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1. Group 2 proportion n2 at time t Hazard ratio h In n1 in n2 Survival time assumption Exponential Survival Exponential Survival Exponential Survival Exponential Survival Total sample size N Power Number of events Cost per sample unit Total study cost Number of Looks 5 5 5 5 Information times Equally Spaced iy Equally Spaced x Equally Spaced 7 Equally Spaced x MaxTimes iF 1 1 1 Determine bounds Spending Function iy Spending Function Spending Function x Spending Function x Spending function O Brien Fleming 7 O Brien Fleming O Brien Fleming O Brien Fleming x A Truncate bounds No No x No z No Truncate at Futility boundaries Don t Calculate Don t Calculate iy Don t Calculate Don t Calculate Spending function O Brien Fleming 7 O Brien Fleming 7 O Brien Fleming O Brien Fleming x aoe SSS Calculate required sample sizes for given power v All columns Figure 3 3 2 Survival Test Table 8 Once all values have been entered select Calculate required sample size for given power from the drop down menu and click Run 48 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size GST Survival 1 Test significance level a 0 05 1 or 2 sided test 2 Ge cE GE Group 1 proportion ni at time t 0 3 Group 2 proportion n
2. cccccessecceeeeeeeeees 57 A LL MVOC CL VOM enn e E E E A E E E 57 Ae MENO OOE Voire E A E EE E 58 IY BY ec Be TAINO A educa E E E E E T 59 4 2 Repeated Measures Design for TWO M AaNS sccccsssececessccceesecccesececseeceeseseceseaecetenaes 71 Be TOUCA EE EE E 71 OF PD MENO ONO OY oa mora e E EE E E E E E wack tanecees 72 A EE e A E A E A EE 73 4 3 Repeated Measure for TWO Proportions ssssssseeseseeessreessrrsssrresrersseeesereserreseersseesseee 87 A Sd ATOU ON ee E E E E E E E E 87 43 2 Methodology crna ie E A EEEa 88 A o EDOS a E E E E EEA 89 4 4 One Way Analysis of Variance ANOVA ccc csseccccssseceecessececeeseceeseeeeeeseeeeeceseeneeetes 100 AAL Mro dU OW ea E E E E 100 AA MOOO OY erne nE E EET E E E NEEN 101 AA ETNO E E E E 102 4 5 Analysis of Covariance ANCOVA essssssssssseserenserrresrrresrrnserrrsreresrressrresereesrreesrereeeerss 109 Aea le IPO CUIC LION sonera E E 109 A e MENO COO ON aerer ran E NN 110 aea A E E T E E A eee eee ene eee 112 4 6 Multivariate Analysis of Variance MANOVA c cccccccssecccesecsceseseeeneceeeeeseteueeeseees 122 Aa AVEO CN CEO Gaps esspetece E E AE A E E E A N 122 4 O2 MEMOdOlOE Yesica E OT 123 AO E T O a E E E A E A E A 127 Chap or B aoe EE E E E E E E E EE 143 References c cccccccccccececccececscscscscseseececececececececetececscscstscseseseseseneeaecenenecetetetscetscscseseseneaees 143 Chapter 1 Systems Guide IM
3. Power vs Sample Size ee a a a E Power vs Sample Size 70 90 Std Dev 6 Std Dev 4 Std Dev 8 50 60 70 80 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 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 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 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 procedures outlined by Keppel 1991 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
4. 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 11 12 IM 2 5 Selecting an nQuery Advisor Design Table through nTerim A new feature added to nTerim 2 0 is the ability to open an nQuery design table through nTerim This enables the user to seamlessly transition between nTerim and nQuery By opening the Study Goal and Design window using the options section Section 2 4 the user has the full range of design tables and nQuery at their disposal Design Goal Fixed Term Means Interim Proportions C Survival Agreement C Regression a One sample t test Paired t test for difference in Means Univariate one way repeated measures analysis of variance One way repeated measures contrast Univariate one way repeated measures analysis of variance Greenhouse Geisser A Selected test is only available in nQuery Advisor Clicking OK will open the test in nQuery Advisor outlined in the previous available in bot
5. 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 entering values 3 Enter 0 05 for alpha 1 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 36 6 The looks will be equally spaced and the Pocock spending function is to be used There will be no truncation of bounds 7 Itis estimated that the cost per unit is roughly 180 so enter 180 in the Cost per Sample unit row g arrir 2 File Edit View Assistants Plot Tools Window Help _ New Fixed Term Test New Interim Test Plot Power vs Sample Size GST Two Proportions 1 eee ee E 2 1 or 2 sided test ji ee DE BE x Group 1 proportions n1 Group 2 proportions n2 Odds ratio n2 1 n1 n1 1 n2 Group 1 size ni E Group 2 size n2 Ratio n2 n1 1 1 1 1 Continuity correction off off off off Power Cost per sample unit Total study cost Number of looks 5 5 5 5 Information Times il Equally Spaced x Equally Spaced 7 Equally Spaced iy Equally Spaced iy Max times Ji 1 1 1 Determine bounds Spending
6. RM Two Means 1 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 n1 7 Group 2 size n2 Ratio n2 ni Power Cost per sample unit Total study cost pi p p Ferree tee a a 4 i E Ast Calculate required sample sizes for given power X 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 power of 85 so enter 85 in the Power row File Edit View Assistants Plot Tools Window Help J New Fixed Term Test E New Interim Test W Plot Power vs Sample Size RM Two Means 1 2 erat EEA Sa 1 or 2 sided test 2 x2 x2 Number of levels M 5 Difference in means pi p2 40 Standard deviation at each level o 80 Between level correlation p 0 5 Group 1 size ni Group 2 size n2 Ratio n2 ni 1 1 1 Power 85 Figure 4 2 15 Design Entry for Two Means Repeated Measures Study 83 fetin 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
7. Scipi Scale D SQRT Sci Standard deviation at each level o Between level correlation p Effect size A C D o SQRT 1 p Power Cost per sample unit Total study cost ma All columns Figure 4 1 8 One way Repeated Measures Contrast Test Table cipi Scale D SQRT Zc wi Compute Effect Size Assistant aa Specify Multiple Factors ij 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 Pg WORT NT ers a WUC y fr Angin Ann p File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Contrast 1 q ee a 0 05 3 Test significance level a Scale D SQRT Sci Standard deviation at each level
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9. 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 2 0 the set of system requirements are listed in full below Operating System Windows 7 or later Windows Vista Windows XP including NET Framework Service Pack 2 0 or higher Windows Server 2008 R2 or later Windows Server 2008 Windows Server 2003 Processor Either 32 bit or 64 bit processor Minimum of 450MHz processor Hard Disc 100MB for the nTerim software package review when completed 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 loc
10. 67 62 0 8 2 39658 2 39658 0 01655 0 00782 0 04324 14 34 81 96 2 38591 2 38591 0 01704 0 00676 0 05000 8 54 90 50 Ma Looks ba Specify Multiple Factors i Output Figure 3 1 9 Boundary Table for Column 2 29 30 Pocock Boundaries with Alpha 0 05 2 3 Figure 3 1 10 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 3 2 Two Proportions 3 2 1 Introduction nTerim 2 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
11. Figure 4 3 6 Re run calculation for Column 3 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 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 SDeoeqger een a a Power vs Sample Size Power vs Sample Size 2110 3110 Column 1 e Column 2 Column 3 40 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
12. Multivariate Statistical Inference and Applications John Wiley 145 146 North Central South America amp Canada Statistical Solutions Stonehill Corporate Center Suite 104 999 Broadway Saugus MA 01906 Tel 1 781 231 7680 Fax 1 781 231 7684 Toll free 1800 262 1171 Email info statsolusa com E Se SOLUTIONS LL www statistical solutions software com T 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 support statsol ie
13. 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 r is calculated for each group as a proportion of group 1 103 IM Fite Ae A p gt WI lt T FP File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size ANOVA 1 a ey a ey eee Number of groups G Compute Effect Size Assistant eee eae aaa Mean i isna sFOUD 1 N as multiple of ni Sri Sni 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 Effect Size Assistant _ l x pe Transfer W Compute Effect Size Assistant u Specify Multiple Factors Output Figure 4 4 5 Completed Compute Effect Size Assistants Table 104 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 Dev
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15. 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 ri i um Sri of 11 76 assuming that the common standard deviation is 6 105 IM 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 must copy the information 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 Plot Power vs Sample Size ANOVA 1 i 3 4 Select All Copy Cut Paste 0 32667 Fill Right Clear Table Clear Column Clear Selection b Debase ee e e e E Print Table Calculate attainable power with the given sample sizes X 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 New Fixed Term Test New Interim Test W Plot Power vs Sample Size Select All est significance level Number of groups G 3 Copy Vari
16. 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 power 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 109 110 IM 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
17. variables p row as shown in Figure 4 6 3 nQuery nTerim 2 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size Common standard deviation o Between level correlation p Group size n Total sample size N Cost per sample unit Total study cost CS D gt Calculate group size using Wilks lambda All columns x Factor Level Table Factor ABC P m dear wij Factor Level Table u Means Matrix Group Sizes u Covariance Matrix ij Specify Multiple Factors u Output Figure 4 6 2 Multivariate Analysis of Variance Design Window 5 Once a value for the number of response variables p is entered 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 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 128 File Edit View Assistants Plot Tools Window
18. 3 1 below 2 NQuery nTerim 2 0 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size L Open Manual ch Statistical Solutions Support 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 design 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
