions Support Figure 2 4 1 New Design Tabs Using either of the steps outlined above the user will then be presented with the Study Goal and Design window as shown in Figure 2 4 2 below In relation to selecting the term of their designs the user must select either Fixed or Interim The user will then be presented with a list of options to the type of design they require yn a Design Goal No of Groups Analysis Method Fixed Term Means Test D Interim Proportions Confidence Interval Equivalence Agreement Regression Cluster Randomized One way analysis of variance One way analysis of variance Unequal n s Single one way contrast Single one way contrast Unequal n s Two way analysis of variance Multivariate analysis of variance MANOVA Analysis of Covariance ANCOVA Figure 2 4 2 Open New Fixed Term Design The options for Fixed term designs are presented in Figure 2 4 2 For example If you want to choose the Analysis of Covariance ANCOVA table you must first select Means as the Goal gt Two as the No of Groups and Test as the Analysis Method You can then select Analysis of Covariance ANCOVA from the list of tests Once you click OK the design table will be launch
222. ple size for given power from the drop down menu and click Run E EEJ El EJE Run E All columns 25 View Assistants Calculate required sample sizes for given power v Run I File Edit Plot Tools Window Help New Fixed Term Test New Interim Test x Plot Power vs Sample Size LLJ Open Manual Statistical Solutions Support GST Two Means 1 xX _ y 1 2 3 4 Test significance level a 0 05 1 or 2 sided test 2 z zli z z Group 1 mean e1 180 Group 2 mean p2 200 Difference in means p1 p2 E Group 1 standard deviation o1 30 Group 2 standard deviation o2 30 Effect size d 0 667 Group 1 size n1 ja Group 2 size n2 49 Ratio N2 N1 Ja 1 1 1 Power 190 36 Cost per sample unit 250 Total study cost 24500 Number of looks s 5 5 5 Information times P Equally Spaced z Equally Spaced z Equally Spaced z Equally Spaced x Max Times j 1 1 1 Determine bounds Spending Function Spending Function Spending Function Spending Function w Spending function O Brien Fleming a a en B a Truncate bounds No z No x No x No x Truncate at Futility boundaries p z Don t Calculate z Don t Calculate z Don t Calculate z Spending function O Brien Fleming x OBrien Fleming v O Brien Fleming x Phi _ ____ _ _ CE gt All columns Figure
223. r 2 sided test variable dropdown menu select Unpooled from the Test Type option and enter a control group proportion of 0 06 4 Next select Proportions from the Solve Using row enter 0 06 and 0 04 for the value of the test statistic under HO and H1 respectively enter 0 01 for the intracluster correlation enter 100 for both cluster sample size variables and enter 80 for Power 5 Finally enter K1 in the Clusters in Control Group K2 row This will solve so that K1 and K2 must be equal Other ratios between K1 and K2 can be calculated by using arguments for K2 such as 2 K1 or K1 2 to have K2 be twice as large as K1 for example The table will appear as in Figure 5 2 3 224 nQuery nTerim File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Z Plot Power vs Sample Size LU Open Two Proportions Inequality Co F b 2 3 4 Test significance level a 0 05 1or 2 sided test 2 2 2 2 z Test Type Unpooled x Likelihood score z Likelihood score x Likelihood score z Control Group Proportion p2 0 06 Solve using p1 p2 p1 or p1 p2 Proportions iz Differences z Differences z Differences z Test Statistic under HO 0 06 Test Statistic under H1 0 04 Intracluster Correlation ICC 0 01 Clusters in Treatment Group K1 Clusters in Control Group K2 K1 Cluster Sample Size in Treatment Group M1
224. r columns Columns 2 to 5 to see how the sample size is affected by various parameter changes File Window Plot Edit Assistants Tools View Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size ope RM Two Means 1 l A i 2 3 4 Jommm anns Test significance level a 0 05 0 05 0 05 0 05 0 05 1 or 2 sided test 2 x2 x2 x2 z 2 Number of levels M 4 4 4 4 4 Difference in means pi p2 15 15 15 15 15 Standard deviation at each leve o 25 25 25 25 25 Between level correlation p 04 0 4 04 07 0 2 Group 1 size n1 Group 2 size n2 Ratio n2 n1 ja 1 1 1 1 Power 90 85 80 90 90 Cost per sample unit Calculate required sample sizes for given power X Run E All columns Figure 4 2 11 Altered columns for comparison 81 Melim 82 12 Firstly we want to investigate how the sample size will be affected by a change in Power To do this we will enter 85 and 80 in the Power row for Columns 2 and 3 respectively as shown in Figure 4 2 11 13 We also would like to examine how the sample size is affected by an increase or decrease in the between level correlation Therefore we will change the between level correlation to 0 7 and 0 2 in Columns 4 and 5 respectively as shown in Figure 4 2 11 14 As we want to calculated the required sample size to obtain the given power select Calcul
225. r each column In the bottom right corner of the plot indicated the exact values for Power and Sample Size for each identifier on the graph It can be seen in Figure 4 4 11 that Column 2 reaches an acceptable power level much faster than the other two designs as it has the lowest value for Common Standard Deviation This plot also shows us how volatile this study design is to any change in Common Standard Deviation 109 110 4 5 Analysis of Covariance ANCOVA 4 5 1 Introduction This table facilitates the calculation of power and sample size for analysis of covariance ANCOVA designs Calculations are performed using the approximations outlined by Muller and Barton 1989 and Muller LaVange Ramey and Ramey 1992 An analysis of covariance ANCOVA design can be viewed as an extension of the one way analysis of variance ANOVA In ANOVA differences in means between two or more groups are tested on a single response variable An ANCOVA on the other hand does the same analysis while adjusting for covariates These covariates provide a way of statistically controlling the effect of variables one does not want to examine in a study It is assumed that the inclusion of these covariates will increase the statistical power of a design However it must be noted that adding a covariate also reduces the degrees of freedom Therefore adding a covariate that accounts for very little variance in the response variable may actually reduce pow
226. r group 19 The total sample size is also automatically calculated and given in the Total sample size N row 20 The final step is to select which method we want to use In this case we want to use the Wilks Lambda approach 21 In order to do this simply select the Calculate power using Wilks Lambda and the click on Run as shown in Figure 4 6 16 below Calculate group size using Wilks lambda Calculate group size o Hotelling Lav trace l g Calculate power using Pillai Bartlett trace _ Calculate power using Hotelling Lawley trace wij Factor Level Table ui Means Matrix Group Sizes u Covariance Matrix ui Specify Multiple Factors ui Output Figure 4 6 16 Selecting the Wilks Lambda option 22 In order to view the results for Power for each level the power values are displayed in the Factor Level Assistants table as illustrated below in Figure 4 6 17 Factor Level Table x Power 29 99738 29 99738 98 07328 100 66 77364 66 77364 100 ij Factor Level Table a Means Matrix Group Sizes u Covariance Matrix ui Specify Multiple Factors a Output Figure 4 6 17 Output Power values calculated 23 Finally the output statement can be obtained by clicking on the Output tab on the bottom of the nTerim window Output Statement A multivariate analysis of variance design with 3 factors and 3 response variables has 27 groups When the total sample size acros
227. r of 85 so enter 85 in the Power row File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test W Plot Power vs Sample Size RM Two Means 1 S l Fl el 2 Test significance level a 0 05 lor 2sided test 2 z 2 v2 Number of levels M i 5 Difference in means p1 p2 40 Standard deviation at each level o 80 Between level correlation p 0 5 Group 1 size n1 2 Fj Group 2 size n2 Ratio n2 ni if 1 1 Power 85 Cost per sample unit 75 ll Calculate required sample sizes for given power X Run o Figure 4 2 15 Design Entry for Two Means Repeated Measures Study 84 8 The cost per sample unit has been estimated as 75 in this particular study Therefore to calculate the overall cost associated with the sample size enter 75 in the Cost per sample unit row as shown in Figure 4 2 15 9 As we want to try several different parameter values for sample size Ratio R we can use the Fill Right function to fill out multiple columns with the same information entered in Column 1 JASEN File Edit View Assistants Plot Tools Window Help nTerim New Fixed Term Test New Interim Test Z Plot Power vs Sample Size J RM Two Means 1 i 2 3 4 Select All Copy 2 Cut Paste Fill Right Clear Table Clear Column Clear Selection Print Table qP m gt Calculate required sample sizes for given power X Run
228. r options and editing 217 Power vs Sample Size 10 Power 88 52 Sample Size 11 Figure 5 1 7 Power vs Number of Clusters plot 10 Finally by clicking on the desired study design column and going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 5 1 8 Output OUTPUT STATEMENT In a cluster randomised trial comparing two continuous variables a sample size of 3 clusters per group with 100 individuals per cluster achieves 89 322 power to detect a difference of 0 5 between the group means when the standard deviation is 1 and the intracluster correlation is 0 01 using a 2 Sided T test at the 0 05 significance level ug Specify Multiple Factors jug Output Figure 5 1 8 Study design Output statement This Output statement can then be easily transferred directly from the output window into a report 218 Melim 5 2 CRT Two Proportions Inequality Completely Randomized 5 2 1 Introduction Binary data is commonly studied in variety of different fields Clustered data is very common in a wide variety of academic social policy and economic studies This two sample test is used to test hypotheses about the difference between two proportions in a completely randomized cluster randomized trial This table facilitates the calculation of the power and sample size for hypothesis tests