19. 4 6 10 133 134 80 62952 A 3 0 05 50 11546 0 05 0 05 94 19063 0 05 0 05 0 05 u3 Factor Level Table u Means Matrix Group Sizes jus Covariance Matrix ug Specify Multiple Factors jeg Output Figure 4 6 10 Output Power values calculated 22 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 80 63 power to test Factor A if a Pillai Bartlett Trace test statistic is used with 0 05 significance level 50 12 power to test Factor B if a Pillai Bartlett Trace test statistic is used with 0 05 significance level 94 19 power to test Factor AB if a Pillai Bartlett Trace test statistic is used with 0 05 significance level Example 2 Wilks Lambda In this example we will calculate the attainable power given a specified sample size using the Wilks Lambda method 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 Study Design Goal Fixed Term Mea
20. 85 Cost per sample unit 75 75 75 75 Totalstudy cost 7425 8700 10125 iT gt Calculate required sample sizes for given power v All columns Figure 4 2 18 Completed multiple design Repeated Measures for Two Means Table 85 86 Metin 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 title 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 Power vs Sample Size EEE fel Power vs Sample Size Column 1 e Column 2 Column 3 Column 4 40 110 130 150 170 190 210 230 250 270 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 f
21. 88 IM 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 Given two of p4 p2 or Y nTerim will compute the other using the following equation _ p2 1 py ___ _ 4 3 1 p 1 p2 l 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 The formula used to calculate power is Nip N2p2 N1q1 N22 N N1p191 N2p2q2 Power 1 Za 4 3 2 MNT 1 7 1 p M 1 apis 1 n p2q2 where is the standard normal density function and Ny 4 3 4 ary 14 3 4 q 1 p 4 3 5 q2 1 p2 4 3 6 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 al
22. 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 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Two Means 1 i 2 3 4 Test significance level a 0 05 select All po o Copy iz 7 5 40 80 Cut Paste Fill Right Clear Table Clear Column Clear Selection Print Table Calculate required sample sizes for given power v 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 84 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 dow
23. Calculator Power vs Sample Size Plot Spending Function Plot Boundaries Plot Print Main Table to Clipboard Print Looks Table to Clipboard Settings Close All if no test window open Close All List of Open Windows Help About IY 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 below ye nQuery nTerim 2 0 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size LL Open Manual ti Statistical Solutions Support Figure 2 4 1 New Design Tabs By 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 Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term Means Test C Interim Proportions C Confidence Interval Survival 0 Equivalence 0 Agreement C Regression One way analysis of variance One way analysis of variance Unequal n s 2 Single one way contrast
24. Function x Spending Function x Spending Function x Spending Function x Spending function O Brien Fleming iy O Brien Fleming x O Brien Fleming 7 O Brien Fleming 5 _ Truncate bounds No x No x No No x Truncate at Futility boundaries Don t Calculate x Don t Calculate x Don t Calculate x Don t Calculate 7 Spending function O Brien Fleming 7 O Brien Fleming 7 O Brien Fleming 7 O Brien Fleming 7 7 Calculate required sample sizes for given power X _ All columns Figure 3 2 2 Two Proportions Test Table 8 Once all the values have been entered select Calculate required sample size for given power from the drop down menu and click Run 37 File Edit View Assistants Plot Tools Window Help ij lt New Fixed Term Test _ New Interim Test Plot Power vs Sample Size GST Two Proportions 1 P m 2 3 4 Test significance level a 0 05 lor2sidedtest 1 Ge Ge Ge Group 1 proportions m1 04 Group 2 proportions n2 0 6 Odds ratio W n2 1 n1 n1 1 n2 2 25 Groupisze nl Ci Group 2 size n2 129 Ratio n2 n1 1 1 1 1 Continuity correction loff off off off Power 90 12 Cost per sample unit 180 Total study cost 46440 Number of looks 5 5 5 5 Information Times Equally Spaced Equally Spaced x Equally Spaced x Equally Spaced x Max times i 1 1 1 1 De
25. Help New Fixed Term Test New Interim Test Plot Power vs Sample Size MANOVA 1 x Factor Level Table a wij Factor Level Table i Means Matrix Group Sizes W Covariance Matrix ui Specify Multiple Factors u Output Figure 4 6 3 Enter Number of Response variables 9 Once the number of levels for each factor has been specified click the Fill button at the bottom right corner of the Factor Level Table as shown in Figure 4 6 4 10 The word Filled will now be displayed in the main table as shown in Figure 4 6 5 telling you the Factor Level Table has been completed wij Factor Level Table a Means Matrix Group Sizes ij Covariance Matrix u Specify Multiple Factors ui Output Figure 4 6 4 Factor Level Table 129 ein 11 The Means Matrix assistant table will also automatically appear guiding the user to fill out the next step in the MANOVA process Depending on the values entered in the Factor Level table the size of the means matrix will be created 2 nQuery nTerim File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size pa OAA REA A a aaa a E Nu mber c s spons Al abies Calculate group size using Wilks lambda X All columns Means Matrix Group Sizes x A aana aa ma W Factor Level Table wil Means Matrix Group Sizes Covariance Matrix ij Specify Multiple Factors ui Output Fig
26. 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 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 C One Test Interim Proportions O Two Confidence Interval Survival 0 Equivalence Agreement Regression One way analysis of variance One way analysis of variance Unequal n s f Single one way contrast f Single one way contrast Unequal n s f Two way analysis of variance Multivariate analysis of variance MANOVA Analysis of Covariance ANCOVA 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 and the Means Matrix assistant table presented in Figure 4 6 5 127 ein 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
27. 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 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 Edit gt View gt Assistants gt Plot gt Tools gt Windows gt Help gt New Open Fresh Table Save Save As Close Test Exit Fill Right Clear Table Clear Column Clear Selection 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 Specify Multiple Factor Table Compute Effect Size Randomisation Distribution Function Windows
28. Power 90 01 Cost per sample unit 120 Total study cost jas haoi gt Calculate required sample sizes for given power All columns Figure 4 3 4 Completed Repeated Measures Design for Two Proportions 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 has 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 91 92 File Edit View Assistants Plot Tools Window Help New Fixed Term Test _ New Interim Test Plot Power vs Sample Size RM Two Proportions 1 ae 1 2 3 i 4 Test significance level a 0 05 0 05 1 or 2 sided test 2 z 2 2 2 Number of levels M 3 3 Between level correlatio
29. Single one way contrast Unequal n s f 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 launched 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 seal R nQuery nTerim 20 eal ee File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size LLI Open Manual RA Statistical Solutions Support _ ANCOVA 1 xX Test significance level a E CBRE TENG tt EITTIE ersten Shes Number of groups G Variance 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 Calculate required sample sizes for given power Compute Effect Size Assistant Variance of means V Total sample size N N as multiple of n1 Sri Sni ni x ri ni ni v Run E An columns Hel
30. Size rox LL Open Manual d Statistical Solutions Support GST Two Means 1 1 or 2 sided test Group 1 mean p1 Group 2 mean p2 Difference in means p1 p2 Group 1 standard deviation o1 Group 2 standard deviation o2 _ Effect size 3 Group isize ni Group 2 size n2 Ratio n2 n1 Power Cost per sample unit _ Total study cost __ Number of looks Ts 5 5 l Information times __MaxTimes Determine bounds 4 mo ooo Equally Spaced Equally Spaced Equally Spaced 1 1 1 Spending Function Spending Function x Spending Function x O Brien Fleming O Brien Fleming 7 O Brien Fleming zi Calculate required sample sizes for given power v C All columns Looks C l O _ Lower bound X Futility bound _ Nominalalpha __Incremental alpha Cumulative alpha 0 2 0 4 wi Looks a Specify Multiple Factors w ij Output xX Help x Two group z test for the 5 difference between independent means Enter a value for alpha a the significance level for the test and select a one or two sided test Specify two of effect size power and sample size and nTerim will compute the third Test significance level a Alpha is the probability of rejecting the null hypothesis of equal means when it is true the probability of a Type error input Advice Enter 0 05 a frequent standard Entry Options 0 001 to 0 20 Boundaries Graph
31. 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 Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term Means E Test Interim Proportions Two Survival 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 47 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 i ween lr nQuery nTerim 2 Ol X File Edit View Assistants Plot Tools Window Help 7 lt New Fixed Term Test _ New Interim Test W Plot Power vs Sample Size LLI Open Manual d Statistical Solutions Support GST Survival 1 YX Group 1 proportion ni at time t 1 or 2 sided test p vil x 1 1 x
32. Times 1 1 1 1 Determine bounds Spending Function Spending Function Spending Function Spending Function Spending function O Brien Fleming O Brien Fleming gt O Brien Fleming gt O Brien Fleming a Truncate bounds No z No z No z No Truncate at Futility boundaries Dont Calculate x Don t Calculate Don t Calculate Spending function x O Brien Fleming x O Brien Fleming x O Brien Fleming x Phi FERFIT TEAT IN gt Calculate required sample sizes for given power v C All columns Figure 3 1 3 Completed Two Means Test Table The boundaries calculated are shown in Figure 3 1 4 Looks 0 4 0 6 0 8 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 0 00000 0 00079 0 00683 0 01681 0 02558 Cumulative alpha 0 00000 0 00079 0 00762 0 02442 0 05000 Exit probability 0 03 10 17 35 07 29 92 15 17 Cumulative exit probability 0 03 10 21 4527 75 19 90 36 Nominal beta j Incremental beta Cumulative beta Exit probability under HO oo Cumulative exit probability under HO 3 Looks a Specify Multiple Factors u Output Figure 3 1 4 Boundary Table for Two Means Test 25 fet 26 10 Finally the boundaries calculated in the table in Figure
33. alpha and beta spending functions are the same In Table 3 1 1 we list the alpha spending functions available in nTerim 2 0 Table 3 1 1 Soending Function Equations Vict Za 2 O Brien Fleming a t 2 1 o VT a t aln 1 e 1 T l e Hwang Shih DeCani a t a ae 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 19 20 IM 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 2 0 represent the critical values at each look These boundaries are constructed using the alpha and beta spending functions Users in nTerim 2 0 are given the option to generate boundaries for early rejection of the null hypothesis Hj using the alpha spending function or to generate boundaries for early rejection of either the null or alternative hypothesis Hy 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 Hy was first proposed by Lan and DeMets 1983 we refer to such boundaries in nTerim 2 0 as efficacy boundaries Building on the work of Lan
34. 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 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 H4 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
35. compute effect size assistant nTerim will then calculate the variance of means using the formula ba rj uj n V 7 4 5 1 ima li where i 2 a 4 5 2 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 r 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 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 a The non centrality parameter is then calculated using the equation o o V A nG 4 5 3 Og where N n 4 5 4 G and o 1 p o 4 5 5 where oZ is the within group variance after considering the covariates and p is the coefficient of multiple determination estimated by R7 Using these two values nTerim calculates the power of this design as the probability of being greater tha
36. determine the test boundaries 59 56 Chapter 4 Fixed Term Design 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 to overall tests This usually makes tests of planned contrasts easier to interpret and more powerful 57 5
37. for Column 1 is given below 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 Power vs Sample Size 1150 Group 1 rr Group 2 750 950 Power 90 83 Sample Size N1 N2 Sample Size 915 Figure 3 3 8 Power vs Sample Size Plot 51 52 IM 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 is 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 us
38. 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 New Interim Test Plot Power vs Sample Size ANCOVA 1 Ey a ey ee eee Test significance level a 0 05 Number of groups G 3 farane of means r Jo Common standard deviation o Number of covariates c Total sample size N Cost per sample unit Total study cost Calculate attainable power with the given sample sizes v All columns Compute Effect Size Assistant x Group 2 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 l Compute Effect Size Assistant x Variance of means V 15 972 Total sample size N 120 3 Compute Effect Size Assistant jug Specify Multiple Factors ui 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
39. has been completed Factor Level Table x eves Alpha Power wij Factor Level Table u Means Matrix Group Sizes ui Covariance Matrix ui Specify Multiple Factors ij Output Figure 4 6 14 Factor Level Table 137 fein 11 The Means Matrix assistant table will also automatically appear guiding the user to fill out the next step in the MANOVA process Depending on the values entered in the Factor Level table the size of the means matrix will be created nQu y aT gt File Edit View Assistants Plot Tools Window Help E New Fixed Term Test _ New Interim Test W Plot Power vs Sample Size MANOVA 1 a 2 3 4 Number of response variables p 3 Factor leveltable Filled Means matrix Between level correlation p E Total sample size N ts Cost per sample unit Total study cost Calculate group size using Wilks lambda All columns Means Matrix Group Sizes x 7 A 1 2 3 4 5 6 7 8 2 3 n W Factor Level Table ij Means Matrix Group Sizes Covariance Matrix ui Specify Multiple Factors ui Output Figure 4 6 15 Means Matrix Group Sizes Assistants Table 8 12 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 13 Th
40. is calculated using the formula PBT tr HT7 The transformation of this test statistic to an approximate F is given by p ifi WWE 1 n df PBT n s min a p a q 1 df ap df s W r p s Hotelling Lawley Trace The test statistic for Hotelling Lawley trace is calculated using the formula HLT tr HE The transformation of this test statistic to an approximate F is given by p df ta 1 n df HLT s YT Hrs df ap df s N r p 1 2 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 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 Faz af a and then the noncentrality parameter A Where A ndf 4 6 26 ITY 125 IM Using these two values nTerim will calculate the power of this design as the probability of being greater than Far afo ON a non central F distribution with non centrality parameter 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 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 126 4 6 3 Examples Example 1 Pillai Bartlett Trace
41. ka Analysis of Covariance ANCOVA 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 3 for the number of groups G as shown in Figure 4 4 4 102 ne juery nlerir File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size ANOVA 1 T es ee es ee ficance level a i Variance of means V N as multiple of n1 Sri Xni ni Total sample size N Cost per sample unit Total study cost TT ak byt eee st ciar s i Aoi PEA q 7 Se ee Calculate required sample sizes for given power v All columns Figure 4 4 2 One way Analysis of Variance Test Table Compute Effect Size Assistant x Total sample size N N as multiple of ni Sri Sni ni Tae wi Compute Effect Size Assistant i Specify Multiple Factors ui Output Figure 4 4 3 Compute Effect Size Assistant Window 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
42. 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 been estimated that it will cost S80 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 J ANCOVA 1 L i a 3 4 moos SERA Copy Cut Paste 15 97222 Fill Right Clear Table Clear Column 80 Clear Selection 9600 Print Table oo gt Calculate attainable power with the given sample sizes w All columns Figure 4 5 12 Fill Right Shortcut Feature elim 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 T
43. of the window A Study Goal and Design window will appear Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term Means One Test C Interim Proportions C Confidence Interval Survival 0 Equivalence 0 Agreement C Regression One sample t test i Paired t test for difference in Means Univariate one way repeated measures analysis of variance a One way repeated measures contrast Univariate one way repeated measures analysis of variance Greenhouse Geisser Figure 4 1 7 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 8 and the effect size assistant table shown in Figure 4 1 9 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 10 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 63 64 mM Sy apenas VOL NM ral we Ps piers File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Contrast 1 ea i i p J z j gt 1 s Ay a Number of levels Contrast C
44. power x 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 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 ui of 2 with a scale D SQRT Sci of 1 41421 assuming a standard deviation at each level of 6 and a between level correlation of 0 2 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
45. stages prior to the conclusion of the trial As a result the alpha and beta values applied at each analysis or look an adjusted is needed to preserve the overall type 1 and type 2 errors The alpha and beta values used at each look are calculated based upon the test hypothesis the spending function chosen the number of looks to be taken during the course of the study as well as the overall type 1 and type 2 error rates 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 2 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 Functionally the alpha and beta spending functions are the same In Table 3 1 1 we list the alpha spending functions available in nTerim 2 0 Table 3 1 1 Soending Function Equations Vict Za 2 O Brien Fleming a t 2 1 o VT a t aln 1 e 1 T l e Hwang Shih DeCani a t a ae 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 43 44 IM The common element among most of the different spending functions is to use lower error values f
46. test continuous outcome f Wilcoxon Mann Whitney rank sum test ordered categories Two group univariate repeated measures ANOVA Greenhouse Geisser correction f 2x2 Crossover Design a Repeated Measures for two means 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 deviation 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 78 MaA d anm 2 File Edit View Assistants Plot Tools Window Help i _ New Fixed Term Test i New Interim Test W Plot Power vs Sample Size RM Two Means 1 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 n1 Group 2 size n2 Ratio n2 n1 1 1 1 Bi SSS Power TARTINE a Tiri as l i U Co Calculate required sample sizes for giv
47. 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 Proportion row 89 File Edit View Assistants Plot Tools Window Help ij New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Two Proportions 1 Number of levels M Between level correlation p Group 1 proportion p1 ___Group2proportion p2 0 Odds ratio W p2 1 p1 pi 1 p2 Group 1 size ni Group 2 size n2 Ratio n2 n1 1 1 1 1 Power Cost per sample unit Total study cost F apa Calculate required sample sizes for given power v AH 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 New Fixed
48. 1681 0 02558 Cumulative alpha e 0 00079 0 00762 0 02442 0 05000 Exit probability 0 03 9 98 34 73 29 96 15 36 Cumulative exit probability 0 03 10 01 44 75 74 71 90 07 Nominal beta Incremental beta Cumulative beta Exit probability under HO ra Cumulative exit probability under HO a Looks ha Specify Multiple Factors u Output Figure 3 3 4 Boundary Table for Column 1 50 In the second column enter the same parameters as above but change the Group 2 proportion to 0 40 Select Run File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test a Plot Power vs Sample Size Lu Open Manual cf Statistical Solutions Support GST Survival 1 xX 1 2 3 4 Test significance level a 0 05 0 05 T 1 or 2 sided test 2 x2 1 x Group 1 proportion ni attimet 0 3 0 3 Group 2 proportion n2 at timet 0 45 0 4 Hazard ratio h In n1 In n2 1 508 1 314 Survival time assumption Exponential Survival Exponential Survival Exponential Survival Exponential Survival Total sample size N 1409 888 Power 90 07 90 02 Number of events 256 578 Cost per sample unit 100 100 Total study cos 40900 88800 Number of Looks 5 5 5 5 Information times Equally Spaced x Equally Spaced 7 Equally Spaced x Equally Spaced x Max Times 1 1 1 1 Determine bounds Spending Function x Spending Function 7 Spend