229. rd Deviation of X1 variable enter 0 1837 for the R squared of X1 and other X s variable enter 1 for the Log Hazard Ratio enter 0 738 for the Overall Event Rate and finally enter 80 in the Power row The table will appear as per Figure 4 11 3 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Cox Regression 1 P 2 7 Test significance level a 0 05 1 or 2 sided test 1 y 2 z 2 2 Standard Deviation of X1 0 0 3126 R squared of X1 and other X s 0 1837 Log Hazard Ratio B 1 Overall Event Rate P 0 738 Sample Size N 80 Cost per sample unit Total study cost pP m gt Run E An columns Figure 4 11 3 Values entered for Cox Regression design 179 Melim 180 5 Select Calculate required sample size for given power from the dropdown menu beside the Run button then click Run This will give a result of 106 for the sample size as in Figure 4 11 4 The result presented in the paper by Hsieh and Lavori is 107 however this is due to them rounding the unadjusted for R sample size as an intermediate step The nQuery nTerim calculation treats the interim sample size as an unrounded figure File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size Cox Regression 1
230. re for Example 1 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear study Goal And Desi n E Design Goal No of Groups Analysis Method Fixed Term Means D One Test Proportions O Two Confidence Interval 5 Survival Equivalence Agreement Regression p One way analysis of variance One way analysis of variance Unequal n s Single one way contrast Single one way contrast Unequal n s Two way analysis of variance Multivariate analysis of variance MANOVA Analysis of Covariance ANCOVA OK Cancel Figure 4 6 1 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear This window is illustrated in Figure 4 6 2 3 There are several tables required for this test including the main test table shown in Figure 4 6 2 the Factor Level table illustrated in Figure 4 6 4 the Means Matrix assistant table presented in Figure 4 6 5 and the Covariance Matrix assistant table shown in Figure 4 6 7 128 4 To begin we first need to specify the number of response variables to be used in the study In this example we are using 2 so enter 2 in the Number of response variables p row as shown in Figure 4 6 3 File Edit View Assistants
231. red of 0 The sample size was adjusted for an anticipated event rate of 0 1 aJ Specify Multiple Factors Output Figure 4 11 8 Study design Output Statement 4 12 Poisson Regression 4 12 1 Introduction The Poisson Regression Model is a method used to analyse the relationship between a dependent variable which is a count and one or more independent variables Count data is common in clinical and epidemiological studies and thus the Poisson Regression Model is widely used in these areas This table facilitates the calculation of the power and sample size for an independent variable both in a single variable model and given its relationship with other variables in a multivariable model Power and sample size is computed using the method outlined by Signorini 1991 183 184 Im 4 12 2 Methodology The Poisson Regression Model assumes that the dependent variable Y follows the Poisson distribution The Poisson distribution models the probability of y events using the following formula ut 4 Y P Y y p t 4 12 1 where u is the mean number of events per unit time of the Poisson distribution and t is length of time of the study in the units of the mean In Poisson regression we assume the mean u is determined by k independent variables by the following relationship u exp Bo P1X1 B2X2 BeXx 4 12 2 where f f2 Bx are the regression coefficients of the independent varia
232. ression design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design Regression as the Goal One as the Number of Groups and Test as the Analysis Method Then click OK and the test window will appear as per Figure 4 13 2 202 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test I NewInterimTest Plot Power vs Sample Size l Probit Regression 1 Vi aaa 2a aa Baal 4 5 5 5 5 Number of Dose Levels Sum of Weights Slope of Probit Regression B1 Relative Potency p Sample Size per Group N Power Total Number of Subjects Cost per sample Total study cost Mam ra um E An cota Figure 4 13 2 Probit Regression Test Table At the bottom of the screen will be the Probit Regression Side Table This will appear as in Figure 4 13 3 below The number of columns in the Probit Regression Side Table is defined by the Number of Dose Levels variable in the main table where the default is set to 5 W Specify Multiple Factors Mil Probit Regression Side Table u Output Figure 4 13 3 Probit Regression Assistant Table The first calculation will be for Sample Size from Table 2 of Kodell et al 2010 for a five dose study design 3 First enter 0 05 for the Test Significance level row then enter 23 25 for Slope of Probit Regression and enter 1 1 for Relative Potency 4 Finally enter
233. review and Page Setup Select Save Image from this list to save the plot Power vs Sample Size Power vs Sample Size Save Image Print Print Preview Page Setup Disable Legend Disable Title Copy to Clipboard Column 1 Column 2 Column 3 50 60 70 80 90 100 110 120 Total sample size N Figure 2 7 5 Saving a plot A separate window will appear prompting the user to select the folder in which they would like to save the plot Once the user has chosen the folder to save the plot in they can select what format to save in The format options available to save a plot are in a JPEG or PNG format Once the location and format have been selected by the user simply click Save to save the plot This image can now be imported to many Microsoft applications such as MS Word for reporting or MS Powerpoint for presentation purposes 17 18 Im 2 8 Help and Support For issues pertaining to the methodology and calculations of each test in nTerim there is a brief outline of how each test is calculated in the Methodology section of each test chapter of the manual There are accompanying references for each test throughout the text and these can be located in the References section of the manual In the nTerim window there are two useful shortcuts that have been added to the tool bar The first shortcut is the Open Manual button which has been added to help the user find
234. riates c R Squared with covariates R Power Total sample size N Cost per sample unit Total study cost por e ra Calculate required sample sizes for given power X All columns Figure 4 5 2 Analysis of Covariance Test Table N as multiple of n1 Sri 5ni n1 wi Compute Effect Size Assistant ui Specify Multiple Factors w Output Figure 4 5 3 Compute Effect size Assistant Window 5 Once you enter a value for the number of groups G the Compute Effect Size Assistant table automatically updates as shown in Figure 4 5 4 6 In order to calculate a value for Effect Size the Variance of Means V needs to be calculated first 7 The mean for each level and the corresponding sample size need to be entered in the Compute Effect Size Assistant table 8 For the Mean values for each group enter 15 for group 1 20 for group 2 25 for group 3 and 18 for group 4 9 For the group sample size n values for each group enter 30 for group 1 45 for group 2 45 for group 3 and 30 for group 4 As a result the ratio 7 is calculated for each group as a proportion of group 1 nQuery nterim Z File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size ANCOVA 1 a 2 mi 3 4 Test significance level a 0 05 Number of groups G 4 Common standard deviation o Number of covariates c R Squared
235. rim Test Tools Window Help a Plot Power vs Sample Size LLI Open Manual Statistical Solutions Support GST Survival 1 1 or 2 sided test Group 1 proportion ni at time t Snop ee Hazard ratio h In n1 In n2 Survival time assumption _ Sy Power Number of events j Cost per sample unit Total study cost Number of Looks Equally Spaced O Brien Fleming Calculate required sample sizes for given power X Xx 5 5 z Equally Spaced 1 1 Spending Function x Spending Function x O Brien Fleming No x No Don t Calculate B Don t Calculate O Brien Fleming O Brien Fleming 5 5 z Equally Spaced 1 1 y O Brien Fleming x No No Don t Calculate O Brien Fleming Figure 3 3 2 Survival Test Table Exponential Survival Exponential Survival Exponential Survival y Exponential Survival z Equally Spaced z Spending Function x Spending Function x O Brien Fleming z lz z Dont Calculate x x x O Brien Fleming D _ All columns 8 Once all values have been entered select Calculate required sample size for given power from the drop down menu and click Run 49 50 Im nQuery nTerim 2 0 els meas File
236. rithm calculates power at various values of the rate ratio until the desired power is reached 161 Im 4 9 3 Examples Example 1 Validation example calculating required sample size for a given power The following example is taken from Table of Zhu and Lakkis 2014 where a sample size calculation problem is conducted to show the effect of changing the mean rate for the control group and the method for calculating the variance under the null hypothesis The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 4 9 1 below Study Goal And Design a eS Goal No of Groups Analysis Method Fixed Term Means One Test D Interim Proportions Two Confidence Interval Survival gt Two Equivalence Agreement Regression Cluster Randomized Two sample t test Wilcoxon Mann Whitney rank sum test continuous outcome Wilcoxon Mann Whitney rank sum test ordered categories Wilcoxon Mann Whitney rank sum test ordered categories Unequal n s Two group univariate repeated measures ANOVA Greenhouse Geisser correction 2x2 Crossover Design Repeated measures for two means Two Incidence Rates Two Poisson Means OK Cancel Figure 4 9 1 Study Goal and Design Window 2 In or
237. roportions Equivalence Completely Randomized ccccccccsssssseeeeees 231 53 1 Introducti ssiosctunen acannon rca a aaa tune aa aeaa aa a manele 231 53 2 Methodology sisseseade rE anaE a iaia aei 232 5 33 EXAM Ple Siessen asa a AE EE aa Aa reS Eea EENAA Ea EA 235 5 4 CRT Two Proportions Non Inferiority Completely Randomized cceccsceeeees 240 DAL Intr dUCtiO Nesse aaea a R Eaa EAE EAE EEKE EERE a Raa 240 5 4 2 Methodolo pV rrisoississsiuneneieniai nenns snieni iiaa nainii unnai laani 241 54 3 Exa Mple Sesen a AE EENE EN 244 5 5 CRT Two Proportions Superiority Completely Randomized c sececeesseeeeeeeteeees 250 SS LANtrodUCtiON sosiete g e amasaadebialacieanmdaaneangtieumacienuad 250 5 5 2 Methodology creisis siuusirreisnis taen iaeiae ennei o ioien binii aaie 251 Im Be SE ASS asese sc strirsecresininserantpiteutsredcntidesiiedeulbieddeinasatdricane a aaa EE N ENEE Chapter 6 References Melim Chapter 1 Systems Guide Im 1 1 System Requirements As with most software packages there are a set of requirements on the various aspects of the users machine in order to achieve full functionality For nTerim 3 0 the set of system requirements are listed in full below Operating System Windows 8 or later Windows 7 Windows Vista Windows Server 2012 R2 or later Windows Server 2012 Windows Server 2008 R2 or later Windows Server 2008 Windows Server 2003 Processor Either 32