49. 2 at timet 0 45 Hazard ratio h In n1 In n2 1 508 Survival time assumption Exponential Survival Exponential Survival Exponential Survival Exponential Survival Total sample size N 409 Power 90 07 Number of events S o 256 Cost per sample unit 100 otal study co 40900 Number of Looks 5 5 5 5 Information times Equally Spaced iy Equally Spaced x Equally Spaced x Equally Spaced Max Times 1 1 1 1 Determine bounds Spending Function x Spending Function Spending Function x Spending Function x Spending function O Brien Fleming x O Brien Fleming x O Brien Fleming 7 O Brien Fleming x Phi Truncate bounds No x No z No x No Truncate at Futility boundaries Don t Calculate Don t Calculate Don t Calculate Spending function O Brien Fleming O Brien Fleming O Brien Fleming oe r a rs 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 O fit 2 3 S 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 0 00000 0 00079 0 00683 0 0
50. 3 1 4 are automatically plotted as illustrated in Figure 3 1 5 Boundaries Graph O Brien Fleming Boundaries with Alpha 0 05 1 2 3 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 u1 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 O Brien Fleming spending function is used to determine the test boundaries Drift 3 29983 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 Study Goal And Design pr Som y Eo Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two C Survival Agreement se Group Sequential Test of Two Means Figure 3 1
51. 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 the bottom right corner of the plot indicated the exact values for Power and Sample Size for each identifier on the graph 121 122 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
52. 5 0 55 0 45 0 55 Group 2 proportion p2 0 39 0 39 0 51 0 51 Odds ratio W p2 1 p1 pi 4 p2 0 78142 0 5231 1 27211 0 85158 Group 1 sie ni Group 2 size n2 Ratio n2 n1 1 1 1 1 Power 30 90 90 90 Cost per sample unit 100 100 100 100 Total study cost iaj o mo oo D Figure 4 3 11 Design Entry for Multiple columns 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 appropriate 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 View Assistants New Fixed Term Test New Interim Test Plot Tools Window Help Plot Power vs Sample Size RM Two Proportions 1 3 a a 2 Test significance level a 0 05 0 05 0 05 0 05 1 or 2 sided test B2 7 2 x 2 x 2 x Number of levels M 3 3 3 3 Between level correlation p 0 4 0 4 0 4 0 4 Group i preportion pt 0 45 0 55 0 45 0 55 Group 2 proportion p2 0 39 0 39 0 51 0 51 Odds ratio W p2 1 p1 pi 1 p2 0 78 142 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 90 Cost per sample unit 100 100 100 100 Total study cost 170400 24200 174600 392400 BLE m g
53. 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 27 28 gt nQuery nTerim 2 0 File Edit View Assistants Plot Tools Window Help New Fixed Term Test _ New Interim Test W Plot Power vs Sample Size LLI Open Manual h Statistical Solutions Support a 3 Test significance level a 0 05 1 or 2 sided test 2 2 2 2 x Group 1 mean pi 220 Group 2 mean p2 200 Difference in means pi p2 20 i Group 1 standard deviation o1 30 Group 2 standard deviation a2 30 Effect size 5 0 667 Group 1 size n1 57 Group 2 size n2 57 Ratio N2 N1 1 1 1 1 Power 90 33 Cost per sample unit 250 f Total study cost 28500 i Number of looks 5 5 5 5 Information times Equally Spaced Equally Spaced x Equally Spaced x Equally Spaced 7 Max Times 1 1 1 1 Determine bounds Spending Function x Spending Function M Spending Function x Spending Function x Spending function See Pocock x O Brien Fleming x O Brien Fleming x O Brien Fleming x Phi 1 Truncate bounds No z No z No z No x Truncate at Futility boundaries Don t Calculate x Dont Calculate x Don t Calculate x Don t Calculate iy Spending function O Brien Fleming x O Brien Fleming 7 O Brien Fleming O Brien Fleming 7 Phi ke j gt Figure 3 1 7 Complete Two Means Te
54. 8 IM 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 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 ea 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 o 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 C A Dov 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 degrees of freedom 4 1 3 The non centrality parameter A is then calculated usin
55. Assistant table in order to calculate values for the Contrast C and Scale D parameters 67 68 IM ag yes d Newey d File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Contrast 1 Number of levels J Contrast C Scipi Scale D SQRT Sci Standard deviation at each level o Between level correlation p Effect size A C D o SQRT 1 p Power Total study cost i l A M zl l g j ce F rs Calculate required sample sizes for given power All columns 10 Figure 4 1 14 One way Repeated Measures Contrast Test Table Compute Effect Size Assistant x trast C Jci pi Scale D SQRT Sci wi Compute Effect Size Assistant l3 Specify Multiple Factors u Output Figure 4 1 15 Compute Effect Size Assistant Table 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 In order to calculate a value for Effect Size two parameters need to be calculated first the Contrast C and Scale D The mean for each level and the corresponding coefficient value need to be entered in the Compute effect Size Assistant table 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 For the Coefficient values f
56. C All columns Figure 4 2 11 Altered columns for comparison 80 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 Calculate 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 a afat File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size _ RM Two Means 1 2 3 O Test significance level a 0 05 0 05 0 05 0 05 0 05 1 or 2 sided test 2 2 2 2 x 2 7 Number of levels M 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 ni 33 28 24 46 24 Group 2 size n2 33 28 24 46 24 Ratio n2 ni 1 1 1 1 1 Power 90 85 80 90 90 Cost per sampl
57. 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 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 Hwang I K Shih W J and deCani J S 1990 Group Sequential Designs using a Family Type Error Probability Soending 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 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 Calculation
58. Goal And Design es Analysis Method Test C Confidence Interval 0 Equivalence Two sample t test Student s t test equal variances f 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 f Wilcoxon Mann Whitney rank sum test ordered categories Two group univariate repeated measures ANOVA Greenhouse Geisser correction f 2x2 Crossover Design a Repeated Measures for two means 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 82 nQuery nTerim z File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size
59. 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 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 and beta values applied at each analysis or look an adjusted is needed to preserve the overall type 1 and type 2 errors The alpha and beta values used at each look are calculated based upon the test hypothesis the spending function chosen the number of looks to be taken during the course of the study as well as the overall type 1 and type 2 error rates 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 2 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 Functionally the alpha and beta spending functions are the same In Table 3 1 1 we list the alpha spending functions availa
60. 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 improvements in depression life satisfaction and physical health are the response variables and our null hypothesis is that a subject s treatment has no effect on any of the three different ratings As there are three response variables MANOVA is used to test this hypothesis 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 field
61. STATISTICAL Y SOLUTIONS Version 2 0 User Manual nTerim 2 0 4500 Airport Business Park Cork Ireland Web www statsol ie Email support statsol ie Tel 353 21 4839100 Fax 353 21 4840026 Printed in the Republic of Ireland User Manual Statistical Solutions Ltd Stonehill Corporate Center Suite 104 999 Broadway Saugues MA 01906 Web www statsolusa com Email info statsolusa com Tel 1 781 231 7680 Fax 1 781 231 7684 No part of this manual may be reproduced stored in a retrieval system transmitted translated into any other language or distributed in any fo rm by any means without prior permission of Statistical Solutions Ltd Aali Statistical Solutions Ltd nTerim 2 0 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 sing
62. Study Goal and Design Window 2 Now you have opened the test table as illustrated in Figure 3 1 2 you can begin entering values 3 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 4 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 23 24 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 truncation of bounds nQuery nTerim 2 o Xx File Edit View Assistants Plot Tools Window Help _ New Fixed Term Test New Interim Test W Plot Power vs Sample Size LLI Open Manual Statistical Solutions Support GST Two Means 1 xX Paar fest significance level a bo 1 or 2 sided test 1 Ee 1 BE M Group 1 mean p1 Group 2 mean p2 Difference in means pi p2 Group 1 standard deviation o1 Group 2 standard deviation a2 Effect size 3 Group 1 size ni Group 2 size n2 Ratio N2 N1 1 1 i i Power Cost per sample unit Total study cost Number of looks j5 5 5 5 Information times j Equally Spaced iy Equally Spaced gt Equally Spaced Equally Spaced Max Times 1 1 1 1 Determine bounds Spending Function gt Spending Function x Spending Functio
63. Table wiJ Means Matrix Group Sizes ij Covariance Matrix Specify Multiple Factors ui Output Figure 4 6 8 Covariance Matrix Window 132 18 Now we have entered all the information required to calculate the attainable Power given a specified sample size 19 The final step is to select which method we want to use In this case we want to use the Pillai Bartlett Trace approach 20 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 9 below Qu gt y on eri in File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size MANOVA 1 T ee eee 4 Factor level table Means matrix Number of response variables p common standard devation 2 Total sample size N 61 gt ARE Pee om i i Run AI columns Calculate group size using Wilks lambda Calculate group size using Pillai Bartlett trace x Calculate group size using Hotelling Lawley trace i Calculate power using Wilks lambda a eT Calculate power using Pillai Bartlett trace 4 2 Calculate power using Hotelling Lawley trace 1 4 J6 8 7 5 6 a Figure 4 6 9 Selecting Type of Test to Run 21 Once the Run button is clicked the Factor Level Table appears again in the Assistants window This is where the output Power values are displayed as illustrated below in Figure
64. Term Test New Interim Test V Plot Power vs Sample Size RM Two Proportions 1 Pho 2 st l P Test significance level a 0 05 1 or 2 sided test 2 2 w 2 x2 x Number of levels M 3 Between level correlation p 05 Group 1 proportion p1 0 45 Group 2 proportion p2 0 55 Odds ratio W p2 1 p1 pi 1 p2 1 49383 Group 1 size ni Group 2 size n2 Ratio n2 n1 it i Power 90 Cost per sample unit LHE um D Figure 4 3 3 Design Entry for Two Proportions Repeated Measures Study 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 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size z 2 4 Test significance level a 0 05 1 or 2 sided test 2 2 2 2 Number of levels M 3 i Between level correlation p 0 5 Group 1 proportion p1 0 45 i Group 2 proportion p2 0 55 Odds ratio W p2 1 p1 pi 1 p2 1 49383 Group 1 size n1 349 Group 2 sie n2 349 Ratio n2 ni 1 1 1 1