238. roup as in Figure 4 13 6 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test New Interim Test Plot Power vs Sample Size Probit Regression 1 i Di Test significance level a Number of Dose Levels Sum of Weights Slope of Probit Regression B1 Relative Potency p Sample Size per Group N Total Number of Subjects Cost per sample Total study cost Figure 4 13 6 Completed Probit Regression study design 0 05 5 2 2013513108 23 25 11 11 90 5379299894 110 pom m p C Run E an columns The next calculation is a sensitivity analysis for sample size where we change the power and the relative potency and see its effect on sample size These values are taken from Table 2 of Kodell et al 2010 8 First enter 90 for Power in column 1 and delete the sample size values Then copy the values in column 1 across to columns 2 to 4 Since the variables defined by the side table are the same for these 4 columns the user can copy the previously calculated value rather than recalculating for each column separately Then change the value for Power to 80 in column 2 and 4 and change Relative Potency to 1 16 in column 3 and 4 This will give a table as per Figure 4 13 7 File Edit View Assistants Plot Tools Window Help I New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Probit Regression 1 Pi Bi Test s
239. rs of 26 8 89 322 36 991 and 97 147 sequentially as in Figure 5 1 6 Query nTe File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Ga Plot Power vs Sample Size _ Means Completely Randomised 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 2 2 2 x2 x Difference Between Means X1 X2 0 2 0 5 0 2 0 5 Standard Deviation 0 1 1 1 1 Intracluster Correlation ICC 0 01 0 01 0 01 0 01 Number of Clusters per Group m 3 3 3 3 Sample Size per Cluster N 100 100 300 300 26 8 89 322 36 991 97 147 Cost per sample Total study cost W o m gt Calculate attainable power with the given sample size 7 All columns Figure 5 1 6 Results from Sensitivity analysis The effect of changing these parameters on the relationship between Power and Sample size can be explored further using the Plot Power vs Sample Size button at the top of the screen 8 Select the first column by clicking the 1 at the top of column 1 Then hold down the Shift key and click the 4 at the top of column 4 All four columns will now be highlighted 9 Click the Plot Power vs Sample Size button at the top of the screen This will give you a plot as displayed in Figure 5 1 7 which will show the relationship between power and number of clusters for each column Right click to add features such as a legend to the graph and double click elements for use
240. rtion and 0 50 for the Group 2 proportion This is displayed in Figure 4 3 6 File Edit View Assistants Plot Tools Window Help _ New Fixed Term Test New Interim Test a Plot Power vs Sample Size RM Two Proportions 1 eee Test significance level a 0 05 0 05 0 05 1 or 2 sided test x 2 2 x 2 3 3 3 leve 0 5 0 5 0 5 0 45 0 4 0 45 0 55 0 55 0 5 Odds ratio W p2 1 p1 p1 1 p2 1 49383 1 83333 1 22222 Group 1 size n1 349 154 1396 Group 2 size n2 1349 154 1396 Ratio n2 ni 1 1 1 Power pg 190 01 90 02 90 Cost per sample unit 120 120 120 Total study cost 83760 36960 335040 Se non Calculate required sample sizes for given power X E All columns Figure 4 3 6 Re run calculation for Column 3 93 94 im It can be seen from Figure 4 3 6 that when the Group 1 Proportion was reduced Column 2 the difference between the two groups increased the odds ratio in turn increased and the sample size was dramatically reduced When the Group 2 Proportion was reduced Column 3 the difference between the two groups reduced and the odds ratio in turn was reduced The sample size was subsequently increased quite substantially This all had an knock on effect on the total study cost associate with the sample size 17 Another feature that enables us to compare designs side by side is by using the Power vs Sample Size plot Multiple columns can be plotted together by simply
241. runcation of bounds z uE 3 nQuery nTerim 2 0 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size LUJ Open Manual Statistical Solutions Support GST Two Means 1 x ce level a a EE Group 1 mean p1 Group 2 mean p2 Difference in means pl p2 Group 1 standard deviation o1 Group 2 standard deviation 02 Effect size 5 Group 1 size ni Group 2 size n2 Ratio N2 N1 1 1 1 1 Power Cost per sample unit Total study cost Number of looks Information times Max Times Determine bounds Spending function Phi Truncate bounds Truncate at Spending function Phi 5 Equally Spaced 1 Spending Function O Brien Fleming No Dont Calculate O Brien Fleming 5 x Equally Spaced 1 x Spending Function x O Brien Fleming No x Don t Calculate x O Brien Fleming 5 FE Equally Spaced 1 Spending Function O Brien Fleming No EJ Don t Calculate Is x O Brien Fleming 5 Equally Spaced 1 Spending Function r O Brien Fleming x No gt Don t Calculate x O Brien Fleming Calculate required sample sizes for given power X Figure 3 1 2 Two Means Test Table It is estimated that the cost per unit is roughly 250 so enter 250 in the Cost per sample unit row Once all the values have been entered select Calculate required sam
242. rvival gt Two Equivalence Agreement Regression Two sample t test Student s t test equal variances Satterwaithe s t test unequal variances Two group t test for fold change assuming log normal distribution Two group t test of equal fold change with fold change threshold Wilcoxon Mann Whitney rank sum test continuous outcome Wilcoxon Mann Whitney rank sum test ordered categories Two group univariate repeated measures ANOVA Greenhouse Geisser correction 2x2 Crossover Design i Repeated Measures for two means cancer Figure 4 2 1 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear This test table is illustrated in Figure 4 2 2 3 Enter 0 05 for alpha the desired significance level and enter 4 for the number of levels M as shown in Figure 4 2 4 4 Two sided test is the default setting in nTerim as well as a Ratio value of 1 for the group sizes 5 In this example we will examine a study where the difference in means is 10 and the standard deviation at each level is 20 Therefore enter a value of 10 in the Difference in Means row and a value of 20 in the Standard deviation at each level row File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size RM Two Means 1 oooO O MlMlMlMl 2 i E 1or 2 side
243. s 22 3 1 3 ER GIGS snina niian aa iach acenetaaunnlsaealeaaatiaeexin he norma 24 3 2 TWO PROPOMtiOMS sorcais ai ie E fe feted Gennes eae les Pe E E 32 3 2 1 Introductio Nieee A E E E A A 32 3 2 2 M thodoloy iiion an ana aiaa e iaai aS 34 3 2 3 EXAM ple Snina EA 37 3 SUNIN a eeaeee a a a Ae ene G arani 44 3 3 1 Introductio M sssri ranana randina Kae ia ra aea paaa NEKEEN pneiandentbunanevenbenee 44 3 3 2 Methodology occa cas aca vane iei eea casis enoi annia ei aaae een SNE an a Einn aasa EAA 46 3 3 3 Example Senee A EE E E EEEE EEE AE 48 Chantera ae E EEE E E ES 57 Ae TERM DIEE BIN E E E E EA 57 4 1 One Way Repeated Measures Contrast Constant Correlation c ccccsccecsseeesseeeeees 58 4IL IMFO ONC ENA Meisene e e aeaaea eea anae adarei 58 4 1 2 M thodOlOgy ccccccsssccccccecesseseecececeescseseseeseseeeeeeesseseaaeseeeeseseeesesaeaeeeeeeeessseseaaeas 59 Im 41 3 EXIMplE Sencesa a EG Saien E NAE Eo 60 4 2 Repeated Measures Design for Two M an ccccsssssccecssstececssseeeeeeeseneeceessaeeeeseseeees 72 4 2 1 ntrodUCtiOM ipea E E T EAEE 72 4 2 2 M thOCOlOBy cccccsscccecceccessesssececeeeeuceesesaesececeesesseesaaeeeeeeeesceeseaeaeeeseeseseneseaaeas 73 4 2 3 ERAN PICS isa saccicausicacdivansaacaaiaasatssaayaidansscaia een saianevaacaae ead eaatae ne ERAR EE 74 4 3 Repeated Measure for TWO Proportions cccccccccessssssssseceseeecessessaeseeeeeesesessssnaeeeeees 88 4 3 1 VOMMOPOWCEIO Masi utensas
244. s in Treatment Group K1 Clusters in Control Group K2 K1 2 K1 4 K1 30 Cluster Sample Size in Treatment Group M1 100 100 100 100 Cluster Sample Size in Control Group M2 100 100 100 100 C E 80 so Cost per sample __ Total study cost _ _ Mal a m gt Calculate required treatment group clusters K1 given power and sample size v Run V All columns Figure 5 2 5 Sensitivity analysis around the Control Group Number of Clusters 9 Select Calculate required treatment group clusters K1 given power and sample size from the dropdown menu beside the Run button Then tick the box to run All Columns Then click Run This will give the resultant values of K1 of 38 27 21 and 57 sequentially with the values of K2 updating automatically to reflect the desired ratio between K1 and K2 as in Figure 5 2 6 226 nQuery nTerim File Edit View Assistants Plot Tools Window Help lt New Fixed Term Test New Interim Test a Plot Power vs Sample Size wW Open Two Proportions Inequality Co a 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 2 2 z 2 x2 ix Test Type Unpooled x Unpooled Unpooled x Unpooled z Control Group Proportion p2 0 06 0 06 0 06 0 06 Solve using p1 p2 p1 or p1 p2 Proportions z Proportions z Proportions z Proportions z Test Statistic under HO 0 06 0 06 0 06 0 0
245. s on sample size and perhaps total study cost 12 In Column 2 enter the same information for level of significance number of levels between level correlation Group 2 proportion power and cost per sample unit 13 Now enter 0 4 for Group 1 Proportion in the Group 1 Proportions row 14 Select Calculate required sample size for given power from the drop down menu below the main table and click Run This is displayed in Figure 4 3 5 duery File Edit View Assistants Plot Tools Window Help _ New Fixed Term Test New Interim Test a Plot Power vs Sample Size RM Two Proportions 1 Eee 3 r Test significance level a 0 05 0 05 lor 2sided test 2 2 2 2 3 Number of levels M 3 Between level correlation p los 0 5 Group 1 proportion p1 0 45 04 Group 2 proportion p2 0 55 0 55 Odds ratio W p2 1 p1 p1 1 p2 1 49383 1 83333 Groupisze ni 349 154 Group 2 size n2 1349 154 Ratio n2 ni 1 1 1 1 Power 90 01 90 02 120 120 83760 36960 n Calculate required sample sizes for given power gt E All columns Figure 4 3 5 Re run calculation for Column 2 15 Figure 4 3 5 illustrates the impact of reducing Group 1 proportion We would also like to see the effect of altering the Group 2 proportion 16 Similar to step 12 enter the same information from Column 1 into Column 3 This time enter 0 45 for Group 1 propo