65. Window Help a New Fixed Term Test New Power vs Sample Size Plot i2 LL Open Manual fy 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 i al O Brien Fleming Boundaries with Alpha 0 05 1 2 3 Figure 2 7 2 Example of a Boundary Plot 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 Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size LLI Open Manual d 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 70 90 Total sample size N Sample Size 59 Figure 2 7 4 Pow
66. aenszel 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 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 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 94 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Two Proportions 1 est significance level a 1 or 2 sided test 2 2 2 2 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 Group 1 size ni Group 2 size n2 Ratio n2 ni 1 1 i i Power i Cost per sample unit Total study cost BHL m r Calculate r
67. alculate the variance of means using the formula D n n V 7 4 4 1 i 1 i where i 2 E 4 4 2 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 r 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 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 V 4 4 3 g2 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 pr 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 NAA 4 4 4 Using these two values nTerim calculates the power of this design as the probability of being greater than For DF a ON a non central F distribution with non centrality parameter In order to calculate sample size a value for power must be specified nTerim does not use a closed form eq
68. ample 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 bottom right corner of the plot indicated the exact values for Power and Sample Size for each identifier on the graph 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 x 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 wi Specify Multiple Factors u3 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 99 100 IM 4 4 One Way Analysis of Variance ANOVA 4 4 1 Introduction This table facili
69. ance of means V 11 76 Cut Common standard deviation o 6 Paste Effect size A V g 0 32667 Power 94 82 Fill Right N as multiple of n1 Sri Sni ni 25 Clear Table Total sample size N 50 Clear Column Cost per sample unit 85 Clear Selection Total study cost 4250 io Print Table me Calculate attainable power with the given sample sizes v All columns Figure 4 4 8 Paste contents of Column 1 into Column 2 106 3 Once the contents of Column 1 have been copied over to Column 2 you can change the value of the Common 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 File Edit View Assistants Plot Tools Window Help lt New Fixed Term Test New Interim Test Plot Power vs Sample Size ANOVA 1 OT eem I 1 Test significance level a 0 05 0 05 Number of groups G i 3 3 Variance of means V 11 76 11 76 Common standard deviation J6 4 Effect size A V o 0 32667 0 735 Power 94 82 99 98 Nas multiple of ni Sri Sni ni 2 5 2 5 Total sample size N 50 50 Cost per sample unit i85 85 a otal study yst 4250 4250 VN eenod Calculate attainable power with the given sample sizes v 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 dis
70. ated in the References section of the manual If further clarification is required please contact our support statisticians by email at support statsol ie 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 statistical solutions software com and iii you can email us for technical help at support statsol ie 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 nTerim 2 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 th
71. ation o2 30 30 Effect size 5 0 667 0 667 Group 1 size n1 57 43 Group 2 size n2 57 86 Ratio N2 N1 Ji 2 Power 90 33 90 5 Cost per sample unit 250 250 Total study cost 28500 32250 Number of looks 5 5 Information times Max Times 1 1 Determine bounds Spending function Pocock iy Pocock Truncate bounds No x No Truncateat Futility boundaries Dont Calculate Dont Calculate Spending function O Brien Fleming O Brien Fleming HE m J 2 5 1 Spending Function gt Spending Function x Spending Function x Spending Function x O Brien Fleming z No Don t Calculate O Brien Fleming 5 Equally Spaced x Equally Spaced 7 Equally Spaced x Equally Spaced x 1 x O Brien Fleming No Dont Calculate O Brien Fleming zl ix gt d All columns Figure 3 1 8 Comparison of two separate Means Tests 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 Looks 0 4 2 43798 2 42677 2 43798 2 42677 Nominal alpha 0 01477 0 01523 Incremental alpha 0 01477 0 01139 Cumulative alpha 0 01477 0 02616 Exit probability _ 20 00 26 18 Cumulative exit probability 20 00 46 19 Nominal beta Incremental beta _ Cumulative beta Exit probability under HO 0 6 2 41014 2 41014 0 01595 0 00927 0 03545 21 44
72. ble in nTerim 2 0 Table 3 1 1 Soending Function Equations Vict Za 2 O Brien Fleming a t 2 1 o VT a t aln 1 e 1 T l e Hwang Shih DeCani a t a ae 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 31 32 IM 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 2 0 represent the critical values at each look These boundaries are constructed using the alpha and beta spending functions Users in nTerim 2 0 are given the option to generate boundaries for early rejection of the null hypothesis Hj using the alpha spending function or to generate boundaries for early rejection of either the null or alternative hypothesis Hy 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 Hy was first proposed by Lan and DeMets 1983 we refer to such boundaries in nTerim 2 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 spend
73. boundaries calculated are shown in Figure 3 3 11 Looks 0 6 2 65511 2 62320 2 58958 2 34880 2 27923 2 65511 2 62320 2 58958 2 34880 2 27923 Futility bound Nominal alpha 0 00793 0 00871 0 00961 0 01883 0 02265 Incremental alpha 0 00793 0 00684 0 00602 0 01464 0 01457 Cumulative alpha 0 00793 0 01477 0 02079 0 03543 0 05000 Exit probability 5 20 8 79 10 68 34 58 26 06 Cumulative exit probability 5 20 13 99 24 68 59 26 85 32 Nominal beta Incremental beta Cumulative exit probability under HO ud Looks a Specify Multiple Factors i Output Figure 3 3 11 Boundary Table for Pocock Spending Function 54 10 Finally the boundaries calculated in the table displayed in Figure 3 3 11 are automatically plotted as illustrated in Figure 3 3 12 sources Graph a Pocock Boundaries with Alpha 0 05 2 3 4 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 proportional These results assume that 5 sequential tests are made and the Pocock spending function is used to
74. ck on Transfer to automatically transfer these values to the main table Compute Effect Size Assistant x 4 kevet levetz kevet3 wij Compute Effect Size Assistant Specify Multiple Factors ij Output Figure 4 1 5 Completed Compute Effect Size Assistant Table 61 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 Plot Power vs Sample Size EE 2 3 4 Test significance level a 0 05 Number of levels M 3 Contrast C Scipi 2 Scale D SQRT Sci2 1 41421 Standard deviation at each level o 6 Between level correlation p 0 2 Effect size A C D o SQRT 1 p 0 26352 Power 89 95 Group size N 152 Cost per sample unit 100 Total study cost 45600 TT LJ Calculate required sample sizes for given
75. correction The steps are e Obtain O by solving Equation 3 2 2 given that N4 R p4 p2 p are known e Obtain power using the algorithm by Reboussin et al 1992 and Jennison amp Turnbull 2000 Calculate missing proportion given N Nz a power and the other proportion Calculate p given p gt 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 p the roots of which give p Similarly if pis specified the roots give the values of pp Calculate p given pz with 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 p the roots of which give p Similarly if pis specified the roots give the values of pp 35 IM 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 Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term Means i Test Interim Proportions Two Survival Group Sequential Test of Two Proportions
76. dow Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size _ RM Two Means 1 3 4 C Test significance level a 0 05 0 025 1or 2sided test 2 2 2 2 x Number of levels M 4 Difference in means pi p2 10 10 Standard deviation at each level a 20 20 Between level correlation p 0 5 0 5 Group 1 size ni 53 63 i Group 2 size n2 53 63 Ratio n2 ni 1 1 1 1 Power 90 90 i Cost per sample unit 90 90 Total study cost 9540 11340 p s D Calculate required sample sizes for given power v All columns Figure 4 2 4 Re run calculations to update Column 2 10 Now we are going 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 75 IM 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 b
77. e 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 ag WEEE aS gt WJUCTY Age oy gt a File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Contrast 1 d C E eee Test significance level a 0 05 3 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 Calculate required sample sizes for given power Compute Effect Size Assistant Mean Coefficient v i uw i Contrast C Sci pi Scale D SQRT Zci 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 cli
78. e u Means Matrix Group Sizes u4 Covariance Matrix jus Specify Multiple Factors ug Output Figure 4 6 20 Output Power values calculated 21 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 across 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 Chapter 5 References 143 144 IM Chow S C Shao J and Wang H 2008 Sample Size Calculations in Clinical Research Second Edition Chapman amp Hall DeMets D L and Lan K K
79. e 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 oco stn i O File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size o 3 a A e Tect cinnificance level o lest SHMNIKANCE EVE U NITO oD 5 i n i mE r eae gt RUE ZS ted tes io K a K Number of levels M Sie Standard deviation at each level o Between level correlation p Group 1 size n1 63 Group 2 size n2 63 Ratio n2 ni a kd 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 76 Power vs Sample Size Power vs Sample Size Column 1 re 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 represent
80. e 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 simply clicking on a column the output statement will be presented 10 IM 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 Study Goal And Design Design Goal No of Groups Analysis Method O Fixed Term Means Test Interim Proportions Survival cE Group Sequential Test of Two Means Cancel Figure 2 4 4 Open New Interim Design The options for Interim term designs are presented in Fig