246. s required to achieve 90 07 power to detect a hazard ratio of 1 508 for survival rates of 0 3 in group 1 and 0 45 in group 2 using a 2 sided log rank test with 0 05 significance level assuming that the survival times are exponential These results assume that 5 sequential tests are made and the O Brien Fleming spending function is used to determine the test boundaries 53 54 Example 2 Pocock Spending Function with Non equally Spaced Looks 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Interim Test from the menu bar at the top of the window A Study Goal and Design window will appear as shown below Select the options as mapped out in Figure 3 3 9 then Click OK p Study Goal And Design ax Design Goal No of Groups Analysis Method 5 Fixed Term Means Test Interim Proportions Two Survival Group Sequential Test of Two Survivals Cancel Figure 3 3 9 Study Goal and design Window 2 Enter 0 05 for alpha 2 sided 0 5 for Group 1 proportion 0 4 for Group 2 proportion The hazard ratio is calculated as 0 756 3 Select Proportional Hazards for the Survival Time Assumption We are interested in solving for power given a sample size of 1000 so enter 1000 in the Total Sample Size row 4 This study planned for 4 interim analyses Including the final analysis this requires Number of Looks to b
247. s the following 4 8 13 VNivit1 1 c eS ee c p c A simulation study by Gu et al 2008 indicated that the variance stabilized statistic had the most reliable performance Powery e 1 o 151 im A closed form equation is not used to calculate the sample size or rate ratio Instead a search algorithm is used The search algorithm calculates power at various values for the sample size or rate ratio until the desired power is reached 152 4 8 3 Examples Example 1 Validation example calculating required sample size for a given power The following example is taken from Table 6 of Gu et al 2008 where a sample size calculation problem is conducted comparing the 5 test statistics The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 4 8 1 below Study Goal And Design T No of Groups Analysis Method One Test Two Confidence Interval gt Two Equivalence Regression 5 Cluster Randomized Two sample t test Wilcoxon Mann Whitney rank sum test continuous outcome Wilcoxon Mann Whitney rank sum test ordered categories Wilcoxon Mann Whitney rank sum test ordered categories Unequal n s Two group univariate repeated measures ANOVA Greenhouse Geisser correction 2x2 Crossover Desi
248. s the 27 groups is 108 distributed across the groups as specified a multivariate analysis of variance will have 30 power to test Factor A if a Wilks Lambda test statistic is used with 0 05 significance level 30 power to test Factor B if a Wilks Lambda test statistic is used with 0 05 significance level 98 07 power to test Factor C if a Wilks Lambda test statistic is used with 0 05 significance level 100 power to test Factor AB if a Wilks Lambda test statistic is used with 0 05 significance level 66 77 power to test Factor AC if a Wilks Lambda test statistic is used with 0 05 significance level 66 77 power to test Factor BC if a Wilks Lambda test statistic is used with 0 05 significance level 100 power to test Factor ABC if a Wilks Lambda test statistic is used with 0 05 significance level 139 140 4 7 One Poisson Mean 4 7 1 Introduction Count data is obtained in a variety of clinical and commercial activities such as the number of accidents at a junction or number of occurrences of a disease in a year The most common distribution used to model count data is the Poisson distribution The one sample test is used to test hypotheses about the mean rate of a Poisson distributed sample against an alternative specified value This table facilitates the calculation of the power and sample size for hypothesis tests of the mean of single Poisson distributed sample Power and sample size is computed using the meth
249. sample size and number of clusters Run __ V all columns Figure 5 2 9 Results from Sensitivity Analysis 5 3 CRT Two Proportions Equivalence Completely Randomized 5 3 1 Introduction Binary data is commonly studied in variety of different fields Equivalence trials are commonly used to assess whether a treatment is equivalent to another treatment in the clinical setting e g comparing two competitor drugs Clustered data is very common in a wide variety of academic social policy and economic studies This two sample test is used to test hypotheses about the equivalence of two treatments in a completely randomized cluster randomized trial This table facilitates the calculation of the power and sample size for equivalence hypothesis tests comparing proportions in a completely randomized cluster randomized trial Power and sample size is computed using the method outlined by Donner and Klar 2000 231 232 Im 5 3 2 Methodology This table provides sample size and power calculations for studies which will be conducting an equivalence trial between proportions in trials which use a completely randomized cluster randomization study design Equivalence trials are those in which the researcher is testing that two treatments have an equivalent effect A completely randomized design assigns clusters randomly to control and treatment groups The sample size calculation for cluster randomized proportions is taken from Donner an
250. samples Poisson rates under the alternative hypothesis R the fixed observation times for group 1 t and group 2 t the mean Poisson rate in group 1 y1 one of the group sample sizes N or Nz or the sample size ratio N N and the test statistic that is being used to test the difference in ratios The formulas for the power which correspond to each of the statistics outlined above are as follows 1 Unconstrained Maximum Likelihood Estimate MLE VNi 1t1 1 c Zi a Vc p c 2 Constrained Maximum Likelihood Estimate CMLE Poweryg 4 8 8 Powercmg Es Z ay c p 1 z 4 8 9 3 Log Unconstrained Maximum Likelihood Estimate In MLE Powen Mitac Z In MLE 7 AA Z4 a c p 1 4 Log Constrained Maximum Likelihood Estimate In CMLE k 7 pin Inc Ve p 2p 1 OWeNn CMLE 7 feel Z1 a c p 1 ct p 5 Variance Stabilized Estimate 4 8 10 4 8 11 3 ETE 2 1 vc N y t 3 Z a c p c Powers gt 2 4 8 12 Vc p 1 where c R R and is the inverse normal cumulative distribution To calculate the power when the test for the null hypothesis is Hp 2 R versus the 1 specified alternative hypothesis of H 2 R lt Ry the power calculation is the same 1 except Zg is used instead of z _ and the power is equal to one minus the normal cumulative distribution For example the power formula for the MLE become
251. score z Likelihood score z Higher Proportions Better Worse Better x Better x Better z Better Control Group Proportion p2 0 5 0 5 0 5 0 5 Solve using p1 p2 p1 or p1 p2 Differences z Differences z Differences z Differences z Superiority Test Statistic 0 1 0 1 0 1 0 1 Actual Value of Test Statistic 0 15 0 15 0 15 0 15 Intracluster Correlation ICC 0 001 0 001 0 001 0 001 Clusters in Treatment Group K1 14 10 9 9 Clusters in Control Group K2 14 20 36 30 Cluster Sample Size in Treatment Group M1 100 100 100 100 Cluster Sample Size in Control Group M2 100 100 100 100 Calculate required treatment group clusters K1 given power and sample size v Run v All columns Figure 5 5 6 Results from Sensitivity analysis 10 By clicking on the desired study design column and going to the Output tab at the bottom of the screen you can get an output statement for the results given by the current calculation as per Figure 5 5 7 Output OUTPUT STATEMENT In a superiority cluster randomized trial comparing two binary variables a sample size of 14 clusters with 100 individuals per cluster in the treatment group and a sample size of 14 clusters per group with 100 individuals per cluster in the control group achieves 82 313 power to detect superiority between two proportions when the Differences for the superiority margin and actual value are 0 1 and 0 15 respectively the intracl
252. seline Response Rate eB0 1 Response Rate Ratio eB1 eB0 1 281 Mean Exposure Time pT 1 Overdispersion Parameter 1 Distribution of X1 sidetable required Normal z Normal z Normal z Normal z Variance of b1 Null Hypothesis Variance of b1 Alternative Hypothesis Sample Size N 100 Power 80 011 Cost per sample Total study cost ee i gt Calculate Response Rate Ratio gt 1 given power and sample size v Run E All columns Figure 4 12 19 Completed Poisson Regression study design 197 198 im Note that for this example no values are transferred into the main table as the Variance of b1 Null Hypothesis and Variance of b1 Alternative Hypothesis statistics require the response rate ratio However these figures will be saved in the system memory for calculation purposes for that column while it is selected 4 13 Probit Regression 4 13 1 Introduction The Probit Regression Model is an analysis method often used to model the relationship between a dependent variable which is a proportion and one or more independent variables in clinical trials testing subject exposure e g to drugs radiation in lethal dose 50 LDso trials These are common in early stage animal trials for example This table facilitates the calculation of the power and sample size for an LDso using probit analysis Power and sample size is computed using the method outlined by Kodell et al 201
253. sistant Table 62 12 Now that values for Contrast C and Scale D have been computed we can continue with filling in the main table For the Standard Deviation enter a value of 6 For the between level correlation enter a value of 0 2 13 We want to calculate the sample size required obtain a power of 90 Therefore enter 90 in the Power row 14 It has been estimated that it will cost 100 per sample unit in this study Therefore enter 100 in the Cost per sample unit row 15 Select Calculate required sample size for given power from the drop down menu below the main table and click Run This is displayed in Figure 4 1 6 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size RM Contrast 1 P 2 3 s Test significance level a 0 05 Number of levels M l3 Contrast C Jci pi 2 Scale D SQRT ici Standard deviation at each level o Between level correlation p Power Group size N 1 41421 6 0 2 Effect size A C D o SQRT 1 p 0 26352 89 95 152 100 45600 Calculate required sample sizes for given power x F All columns Figure 4 1 6 Completed One way Repeated Measures Contrast Table It can be seen from Figure 4 1 6 that a sample size of 152 per group for each of the three groups thus a total sample size N of 456 is required to obtain a power of 89 95 Due to
254. size per cluster in the treatment group Dry is the maximum likelihood estimator for each group proportion and IF is the inflation factor for the effect of clustering in the treatment and control groups respectively IF is defined as follows IF 1 ICC M 1 i 1 2 5 2 3 The constrained maximum likelihood where p pz A estimator of the two proportions is calculated using the following calculations K M IF gt 5 2 4 KM IF oe a 1 t 5 2 5 b 1 p tp A t 2 5 2 6 c A A 2p t 1 p tpz 5 2 7 d p A 1 A 5 2 8 b be d v 3 c2 ae a u sign v e oa 5 2 10 g 3a 3a Vv ete ha coal 3 Prm1 2ucos w b 3a 5 2 12 Prmz Prm1 A 5 2 13 Im 221 222 Metin 2 Unpooled Test Statistic This test statistic uses the estimated group proportions to calculate the standard error Its formula is as follows a Pil pi IF p201 p2 IFz Unpool KiM KM 5 2 14 where p and p are the estimated mean proportions for the two groups 3 Pooled Test Statistic The pooled test statistic uses the weighted average of the two proportions to calculate the standard error Its formula is as follows n 1 DIF n 1 p IF OPool a oe 1 pO SPE PIF 5 2 15 K M KM2 where K M K M j M4 P1 2M2P2 5 2 16 K M K M For the one sided test Z4 q 2 is replaced with z __ and the following equations are used depending on the val
255. staaebeeseedens wih rtei tek devaesbaistenaeneaneens 140 AT AI Vy NUNN Mio e E e E N E EE E 140 4 7 2 M thOGOlO BY ccnn anaa E EE E RE 141 4 7 3 Examples eee 143 4 8 TWO POISSON MEANS siiis a E E N 148 AL MARPAC LON a ausisses stinde e ana a aaaea 148 4 8 2 Methodology irc sstacreserascnasteyiniiec uspaccensndeandneseuadsetasabaceusiaduasn Cendeemevendaulldubecuinnedee NAAA Taa 149 4 8 EXAMPLES mee oar a A E E Sauk coieatesnnars 153 4 9 Two Negative Binomial Rates wiciiscctscsisprcrnaticodanugpdteasipnetsnsesnicresineabeniadatreneauelaesleeniean 158 49 L MARIE COM esiis ienien aaaeeeaei iaaiaee 158 4 92 Methodology scires iiniosiriiirr iiit iar isasi E E ARER aK veumeuicoucesevendente 159 Pa Ro EAI 5 E E A EE E E ndsedcin ated taneaasenndransentuaageandenteoes 162 4 10 Two Incidence Rates a ceraisaczerannaedannnnannetanagssdtnesdeaadsnnhederansacdetonlatedddeandaesanns Cbsneiaietebians 168 4 10 1 introduction serensssiriraererr seisine ri Pe ENARE EAEEREN EEE NAE TETEE NEE SEREEN 168 ALO INMCRIO OG OB sist coccainarnncsicnstoncietiteutanninnb waudainvatdstneumacececedanvenevdanvendideseensiauieceweees 169 4 10 3 Exa MPpl Ssss serari cosa niguandavavautsietsssutacvapeduais AIEEE EREKET SKETSE 170 4 11 COX RESPESSIOM iss cadvascncecetaceckescetaase ord a T E E E E A A EE A a 175 4 111 NAF OOMC HOM ssela e eaae a de aie eT taaie 175 4 11 2 Methodology iraniar aeina aa EEEE EEEE 176 411 3 EXAM Pls nre n a a T A E A E e 178 4 12 POISSON RESreSSIO
256. t for the results given by the current calculation as per Figure 5 4 7 Output x OUTPUT STATEMENT In a non inferiority cluster randomised trial comparing two binary variables a sample size of 14 clusters with 100 individuals per cluster in the treatment group and a sample size of 14 clusters per group with 100 individuals per cluster in the control group achieves 81 199 power to detect non inferiority between two proportions when the Differences for the non inferiority margin and actual value are 0 1 and 0 05 respectively the intracluster correlation is 0 001 and when testing at the 0 05 significance level using the Unpooled statistic ug Specify Multiple Factors W Output Figure 5 4 7 Study design Output statement This Output statement can then be easily transferred directly from the output window into a report 248 Example 2 Validation example calculating clusters when higher proportions are worse A calculation is conducted to show the effect of selecting Worse for the Higher Proportions are Better Worse option If Worse is selected then higher values for the proportion are considered worse from the study s perspective and thus positive differences would be used to test non inferiority 11 Return the table to its values before the sensitivity analysis Then replace the 0 1 with 0 1 and 0 05 with 0 05 in the Non Inferiority Test Statistic and Actual Value Test Statistic rows respectively Set the
257. t rates y y2 are used to specify the distribution of each sample 4 8 2 Methodology As suggested by Gu et al 2008 there are five test statistics which could be used to test statistical hypotheses comparing two Poisson mean rates The statistics proposed tested the null hypothesis of Ho 2 Ro usually R 1 against 1 the specified alternative hypothesis of H 2 R gt Ro These are given by the 1 following equations 1 Unconstrained Maximum Likelihood Estimate MLE The unconstrained maximum likelihood estimator is given by f X t i 1 2 The null hypothesis can be re arranged as follows to yield a new statistic Ho WwW Y RoV2 4 8 1 The statistic W is asymptotically normally distributed with a mean of uw f RV2 and variance of o A t R A2 t2 Thus the test statistic of uy o can be used for tests of Ho For the unconstrained maximum likelihood estimate this yields the following formulation X t RX2 t2 _ X X2p Wmr Xi X2 E M 4 8 2 J X t2 X R2 t2 X1 X2p where P Rt N t2Np 2 Constrained Maximum Likelihood Estimate CMLE The constrained maximum likelihood estimators are given by the following X X Xi X Aben 4 8 3 SEG i p KAFA Similar to the unconstrained maximum likelihood estimator these can be used to generate the following statistic for hypothesis testing X t RX2 t2 X X29 Wome Xi X2 4 8 4 J X X2 R2 d R
258. tatistic This test statistic uses the estimated group proportions to calculate the standard error Its formula is as follows p l pi IF p2 1 p2 IFp Unpool KiM KM 5 4 14 where p and p are the estimated mean proportions for the two groups 3 Pooled Test Statistic The pooled test statistic uses the weighted average of the two proportions to calculate the standard error Its formula is as follows n 1 DIF n 1 p IF PEPE PU PIF _ pA PIF 5 4 15 K M KM2 where K M K M p M4 P1 2M2P2 5 4 16 K M K M A closed form equation is not used to calculate the other parameters Instead a search algorithm is used The search algorithm calculates power at various values for the relevant parameter until the desired power is reached 243 244 Im 5 4 3 Examples Example 1 Validation example calculating required number of clusters for a given power The following example will look at a number of clusters calculation problem is conducted and then a sensitivity analysis is conducted to show the effect of changing control group number of clusters The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 5 4 1 below f Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term O Means
259. te until the desired power is reached 169 Im 4 10 3 Examples Example 1 Validation example calculating required sample size for a given power The following example is taken from Table 3 2 on page 55 of Smith and Morrow 1996 where a sample size calculation problem is conducted to show the effect of changing the treatment group incidence rate and power The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 4 10 1 below S EE Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval Survival gt Two Equivalence Agreement Regression Cluster Randomized Two sample t test Wilcoxon Mann Whitney rank sum test continuous outcome Wilcoxon Mann Whitney rank sum test ordered categories Wilcoxon Mann Whitney rank sum test ordered categories Unequal n s Two group univariate repeated measures ANOVA Greenhouse Geisser correction 2x2 Crossover Design Repeated measures for two means Negative Binomial Two Poisson Means Figure 4 10 1 Study Goal and Design Window 2 In order to select the Two Incidence Rates design table navigate through the Study Goal and Design Window by selecting Fixed Term as the Design M
260. that an effect will not be found that the trial can be terminated by rejecting the alternative hypothesis In the case where the user wishes to generate boundaries for early rejection of either the null or alternative hypothesis H or H they are given two options either to have the boundaries binding or non binding With binding boundaries if the test statistic crosses the futility boundary the test must be stopped otherwise the type 1 error may become inflated The reason for this is that there is an interaction between the efficacy and futility boundaries in their calculation that could cause the efficacy boundary to shift In the case of non binding boundaries the efficacy boundaries are calculated as normal that is as if the futility boundaries did not exist This eliminates the danger of inflating the type 1 error when the futility boundary is overruled The downside of the non binding case is that it may increase the required sample size relative to the binding case The boundaries calculated in nTerim 3 0 follow the procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 45 46 Melim 3 3 2 Methodology Sequential Log Rank test of survival in to groups the variables are defined as Symbol Description a Probability of Type error f Probability of Type II error 1 6 Power of the Test S1 S2 Group Survival Proportions Number of Events Sample Siz
261. the Standard Drug we have approximated the reduction in blood pressure as roughly 12mmHg with a standard deviation of 6mmHg Likewise in previous studies the Placebo has resulted in an estimated reduction of SmmHg This example will examine using a One way Analysis of Variance with a 0 05 level of significance The following steps outline the procedure for Example 1 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear a Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term Means Test O Interim Proportions Confidence Interval Survival D Equivalence O Agreement D Regression One way analysis of variance One way analysis of variance Unequal n s Single one way contrast Single one way contrast Unequal n s i Two way analysis of variance Multivariate analysis of variance MANOVA Analysis of Covariance ANCOVA Cancel Figure 4 4 1 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear 3 There are two main tables required for this test the main test table illustrated in Figure 4 4 2 and the effect size assistant table shown in Figure 4 4 3 4 Enter 0 05 for alpha the desired significance level and enter