81. e 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 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 and beta values applied at each analysis or look an adjusted is needed to preserve the overall type 1 and type 2 errors The alpha and beta values used at each look are calculated based upon the test hypothesis the spending function chosen the number of looks to be taken during the course of the study as well as the overall type 1 and type 2 error rates 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 2 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 Functionally the
82. e for selecting 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 Aali Acknowledgements We would like to sincerely thank all those who made the production of Statistical Solutions nTerim 2 0 possible The Statistical Solutions Team Eoghan Murphy Andrew Grannell Brian O Toole Aisling Noonan Brendan Nyhan Diana Scriven Helen Murphy Kevin Connolly Caroline Costello Kevin Sievewright Mary Byrne Special Thanks to Brian Sullivan and Niall Fitzgerald Contents ET a tee no ee ene en een ere 1 Systems CUI arse acess sso esc sa sen everest ep ce vee aan eres eva vrs sates eins E 1 1 1 System Requirements wsicescunsansceauessacnweceomcuneewweesnes crecaeeapdesnundeueendsavsecnewaateandensteawenassecsears 2 E E EE a E EE A E E N E A 2 ES UPOO e E E E N E E EEE 2 CRIP OT aeai E E E E A E 4 Getting Started Guide 2 0 0 0 cccccccssecccesececeeeeceeeececeeecesaeseeseeseceeeeecesseeesseaeceseaecesseneeesges 4 Zk SURV TOU A e voce ouncw send eins seesenase E
83. e 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 alphabetical order You can locate nTerim under the title nQuery Advisor nTerim 2 0 Click on this folder and then select nQuery Advisor nTerim 2 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 New Fixed Term Test New Interim Test Plot Power vs Sample Size Figure 2 2 1 Home Window IM 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
84. e 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 Goa And 0 on il Design Goal No of Groups Analysis Method Fixed Term Means Test Interim Proportions Confidence Interval Survival 0 Equivalence Agreement C 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 f Wilcoxon Mann Whitney rank sum test ordered categories Two group univariate repeated measures ANOVA Greenhouse Geisser correction 2x2 Crossover Design a Repeated Measures for two means 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
85. e 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 1 2 3 4 23 4 2 5 1 M 2 1 2 1 4 1 2 1 42 1 1 4 1 3 6 8 7 4 5 64 4 4 4 5 4 6 12123 42 3 42 5 141 2 2 1 4 12 1 42 1 1 4 122 1 138 File Edit View Assistants Plot Tools Window Help _ New Fixed Term Test New Interim Test Plot Power vs Sample Size LLJ Open Ma MANOVA 1 A R 2 li 3 4 Number of response variables p 3 Means matrix as Common standard deviation o Between level correlation p Group size n Total sample size N Cost per sample unit Total study cost All columns Means Matrix Group Sizes x i dR 2 3 4 _ 5 6 7 8 1 1 1 2 1 2 3 2 2 4 Factor Level Table u Means Matrix Group Sizes Covariance Matrix ui Specify Multiple Factors i Output Figure 4 6 16 Completed Means Matrix Group Sizes Assistant Table 14 Enter this matrix in the Means Matrix Assistant table as illustrated in Figure 4 6 16 and then click the Fill button at the bottom right corner of the Means Matrix assistant table 15 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 that all groups have equal sample size In this event nTerim will automatically updat
86. e power The steps are e Obtain by solving Equation 3 1 1 given that N4 R 44 U2 0 and o are known e Obtain power using the algorithms and procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 21 IM Calculate Means given all other information Given a N group standard deviations 01 o2 R or Nz power 1 f time points and type of spending function The requirement is to obtain either u or u given the other The steps are e Obtain by solving Equation 3 1 1 given that N4 R 4 U2 0 and oz are known e Equation 3 1 1 can be expressed as a quadratic in 4 oru The roots give the unknown LL By default nTerim assumes that 44 lt u and will select the appropriate root 22 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 5 3 a Gee Study Goal And Design a Design Goal No of Groups Analysis Method C Fixed Term Means Test Interim Proportions Two C Survival Group Sequential Test of Two Means Figure 3 1 1
87. e this matrix once we have entered a value for Group Size in the main design table 16 The next step in the MANOVA process is to generate the Covariance Matrix We can do this by to entering values for common standard deviation and correlation so nTerim can create the matrix automatically 139 140 fein 17 In the Common standard deviation row enter a value of 2 In the Between level correlation row enter a value of 0 6 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 The total sample size is also automatically calculated and given in the Total sample size N row Notice that the Means Matrix in Figure 4 6 17 has now been updated with the sample size per group File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size MANOVA 1 ET ae a Number ofresponse variables Factorkveltable Sled 3 Group size n Means Matrix Group Sizes x B bankan ne l NE SN a A N ETN AA A rme eanne N ON w A be fb bhi A AN S 1 1 1 2 a k 4 4 uF Factor Level Table ij Means Matrix Group Sizes W Covariance Matrix ui Specify Multiple Factors a Output Figure 4 6 17 Completed MANOVA Design Table 18 The generated covariance matrix can be viewed in the Covariance Matrix window as shown in Figure 4 6 18
88. e unit 65 65 65 65 65 Total study cost 3640 3120 5980 3120 LE V All columns 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 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 81 IM 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 4 Design Goal No of Groups Fixed Term Means Interim Proportions C Survival 0 Agreement C Regression Study
89. ed to determine the test boundaries 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 Study Goal And Design Design Goal No of Groups Analysis Method O Fixed Term O Means Test Interim Proportions Two Survival Group Sequential Test of Two Survivals OK Cancel Figure 3 3 9 Study Goal and design Window 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 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 This study planned for 4 interim analyses Including the final analysis this requires Number of Looks to be 5 The Pocock spending function is to be used however the looks will not be evenly spaced 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 It is estimated that the cost per unit is roughly 100 so enter 100 in the Cost per Sample unit row 53 I g x File Edit View Assistant
90. en power v 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 nQuery nlenim 2 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Two Means 1 2 Test significance level a 0 05 1 or 2 sided test 2 w 2 2 Number of levels M Difference in means pi p2 15 Standard deviation at each level a 25 Between level correlation p 0 4 Group i size ni Group 2 size n2 Ratio n2 ni 1 1 1 Power 0 Cost per sample unit isase i Calculate required sample sizes for given power v All columns y Figure 4 2 9 Design Entry for Two Means Repeated Measures Study 79 elim 8 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 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 i
91. equired 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 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Contrast 1 2 3 4 Test significance level a 0 05 Number of levels M 4 Contrast C dcrpi 15 Scale D SQRT Sci2 4 47214 Standard deviation at each level a 10 l Between level correlation p 0 7 Effect size A C D o SQRT 1 p 0 61237 Power 90 32 Group size N 29 Cost per sample unit Total study cost 1 a gt Calculate required sample sizes for given power x 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 analys
92. equired sample sizes for given power v 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 Multiple 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 mi Ratio N2 N1 Group 2 Proportions m2 Power Odds Ratio W m2 1 11 m1 1 12 Cost per sample unit Cea Tae 48 Specify Multiple Factors jug 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 95 96 elim 8 We want a 5 level of significance so enter 0 05 in the Test Significance Level box We want an e
93. er 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 15 16 IM 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 Preview and Page Setup Select Save Image from this list to save the plot Power vs Sample Size TTT ES Power vs Sample Size Save Image Print Print Preview Page Setup Disable Legend Disable Title Copy to Clipboard Column 1 r Column 2 Column 3 10 20 30 40 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 2 8 Help and Support For issues pertaining to the methodolog
94. 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 in 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 0 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 Oy C X X C 00 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 r 4 6 5 where 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 afy F aee 4 6 8 Hi Clee Ce where 1 Zane I _ _ ap 4Y 4 6 10 a p2 5 df ap df g N r p a 1 2 ap 2 2 Pillai Bartlett Trace The test statistic for Pillai Bartlett trace
95. g 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 For DF a ON a non central F distribution with non centrality parameter In order to calculate sample size a value for power must be specified 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 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 Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term Means One Test C Interim Proportions Confidence Interval Survival 0 Equivalence _ Agreement C Regression One sample t test i Paired t test for difference in Means Univariate one way repeated measures analysis of variance One way repeated measures contrast Univariate one way repea
96. gorithm is used This search algorithm calculates power at various sample sizes until the desired power is reached 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 Study Goal and Design window will appear l Design Goal No of Groups Analysis Method Fixed Term Means One Test Interim Proportions Two Confidence Interval Survival gt Two Equivalence Agreement Regression Chi squared test to compare two proportions Compute power or sample size Compute one or two proportions 5 Chi squared test continuity corrected Compute power or sample size Compute one or two proportions f 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 Figure 4 3 1 Study Goal and Design Window 2 Once the correct test has been selected click OK and the test window