262. the appropriate chapter of the manual much easier If the user is working in a particular design window for example the MANOVA window and the user clicks on the Open Manual button a PDF of the MANOVA chapter in the manual will automatically open providing the user with the background and technical information on MANOVA as well as examples in nTerim The second shortcut is the Statistical Solutions Support button If further clarification on any aspect of nTerim is required please contact our support statisticians by clicking on this button This shortcut takes the user to the Statistical Solutions support website where queries can be entered and sent directly to our support team These support shortcuts are highlighted in the nTerim tool bar in Figure 2 8 1 below File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test x Plot Power vs Sample Size LLI Open Manual che Statistical Solutions Support Figure 2 8 1 Manual and Support Shortcut Tabs If there are any issues with any aspect of the installation process there are three approaches you can take i you can check the system requirements outline in Section 1 1 of this manual ii look up the installation help and FAQ s on our website http www statsols com products nquery advisor nterim and iii you can email us for technical help at support statsols com In order to help us address your questions in the best way possible the m
263. the bias from a single measure For example an individual s blood pressure is known to be sensitive to many temporary factors such as amount of sleep had the night before mood excitement level exercise etc If there is just a single measurement taken from each patient then comparing the mean blood pressure between two groups could be invalid as there could be a large degree of variation in the single measures of blood pressure levels among patients However by obtaining multiple measurements from each individual and comparing the time averaged difference between the two groups the precision of the experiment is increased This table facilitates the calculation of power and sample size for the time averaged difference between two means in a repeated measures design Power and sample size is computed using the method outlined by Liu and Wu 2005 4 2 2 Methodology Power and sample size are calculated using standard normal distributions and follow the procedures outlined by Liu and Wu 2005 To calculate power and sample size the user must first specify the test significance level a and choose between a one or a two sided test The user must then enter a value for the number of levels M This value corresponds to the number of measurements that will be taken on each subject Values must then be provided for the difference in means d the standard deviation at each level g and the between level correlation p The difference in means
264. tle for Column 1 and drag across to Column 4 17 Then click on the Plot Power vs Sample Size button on the menu bar The multiple column plot is displayed in Figure 4 2 19 Rover Spe See Power vs Sample Size Column 1 Column 2 Column 3 Column 4 40 90 110 130 150 170 190 210 230 250 270 Power 82 67 Sample Size N1 N2 Sample Size 130 Figure 4 2 19 Power vs Sample Size Plot It can be seen from the legend on the left hand side legend can be altered manually that the blue line represents Column 1 the orange line represents Column 2 the red line represents Column 3 and the navy line represents Column 4 The cross on the graph illustrates how the user can identify what the sample size is for a corresponding power value for each column In the bottom right corner of the plot indicated the exact values for Power and Sample Size for each identifier on the graph It can be seen in Figure 4 2 19 that Column 1 reaches an acceptable power level faster than the design in Column 2 3 or 4 The researcher can now make an assessment as to which design they would prefer to use 87 88 im 4 3 Repeated Measure for Two Proportions 4 3 1 Introduction This table facilitates the calculation of power and sample size for the time averaged difference between two proportions in a repeated measures design Power and sample size is computed using the method outlined by Liu and Wu 2005 A r
265. to overall tests This usually makes tests of planned contrasts easier to interpret and more powerful 4 1 2 Methodology Power and sample size is calculated using central and non central F distributions and follows the procedures outlined by Overall and Doyle 1994 To calculate power and sample size the user must specify the test significance level a and the number of levels M The user must then enter values for the contrast C and the Scale D Alternatively the user can enter the expected means at each level and the respective contrast coefficients using the compute effect size assistant nTerim will then calculate the contrast and scale using the following formulas for contrast M C X citi 4 1 1 i 1 and scale 4 1 2 Once the contrast and the scale have been entered the user must input values for the common standard deviation and the between level correlation p The standard deviation at each level is assumed to be the same and the correlation between each pair of levels is assumed to be the same Given these four values nTerim will automatically calculate the effect size using the following formula ICI Aan Doy 1 p In order to calculate power a value for the total sample size N must be entered nTerim then calculates the power of the design by first determining the critical value For DF a Where DF 1 is the numerator degrees of freedom and DF M 1 N 1 is the denominator
266. to repeat the same study design example except we re going to enforce a stricter level of significance In the second column enter 0 025 in the Test Significance Level row Now we are looking for a 2 5 level of significance instead of a 5 level as in the first column 11 We want to see the effects of changing the level of significance has on sample size and perhaps the total study cost 76 12 Enter the same information for number of levels Difference in Means standard deviation at each level between level correlation power and cost per sample unit 13 Select Calculate required sample size for given power from the drop down menu below the main table and click Run This is displayed in Figure 4 2 4 above It can be seen from Figure 4 2 4 that sample size has increase be 20 10 per group and the estimated cost has increased by 1 800 14 Another feature that enables us to compare designs side by side is by using the Power vs Sample Size plot Multiple columns can be plotted together by simply highlighting the desired columns and clicking on the Plot Power vs Sample Size button on the menu bar AZ nQuery nTerim 0 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size RM Two Means 1 3 4 os RE 7 E 2 z 10 20 0 5 63 63 ooo f gt Calculate required sample sizes for given power Run E All columns
267. tom right corner of the plot indicated the exact values for Power and Sample Size for each identifier on the graph 99 100 im 17 Finally by clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation depending on which column you have clicked on Output OUTPUT STATEMENT When the sample size is 852 in group 1 and 852 in group 2 a test for the time averaged difference between two proportions in a repeated measures design with a 0 05 significance level will have 90 power to detect an odds ratio of 0 78142 for proportions of 0 45 in group 1 and 0 39 in group 2 in a design with 3 repeated measurements when the between level correlation is 0 4 ug Specify Multiple Factors ui Output Figure 4 3 14 Output statement The output statement in Figure 4 3 14 is for Column 1 This statement can be copied and pasted into any report 4 4 One Way Analysis of Variance ANOVA 4 4 1 Introduction This table facilitates the calculation of power and sample size for a one way analysis of variance ANOVA design Calculations are performed using the methods outlined by O Brien and Muller 1993 A one way ANOVA compares means from two or more groups in order to determine whether any of those means are significantly different from each other Note if we were to compare just two means using the one way ANOVA then this would be equivalent to a t test for two independent m
268. tput Figure 3 3 6 Boundary Table for Column 2 51 im Finally in terms of output the boundaries that were calculated as shown in Figure 3 3 4 and 3 3 6 were automatically plotted by nTerim the boundary plot for Column 1 is given below O Brien Fleming Boundaries with Alpha 0 05 1 2 3 4 Figure 3 3 7 Boundary Plot for Column 1 10 Click on the column title for Column 1 and drag across to highlight both Columns 1 and 2 11 Select Plot Power Sample Size from the toolbar it may take a moment to generate the plot as multiple calculations are performed PoerssonpeSee 2 BR O Power vs Sample Size 350 550 750 950 1150 1350 1550 eS SS eS Sea Saat Pt 100 A Power 90 83 Sample Size 915 Figure 3 3 8 Power vs Sample Size Plot 52 As it can be seen in Figure 3 3 8 an illustration of the comparison between Column 1 and Column 2 in relation to Power vs Sample Size performance can be created The cross on the graph illustrates how the user can identify what the sample size is for a corresponding power value for each column In the bottom right corner of the plot indicated the exact values for Power and Sample Size for each identifier on the graph Finally by clicking on the Output tab at the bottom of the screen you can see a statement giving details of the calculation Column 1 Output Statement A total sample size of at least 409 256 events i
269. tudy Goal and Design window will appear Study Goal And Design oom fi N Design Goal No of Groups Analysis Method Fixed Term 5 One Test Interim Proportions Two Confidence Interval Survival D gt Two Equivalence D Agreement D Regression 5 Chi squared test to compare two proportions Compute power or sample size Compute one or two proportions Chi squared test continuity corrected Compute power or sample size i Compute one or two proportions i Fisher s exact test Two group Chi square test comparing proportions in C categories Mantel Haenszel Cochran test Mantel Haenszel Cochran test of OR 1 in S strata Mantel Haenszel Cochran test of OR 1 in S strata continuity corrected Repeated Measures for two proportions l Cancel Figure 4 3 1 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear 3 Enter 0 05 for alpha the desired significance level and enter 3 for the number of levels M as shown in Figure 4 3 3 4 Two sided test is the default setting in nTerim as well as a Ratio value of 1 for the group sizes as shown in Figure 4 3 2 5 In this example we will examine a study where the group 1 proportion is estimated as 0 45 and the group 2 proportion is estimated as 0 55 Enter 0 45 in the Group 1 Proportion row and enter 0 55 in the Group 2 Prop