97. h nTerim Analysis Method Test C Confidence Interval C Equivalence 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 2 6 Using the Assistant Tables 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 File Edit View Assistants Plot Tools Window Help New Fred Term Tel Ld ipecity Multi Factor Table j LUJ Open Manual Statistical Solutions Support Compute Hied Sire 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 regu
98. hat 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 2 0 follow the procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 3 3 2 Methodology Sequential Log Rank test of survival in to groups the variables are defined as Probability of Typelerror _ Sosa Group Survival Proportions od Number of Events K 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 h
99. hi 3 Truncate bounds Yes x No x No z No x Truncate at 3 Futility boundaries x Don t Calculate x Don t Calculate x Don t Calculate x function O Brien Fleming O Brien Fleming 7 O Brien Fleming 7 Lk a es Calculate attainable power with the given sample sizes _ All columns 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 a a eV ar 0 4 0 6 0 8 1 Lower bound 3 00000 3 00000 2 67717 2 31962 2 05069 Upper bound 3 00000 3 00000 2 67717 2 31962 2 05069 Futility bound Nominal alpha 0 00270 0 00270 0 00742 0 02036 0 04030 Incremental alpha 0 00270 0 00222 0 00588 0 01480 0 02440 Cumulative alpha 0 00270 0 00492 0 01080 0 02560 0 05000 Exit probability 441 9 19 21 18 27 13 19 26 Cumulative exit probability 4 41 13 60 34 78 61 91 81 17 Nominal beta Incremental beta Cumulative beta Exit probability under HO Cumulative exit probability under HO 3 Looks 3 Specify Multiple Factors u Output Figure 3 2 8 Boundary Table for Power Family Spending Function 41 42 Power Family Boundaries with Phi 3 and Alpha 0 05 1 2 3 4 Figure 3 2 9 Boundary Plot for Power Family Spending Function Finally by clicking on the Output tab at the bo
100. iJ Factor Level Table wij Means Matrix Group Sizes i Covariance Matrix ui Specify Multiple Factors ui Output Figure 4 6 18 Covariance Matrix Window 19 The final step is to select which method we want to use In this case we want to use the Wilks Lambda approach In order to do this simply select the Calculate power using Wilks Lambda and the click on Run as shown in Figure 4 6 19 below nQuery nTerim 2 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size D Runz P An columns x RILATE a j TE f r sts ee Calculate group size using Wilks lambda Calculate group size using Pillai Bartlett trace Calculate group size using Hotelling Lawley trace 7 8 Calculate power using Wilks lambda Calculate power using Pillai Bartlett trace Calculate power using Hotelling Lawley trace wi Factor Level Table ui Means Matrix Group Sizes W Covariance Matrix ui Specify Multiple Factors ui Output Figure 4 6 19 Selecting the Wilks Lambda option OOOO a 20 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 20 141 142 Factor Level Table x Power 3 0 05 29 99738 3 0 05 29 99738 3 0 05 98 07328 0 05 100 0 05 66 77364 0 05 66 77364 Factor ABC 0 05 100 ud Factor Level Tabl
101. iation 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 Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size ANOVA 1 ane ner nn ee EE 2 2 4 Test significance level a 0 05 Number of groups G 3 Variance of means V 11 76 Common standard deviation o 6 Effect size A V g 0 32667 Power 94 82 m t N as multiple of n1 dri Zni n1 2 5 Total sample size N 50 Cost per sample unit 85 Total study cost 4250 DETE mn Calculate attainable power with the given sample sizes v All columns Figure 4 4 6 Completed One Way Analysis of Variance Test Table 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
102. im 2 0 follow the procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 3 2 2 Methodology The variables are defined as a Probability of Type leror pups GroupMeans Gon Group Standard Deviations Group Sample Sizes R Ratio of N to Nz Drift Parameter Number 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 and power 1 the drift parameter 0 can be obtained using algorithms and procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 The test statistic is defined as moel p p PA P 3 2 1 N Ny The user supplies the proportions p p 2 and either R N or N3 N1P1 N2P2 where p Ni Np l No a Since R it follows that a value of R 1 indicates equal sample sizes and that 1 Rx i p reas The approach to solving this problem is dependent on what information the user supplies For the case of continuity correction the formula can be written as 1 R 1 p1 Pal ay R _ 1 0 3 2 2 pa p R 1 N R 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 cond
103. ing Function x Spending Function Spending function O Brien Fleming x O Brien Fleming x O Brien Fleming x O Brien Fleming x Phi Truncate bounds No No No x No x Truncate at Futility boundaries gt gt Dont Calculate Don t Calculate 7 O Brien Fleming O Brien Fleming Looks me Lower bound x gt Calculate required sample sizes for given power 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 a a a ay af _ L 0 2 0 4 0 6 0 8 1 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 0 00000 0 00079 0 00683 0 01681 0 02558 Cumulative alpha 0 00000 0 00079 0 00762 0 02442 0 05000 Exit probability 10 03 9 95 34 68 29 96 15 39 Cumulative exit probability 0 03 9 98 44 67 74 63 90 02 Nominal beta Incremental beta Cumulative beta Exit probability under HO GEI Looks ij Specify Multiple Factors u Output Figure 3 3 6 Boundary Table for Column 2 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
104. ing 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 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 H4 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 nTer
105. 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 sample 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 Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term Means Test C Interim Proportions C Confidence Interval Survival 0 Equivalence Agreement C 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 f Two way analysis of variance Multivariate analysis of variance MANOVA Analysis of Covariance ANCOVA eam 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 3 There are two main tables req
106. is of variance with 4 levels will have 90 32 power to detect a contrast C Yci wi of 15 with a scale D SQRT gt ci of 4 47214 assuming a standard deviation at each level of 10 and a between level correlation of 0 7 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 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 71 72 IM 4 2 2 Methodology Power and sample s
107. it View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size RM Contrast 1 ee eee ee 2 3 4 Test significance level a 0 05 Number of levels M 3 Contrast C Scrpi l6 Scale D SQRT Sci2 2 44949 Standard deviation at each level a 3 677 Between level correlation p 0 Effect size A C D o SQRT 1 p 0 66617 Power 94 82 Group size N 30 Cost per sample unit Total study cost KE a rm Calculate attainable power with the given sample sizes gt ei pucks 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 Scii 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 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 i
108. it may increase the required sample size relative to the binding case The boundaries calculated in nTerim 2 0 follow the procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 3 1 2 Methodology Section The variables are defined as ____ ProbabilityofTypelerror __ B Probability ofTypell error _ Group Standard Deviations N4 N3 Group Sample Sizes Ratio of N to N Drift Parameter Number 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 and the power 1 the drift parameter can be obtained using the algorithms and procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 The test statistic is defined as o A Or 7 ox 3 1 1 N Np The user supplies the means 4 2 the group standard deviations 01 02 and either R N N2 oe or N Since R A it follows that a value of R 1 indicates equal sample sizes The 1 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 u U2 group standard deviations c1 o2 R or N3 time points and type of spending function The requirement is to obtain th
109. itions are met p N gt T aon 1 p2 Nz gt T where T is a predefined threshold 33 34 IM User supplies R only The requirement is to obtain N and N3 Using that N R x N the result from Equation 3 2 2 obtained is _ RPA p PA P Peo t Rp p2 The steps involved are e Obtain e Solve Equation 3 2 4 for N and N R XN 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 TO o 3 2 5 R N p1 p2 to obtain Nicc It follows that Nz is then R X Nicc If the user has NOT selected to use continuity correction then N N andN R X N User specifies N 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 N proportions p4 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 O by solving Equation 3 2 1 given that N4 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
110. ize 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 and the between level correlation p The difference in means 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 Zy d IMN z 1 m Power 1 4 2 1 a 1 p M 1 where is the standard normal density function and N N N 4 2 2 Ny 4 2 3 In order to calculate sample size for a given power the following formula is used Za Zg 1 M 1 p 0 y _ Gat Ze M DP PEP Md n 1 T where p is the probability of a type II error p 1 Power 4 2 5 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 investigat
111. larly 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 N as multiple of ni fri Ini ni wil Compute Effect Size Assistant u Specify Multiple Factors Output Figure 2 6 2 Example of Effect Size Assistant Table 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 13 14 Im 2 7 Plotting A plotting menu has been introduced to nTerim 2 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
112. lculation Sample sizes of at least 129 in group 1 and 129 in group 2 are required to achieve 90 12 power to detect an odds ratio of 2 25 for proportions of 0 4 in group 1 and 0 6 in group 2 using a 1 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 39 IM 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 Design Goal No of Groups Analysis Method Fixed Term Means Test Interim Proportions Two C Survival Group Sequential Test of Two Proportions OK 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 analyse
113. le COMPUTER 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