270. u beside the Run button Then tick the box to run All Columns Then click Run This will give the resultant sample sizes of 1433 1230 1094 and 997 sequentially as in Figure 4 9 6 Plot Tools Window Edit View Assistants Help File New Fixed Term Test New Interim Test VA Plot Power vs Sample Size LL Of Negative Binomial 1 0 05 0 05 0 05 0 05 Test significance level a Mean Rate of Event for Control r0 0 8 1 i 14 Rate Ratio r1 r0 0 85 0 85 0 85 0 85 Average Exposure Time pT 0 75 0 75 0 75 0 75 Dispersion Parameter k 0 7 0 7 0 7 0 7 Rates Variance Reference Group Rate z Reference Group Rate x Reference Group Rate x Reference Group Rate r Sample Size Ratio N1 NO 1 1 1 t Control Group Sample Size N0 1433 1230 1094 997 Treatment Group Sample Size N1 1433 1230 1094 997 80 0115952771 80 0226600741 80 0089495575 80 0024051023 Calculate required sample size for given power v All columns Figure 4 9 6 Results from Sensitivity analysis The effect of changing the mean rate of the event for the control group on the relationship between Power and Sample size can be explored further using the Plot Power vs Sample Size button at the top of the screen 8 Select the first column by clicking the 1 at the top of column 1 Then hold down the Shift key and click the 4 at the top of column 4 All four columns will now be highlighted 9
271. ues of and A A o Ep ao Sunpool Sunpool 6 A o Test al A gt s Ounpool Sunpool Power 5 2 17 A closed form equation is not used to calculate the other parameters Instead a search algorithm is used The search algorithm calculates power at various values for the relevant parameter until the desired power is reached 5 2 3 Examples Example 1 Validation example calculating required number of clusters for a given power The following examples are taken from Donner and Klar 2000 where a number of clusters calculation problem is conducted and then a sensitivity analysis is conducted to show the effect of changing control group number of clusters The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as Figure 5 2 1 below Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval Survival gt Two Equivalence Agreement Regression Cluster Randomized CRT Two Means Completely Randomized CRT Two Proportions Equivalence CRT Two Proportions Non Inferiority CRT Two Proportions Superiority Figure 5 2 1 Study Goal and Design Window 2 In order to select the CRT Two Proportions Completely
272. ultiple Factors x 1 or 2 sided test Test Significance Level a Number of levels M Group 1 size N1 Between level correlation p Group 2 size N2 Group 1 Proportions m1 Ratio N2 N1 Group 2 Proportions m2 Power Odds Ratio Y n2 1 m1 m1 1 12 Cost per sample unit Fill Table Clear Table a Specify Multiple Factors Output Figure 4 3 9 Specify Multiple Factors Table 7 We know that the Group 1 proportion ranges from 0 45 to 0 55 so enter 0 45 0 55 in the Group 1 Proportions box with a space separating the two numbers We also know that the Group 2 proportion ranges from 0 39 to 0 51 so enter 0 39 0 51 in the Group 2 Proportions box These entries are displayed in Figure 4 3 10 below 8 We want a 5 level of significance so enter 0 05 in the Test Significance Level box We want an equal sample size for each group so enter 1 in the Ratio N2 N1 box We would like to obtain 90 power in this study design so enter 90 in the Power box 9 Finally it has been projected that the cost per sample unit will be 100 therefore enter 100 in the Cost per sample unit box Specify Multiple Factors x 1 or 2 sided test 2 Test Significance Level a 0 05 Number of levels M 3 Group 1 size N1 Between level correlation p 0 4 Group 2 size N2 Group 1 Proportions m1 0 45 0 55 Ratio N2 N1 1 Group 2 Proportions m2 0 39 0 51 Power 9 90 Odds Ratio Y m2 1 11 m1 1 n2 C
273. uster correlation is 0 001 and when testing at the 0 05 significance level using the Likelihood score statistic a Specify Multiple Factors Output Figure 5 5 7 Study design Output statement This Output statement can then be easily transferred directly from the output window into a report Example 2 Validation example calculating clusters when higher proportions are worse A calculation is conducted to show the effect of selecting Worse for the Higher Proportions are Better Worse option If Worse is selected then higher values for the proportion are considered worse from the study s perspective and thus positive differences would be used to test non inferiority 11 Return the table to its values before the sensitivity analysis Then replace the 0 1 with 0 1 and 0 15 with 0 15 in the Superiority Test Statistic and Actual Value Test Statistic rows respectively Set the Clusters in Control Group K2 back to K1 K1 2 4 K1 and 30 for columns 1 to 4 respectively Then set the power values back to 80 12 Finally select Calculate required treatment group clusters K1 given power and sample size from the dropdown menu beside the Run button Next tick the box to run All Columns Then click Run This will give Figure 5 5 8 File Edit View Assistants Plot Window Help New Fixed Term Test New Interim Test Z
274. vely as in Figure 4 8 4 The result in Gu et al 2008 is 8527 for the sample size in Group 1 but the authors agreed this was the more accurate result due to their usage of two decimal place rounding nQuery nTerim 3 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test x Plot Power vs Sample Size LU Op Two Poisson Means 1 2 3 4 5 Test significance level a 0 05 Null Poisson Rate Ratio RO yO y1 1 Alt Poisson Rate Ratio R1 y0 y1 4 Test Statistic W1 MLE m W1 MLE W1 MLE w W1 MLE w W1 MLE wi Observation Time for Group 1 t1 2 Observation Time for Group 2 t2 2 Mean Poisson Rate in Group 1 y1 0 0005 Sample Size Allocation Ratio N2 N1 0 5 Sample Size in Group 1 N1 8564 Sample Size in Group 2 N2 4282 r 90 0004577144 Cost per sample Total study cost Ml o ee gt Calculate required Group 1 and 2 sample sizes for given power and sample size allocation ratio x Run F All columns Figure 4 8 4 Completed Two Poisson Means study design 6 The next calculation is a sensitivity analysis for sample size where we change the Test Statistic to investigate the impact this has on the sample size estimate To do this replace the updated power with 90 and then copy the same values across to columns 2 to 5 Then select from the Test Statistic dropdown menu W2 CMLE for column 2
275. ven by the following equation 2 B z _ z p 4 11 2 P 1 R o2B2 The power is calculated by re arrangement of the above formula to give the following equation Power vra R o B N z _ 4 11 3 2 Similar re arrangements yield the following equations for the test significance level and log hazard ratio a 2 1 _ Pa R o B N z g 4 11 4 2 z2 z1 6 eee ee ae PC R2 o2N 4 11 5 For the one sided test Z _ would be used in place of z _ 2 177 Im 4 11 3 Example Example 1 Validation example calculating required sample size for a given power The following examples are taken from Hsieh and Lavori 2000 where a sample size calculation problem is conducted for a multiple myeloma data set page 557 and then a sensitivity analysis is conducted Table 1 page 555 to show the effect of changing the event rate The following steps outline the procedure for this example 1 Open nQuery nTerim 3 0 via the start menu or desktop shortcut Click New Fixed Term Test from the top of the window The Study Goal and Design window will appear as displayed in Figure 4 11 1 below oE No of Groups Analysis Method ler Fixed Term Means One Test 5 Interim Proportions Two Confidence Interval A Survival gt Two Equivalence Agreement Regression Cluster Randomized Logistic Regression one normal covariate Logistic Regress
276. vents 256 578 Cost per sample unit 100 100 Total study cost 40900 88800 Number of Looks 5 5 5 5 Information times Equally Spaced z Equally Spaced z Equally Spaced z Equally Spaced Max Times 1 1 1 1 Determine bounds Spending Function Spending Function Spending Function Spending Function x Spending function O Brien Fleming z O Brien Fleming z O Brien Fleming x O Brien Fleming Phi Truncate bounds No z No x No x No x Truncate at Futility boundaries z z Dont Calculate Dont Calculate X Spending function z z O Brien Fleming E O Brien Fleming E Phi p P Calculate required sample sizes for given power X All columns Figure 3 3 5 Complete Survival Table for Two tests In addition to the sample size and cost output for Column 2 the boundary calculations are also presented as shown below Looks i a ee ee 0 2 0 4 0 6 0 8 1 Lower bound 4 87688 3 35695 2 68026 2 28979 2 03100 Upper bound 4 87688 3 35695 2 68026 2 28979 2 03100 Futility bound Nominal alpha 0 00000 0 00079 0 00736 0 02203 0 04226 Incremental alpha lo ooo00 0 00079 0 00683 0 01681 0 02558 Cumulative alpha 0 00000 0 00079 0 00762 0 02442 0 05000 Exit probability 0 03 9 95 34 68 29 96 15 39 Cumulative exit probability 0 03 9 98 4467 7463 90 02 Nominal beta Incremental beta _ Cumulative beta Exit probability under HO uf Looks i Specify Multiple Factors i Ou
277. w Then copy the first column into columns 2 to 4 8 Enter K1 2 in column 2 for Clusters in Control Group K2 4 K1 in column 3 and 30 in column four This will give a table as per Figure 5 3 5 0 05 x Pooled 0 5 x Differences 0 1 0 1 0 0 001 4 K1 100 100 80 0 05 z Pooled 0 5 z Differences 0 1 30 100 100 80 nQuery nTerim 3 Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size Two Proportions Equivalence 1 __ Test significance level a 0 05 0 05 Test Type Pooled x Pooled Control Group Proportion p2 0 5 0 5 Solve using p1 p2 p1 or p1 p2 Differences x Differences Test Statistic for Upper Equivalence Margin 0 1 0 1 Test Statistic for Lower Equivalence Margin 0 1 0 1 Actual Value of Test Statistic 0 0 Intracluster Correlation ICC 0 001 0 001 Clusters in Treatment Group K1 Clusters in Control Group K2 K1 2 K1 Cluster Sample Size in Treatment Group M1 100 100 Cluster Sample Size in Control Group M2 100 100 80 80 Cost per sample Total study cost MaLo m Calculate required treatment group clusters K1 given power and sample size v Figure 5 3 5 Sensitivity analysis around the Control Group Number of Clusters 9 Select Calculate required treatment group clusters K1 given power and sample size from the dropdown menu beside the Run butto
278. w Fixed Term Test New Interim Test Z Plot Power vs Sample Size w Poisson Regression 1 A 1 2 aes eee Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 1 z z z ix Baseline Response Rate eB0 1 1 1 1 Response Rate Ratio eB1 eB0 0 5 0 9 1 3 2 Mean Exposure Time pT 1 a 1 Overdispersion Parameter 1 1 1 1 Distribution of X1 sidetable required Normal z Normal z Normal z Normal iz 1 1 1 1 Variance of b1 Alternative Hypothesis 0 786 0 994 0 966 0 786 R squared X1 and independent variables 0 0 0 0 Sample Size N Power 95 95 95 95 Cost per sample Total study cost pe Calculate required sample size for given power J a i Figure 4 12 12 Variance of b1 values entered for Sensitivity analysis 10 Select Calculate required sample size for given power from the dropdown menu beside the Run button Next tick the box to run All Columns Then click Run This will give the resultant sample sizes of 21 973 155 and 21 sequentially as in Figure 4 12 13 Similar to the example above the answers for column 2 to 4 differ from Signorini 1991 due to two decimal place rounding File Edit View Assistants Plot Tools Window Help I New Fixed Term Test New Interim Test Plot Power vs Sample Size WwW Poisson Regression 2 ey ee 4 Test significance level a 0 05 0 05 0 05 0 05 1
279. wing steps outline the procedure for Example 1 1 Open nTerim through the Start Menu or by double clicking on the nTerim desktop icon Then click on New Fixed Term Test from the menu bar at the top of the window A Study Goal and Design window will appear Design Analysis Method Fixed Term Means Test D Interim Proportions Confidence Interval 5 Survival Equivalence 5 Agreement Regression p One way analysis of variance One way analysis of variance Unequal n s Single one way contrast Single one way contrast Unequal n s Two way analysis of variance Multivariate analysis of variance MANOVA Analysis of Covariance ANCOVA Cancel Figure 4 5 1 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window will appear 3 There are two main tables required for this test the main test table illustrated in Figure 4 5 2 and the effect size assistant table shown in Figure 4 5 3 4 Enter 0 05 for alpha the desired significance level and enter 4 for the number of groups G as shown in Figure 4 5 4 113 114 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Z Plot Power vs Sample Size ANCOVA 1 i Tes t signifi ca n ce level J i SE Number of groups G Variance of means V Common standard deviation o Number of cova
280. with covariates R Power Total sample size N Cost per sample unit Total study cost gt Calculate required sample sizes for given power Ma All columns Compute Effect Size Assistant x N as multiple of n1 Jri Jni ni Figure 4 5 4 Automatically updated Compute effect size Assistant Window 10 Once the table illustrated in Figure 4 5 5 is completed and the values for Variance of Means V and Total sample size N are computed click on Transfer to automatically transfer these values to the main table 115 Compute Effect Size Assistant x wij Compute Effect Size Assistant Specify Multiple Factors ui Output Figure 4 5 5 Completed Compute Effect size Assistant Window 11 Now that values for Variance of Means V and Total sample size N are computed we can continue with filling in the main table For the Common Standard Deviation enter a value of 25 12 The number of covariates to be used in this study is set at 1 so enter the value 1 in the Number of covariates row Also the R Squared value has been estimated as 0 75 for this study design so enter 0 75 in the R Squared with covariates row 13 We want to calculate the attainable power give the sample size of 150 14 It has been estimated that it will cost 100 per sample unit in this study Therefore enter 100 in the Cost per sample unit row
281. y the Farrington and Manning test statistic is defined as follows 1 IF 1 IF E Prui Prm1 1 Pem2 Pru2 IF2 5 5 2 K M KM2 where K4 is the number of clusters in the treatment group M4 is the sample size per cluster in the treatment group Dry is the maximum likelihood estimator for each group proportion and IF is the inflation factor for the effect of clustering in the treatment and control groups respectively IF is defined as follows IF 1 ICC M 1 i 1 2 5 5 3 The constrained maximum likelihood where p pz A estimator of the two proportions is calculated using the following calculations K M IF gt 5 5 4 K M IF pa a 1 t 5 5 5 b 1 t p tp A t 2 5 5 6 c A A 2p t 1 p tpz 5 5 7 d p A 1 A 5 5 8 b bc d E ul ee aad 5 5 9 j lt 2 52 by c i pale reed 5 5 10 w sign 3a Vv z x cos a 5 5 11 w 3 Prm1 2ucos w b 3a 5 5 12 Prmz Prmi A 5 5 13 2 Unpooled Test Statistic This test statistic uses the estimated group proportions to calculate the standard error Its formula is as follows p p IF p21 p2 IFz K M KM2 OUnpool 5 5 14 where p and p are the estimated mean proportions for the two groups 3 Pooled Test Statistic The pooled test statistic uses the weighted average of the two proportions to calculate the standard error Its formula is as follows n 1
282. you had a design with two response variables and 2 factors Factor A and Factor B each with two levels This design would give a matrix with 2 rows and q 2 2 4columns A B A Bz A2B AB M p M H12 H13 H14 4 6 1 P2 Has H22 H23 H24 Where for example 23 is the mean of the second response of subjects in the third group Note the matrix is in this form for ease of user input The transpose of this inputted matrix is used in the power calculations In the means matrix there is also a row labelled n This row is used to specify the number of subjects per group This row need only be specified when solving for power and it is anticipated that the sample size per group will be unequal The next step for the user is to input values for the standard deviation and the correlation p These two values are used by nTerim to calculate the covariance matrix Co op as Up 2 2 2 x P 7 i Pi 4 6 2 o p op o Where is a p x p matrix Alternatively the user may manually specify the covariance matrix using the Covariance Matrix assistant When values for standard deviation and the correlation are not entered and the covariance matrix has been filled out nTerim will use the specified covariance matrix to compute power and sample size In order to calculate power a value for the group size n must be entered Entering this value in the main table assumes that group sizes are equal If it is expected that the sample sizes i
283. z Plot Power vs Sample Size LLI Open Ma Two Proportions Superiority 1 o Oi 3 I 0 05 0 05 0 05 0 05 Test Type Likelihoodscore Likelihoodscore w Likelihoodscore Likelihoodscore v Higher Proportions Better Worse Worse Worse y Worse Worse I Control Group Proportion p2 0 5 0 5 0 5 0 5 Solve using p1 p2 p1 or p1 p2 Differences x Differences y Differences Differences A Superiority Test Statistic 0 1 0 1 0 1 0 1 o Actual Value of Test Statistic 0 15 0 15 0 15 0 15 Intracluster Correlation ICC 0 001 0 001 0 001 0 001 Clusters in Treatment Group K1 14 10 9 9 Clusters in Control Group K2 14 20 36 30 Cluster Sample Size in Treatment Group M1 100 Cluster Sample Size in Control Group M2 Power Cost per sample Total study cost 100 82 313 100 100 80 936 100 100 83 726 Figure 5 5 8 Results from second Sensitivity analysis Calculate required treatment group clusters K1 given power and sample size x v 100 100 82 38 All columns This will give the same answers as for the above sensitivity calculation due to the control proportion being 0 5 in which case these values lower and higher than 0 5 are symmetric in terms of the calculation If the control group proportion were not 0 5 we would have expected different values for the two calculations 259 260 Chapter 6 References Chow S C Shao J and Wang H 2008 Sample Si
284. ze v Run E All columns Figure 5 2 4 Completed CRT Two Proportions Completely Randomized study design 225 im The next calculation is a sensitivity analysis for the treatment group number of clusters when the control group number of clusters is changed 7 Delete the values for K1 and K2 in the first column then replace the updated power with 80 Enter K1 in the control group clusters row Then copy the first column into columns 2 to 4 8 Other ratios other than K1 being equal to K2 between K1 and K2 can be calculated by using arguments for K2 such as 2 K1 or K1 2 to have K2 be twice as large as K1 for example Enter 2 K1 in column 2 for Clusters in Control Group K2 4 K1 in column 3 and 30 in column four This will give a table as per Figure 5 2 5 Query nTerim 3 File Edit View Assistants Plot Tools Window New Fixed Term Test New Interim Test Ka Plot Power vs Sample Size LUJ Open Two Proportions Inequality Co 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test 2 2 2 x2 m Test Type Unpooled x Unpooled x Unpooled x Unpooled x Control Group Proportion p2 0 06 0 06 0 06 0 06 Solve using p1 p2 p1 or p1 p2 Proportions x Proportions x Proportions x Proportions hd Test Statistic under HO 0 06 0 06 0 06 0 06 Test Statistic under H1 0 04 0 04 0 04 0 04 Intracluster Correlation ICC 0 01 0 01 0 01 0 01 Cluster
285. ze Calculations in Clinical Research Second Edition Chapman amp Hall DeMets D L and Lan K K G 1984 An Overview of Sequential Methods and their Applications in Clinical Trials Communications in Statistics Theory and Methods 13 pp 2315 2338 DeMets D L and Lan K K G 1994 Interim Analysis The Alpha Spending Function Approach Statistics in Medicine 13 pp 1341 1352 Donner A amp Klar N 1996 Statistical Considerations in the Design and Analysis of Community Intervention Trials Journal of Clinical Epidemiology 49 4 pp 435 439 Donner A amp Klar N 2000 Design and Analysis of Cluster Randomization Trials in Health Research Arnold Publishers London Fleiss J L Tytun A Ury S H K 1980 A Simple Approximation for Calculating Sample Sizes for Comparing Independent Proportions Biometrics 36 pp 343 346 Fleiss J L 1981 Statistical Methods for Rates and Proportions Second Edition Wiley Gu K Ng H K T Tang M L amp Schucany W R 2008 Testing the Ratio of Two Poisson Rates Biometrical Journal 50 2 pp 283 298 Guenther W C 1977 Sampling Inspection in Statistical Quality Control Charles Griffin and Company Limited pp 25 30 Hsieh F Y amp Lavori P W 2000 Sample size calculations for the Cox proportional hazards regression model with nonbinary covariates Controlled Clinical Trials 21 6 pp 552 560 Huffm

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