114. n p 0 5 0 5 Group 1 proportion p1 0 45 0 4 Group 2 proportion p2 0 55 0 55 Odds ratio W p2 1 p1 p1 1 p2 1 49383 1 83333 Group 1 size ni 349 154 Group 2 size n2 349 154 Ratio n2 n1 1 1 1 1 Power 90 01 90 02 Cost per sample unit 120 120 Total study cost _ 83760 36960 a D 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 proportion 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 Plot Power vs Sample Size RM Two Proportions 1 4 1 2 4 Test significance level a 0 05 0 05 0 05 1 or 2 sided test 2 x 2 l2 l2 Number of levels M J3 3 3 Between level correlation p 0 5 0 5 0 5 Group 1 proportion p1 0 45 0 4 0 45 Group 2 proportion p2 0 55 0 55 0 5 Odds ratio p2 1 p1 p1 1 p2 1 49383 1 83333 1 22222 Group 1 size ni 349 154 1396 Group 2 size n2 349 154 1396 Ratio n2 ni 1 1 1 1 Power 90 01 90 02 90 Cost per sample unit 120 120 120 Total study co 36960 re Ase Calculate required sample sizes for given power v All columns
115. n x Spending Function x Spending function O Brien Fleming x O Brien Fleming z O Brien Fleming O Brien Fleming 7 Phi Truncate bounds No No No No iy Truncate at Futility boundaries Dont Calculate Don t Calculate Dont Calculate i Dont Calculate x Spending function O Brien Fleming iy O Brien Fleming fa O Brien Fleming x O Brien Fleming x Phi MHL m D 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 sample size for given power from the drop down menu and click Run File Edit View Assistants New Fixed Term Test _ New Interim Test Plot Tools Window Help Plot Power vs Sample Size LJ Open Manual Statistical Solutions Support GST Two Means 1 xX 1 een ened 2 3 4 0 05 2 oF SE g Group 1 mean p1 180 Group 2 mean p2 200 Difference in means pi p2 20 Group 1 standard deviation o1 30 Group 2 standard deviation o2 30 Effect size 5 0 667 Group 1 size ni 49 Group 2 size n2 lag Ratio N2 N1 f 1 1 1 Power 90 36 Cost per sample unit 250 Total study cost 24500 Number of looks _ 5 5 5 5 Information times Equally Spaced x Equally Spaced Equally Spaced x Equally Spaced x Max
116. n Fe _ N G c a ON a non central F distribution with non centrality parameter 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 111 IM 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 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 a p Design Goal Fixed Term Means Interim Proportions C Survival Agreement Regression No of Groups Analysis Method Test O Confidence Interval 0 Equivalence 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 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
117. n 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 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 ke nQuery nTerim 2 File Edit View Assistants Plot Tools Window Help E New Fixed Term Test _ New Interim Test W Plot Power vs Sample Size ANCOVA 1 A 2 3 4 Test significance level a 0 05 Number of groups G 4 Variance of means V 13 29 Common standard deviation o 25 Number of covariates c 1 R Squared with covariates R 0 75 Power 85 37 Total sample size N 150 100 Cost per sample unit Total study cost Calculate attainable power with the given sample sizes v All columns Figure 4 5 6 Completed ANCOVA Test Table 15000 115 116 IY 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
118. n menu below the test table File Edit View Assistants Plot Tools Window Help 3 _ New Fixed Term Test _ New Interim Test Plot Power vs Sample Size RM Two Means 1 Test significance level a 0 05 00 00 0 05 1 or 2 sided test 2 2 x 2 2 Number of levels M 5 5 5 5 Difference in means p1 p2 40 40 40 40 Standard deviation at each level a 80 80 80 80 Between level correlation p 05 0 5 0 5 0 5 Group 1 size ni Group 2 size n2 Ratio n2 n1 1 2 3 4 Power 85 85 85 85 Cost per 75 75 75 75 sample unit oA KE gt Calculate required sample sizes for given power v All columns 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 New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Two Means 1 cement 2 ss z Test significance level a 0 05 0 05 0 05 0 05 1or 2 sided test 2 2 2 2 Number of levels M 5 5 5 5 Difference in means pi p2 40 40 40 40 Standard deviation at eachlevel a 80 80 80 80 Between level correlation p los 0 5 0 5 0 5 Group 1 size ni 44 33 29 27 Group 2 size n2 44 66 87 108 Ratio N2 N1 1 2 3 4 Power 85 85 85
119. nce in means pi p2 10 Standard deviation at each level a 20 Between level correlation p 05 Group 1 size n1 53 Group 2 size n2 53 Ratio n2 ni 1 1 1 1 Power 90 Cost per sample unit otal study cos j mareos naiiai rrr ah Calculate required sample sizes for given power v All columns Figure 4 2 3 Completed Repeated Measures Design for Two Means 74 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 Therefore 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 Win
120. ncrease 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 Study Goal and Design window will appear Study Goal And Design Design Goal No of Groups Analysis Method Fixed Term Means One Test C Interim Proportions L0 Confidence Interval Survival 0 Equivalence 0 Agreement C Regression One sample t test i Paired t test for difference in Means Univariate one way repeated measures analysis of variance One way repeated measures contrast Univariate one way repeated measures analysis of variance Greenhouse Geisser ama 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
121. nformation has been entered click on Edit and Fill Right as shown in Figure 4 2 10 Plot Tools Window Help erim Test W Plot Power vs Sample Size Clear Table R Clear Column 5 r A 3 4 5 Te Clear Selection 0 05 1 or 2 sided test 2 2 2 2 2 Number of levels M 4 Difference in means pi p2 15 Standard deviation at each level a 25 Between level correlation p 0 4 Group 1 size ni Group 2 size n2 Ratio n2 n1 la 1 1 1 1 Power 0 Cost per sample unit 65 pe P 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 other columns Columns 2 to 5 to see how the sample size is affected by various parameter changes File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Two Means 1 4 i 2 3 4 _ Test significance level a 0 05 0 05 0 05 0 05 0 05 1 or 2 sided test 2 x2 x2 x2 x2 x Number of levels M 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 ni Group 2 size n2 Ratio n2 jni 1 1 1 1 1 Power 90 85 80 90 90 Cost per sample unit 65 65 65 65 65 E ta study cost KE gt Calculate required sample sizes for given power v
122. ns Interim Proportions C Survival Agreement C Regression One way analysis of variance fe One way analysis of variance Unequal n s fic Single one way contrast fee Single one way contrast Unequal n s bs Two way analysis of variance Multivariate analysis of variance MANOVA Analysis of Covariance ANCOVA No of Groups Analysis Method One Test C Confidence Interval 0 Equivalence Figure 4 6 11 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 12 There are several tables required for this test including the main test table shown in Figure 4 6 12 the Factor Level table illustrated in Figure 4 6 4 and the Means Matrix assistant table presented in Figure 4 6 5 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 row as shown in Figure 4 6 13 135 136 IM nQuery nTerim 2 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size MANOVA 1 Tu h 4 al SS Calculate group size using Wilks lambda i cols x bevels Alpha Power 0 05 Factor Level Table rit _ Gear wij Factor Level Table u Means Matrix Grou
123. 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 es eee Calculate required sample sizes for given power Compute Effect Size Assistant Scale D SQRT Zci Figure 4 1 10 Automatically Updated Compute Effect Size Assistant Table 11 Once the table in Figure 4 1 11 is completed and values for Contrast C and Scale D are computed click on Transfer to automatically transfer these values to the main table Compute Effect Size Assistant m x E Compute Effect Size Assistant E Specify Multiple Factors i Output Figure 4 1 11 Completed Compute Effect Size Assistant Table 65 66 Aiei 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 Ed
124. o 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 W Plot Power vs Sample Size ANCOVA 1 SSS 1 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 Number of groups G 3 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 Ja 1 1 1 R Squared with covariates R 10 5 0 6 0 7 0 8 Power Total sample size N 120 120 120 120 ample ur 80 80 80 80 Cost per sample unit 9600 9600 9600 9600 o_o EE ra Calculate attainable power with the given sample sizes v All columns Figure 4 5 13 Altered columns for R Squared Comparison 17 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 18 Now select Calculate attainable power given sample size from the drop down menu below the main table and click Run _nQuery nTerim 2 File Edit View Assistants Plot Tools Window Help _ New Fixed Term Test New Interim Test Plot Power vs Sample Size ANCOVAI F l 2 3 4 Test significance level a 0 05 0 05 0 05 0 05 N
125. ols Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size ANCOVA 1 M ee ee eee eee eee eee Test significance level a 0 05 Number of groups G T v Common standard deviation o Number of covariates c R Squared with covariates R2 Total sample size N 2 i 3 Total study cost Calculate required sample sizes for given power All columns Compute Effect Size Assistant x _ Al Mean f ps Variance of means V Total sample size N 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 Compute Effect Size Assistant x Variance of means Total sample size N gpa O UE N gee i P m 315 MIL i oF ia p PIN NP gt p wi Compute Effect Size Assistant i Specify Multiple Factors i 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 bee
126. or 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 4 3 Repeated Measure for Two Proportions 4 3 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 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 proportions in a repeated measures design Power and sample size is computed using the method outlined by Liu and Wu 2005 87
127. or 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 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Contrast 1 Test significance level a 0 05 Number of levels M Scale D SQRT Sci Standard deviation at each level o Effect size A C D o SQRT 1 p Power Group size N Hae weit f Total study cost Calculate attainable power with the given sample sizes e Effect Size Assistant Contrast C Jci pi Scale D SQRT ici 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 Compute Effect Assistant x Transfer 3 Compute Effect Size Assistant 3 Specify Multiple Factors ij Output Figure 4 1 17 Completed Compute Effect Size Assistant Table 69 70 Metin 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 r
128. or 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 2 0 represent the critical values at each look These boundaries are constructed using the alpha and beta spending functions Users in nTerim 2 0 are given the option to generate boundaries for early rejection of the null hypothesis Hj using the alpha spending function or to generate boundaries for early rejection of either the null or alternative hypothesis Hy 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 Hy was first proposed by Lan and DeMets 1983 we refer to such boundaries in nTerim 2 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 probability that an effect will not be found t
129. p Analysis of i covariance ANCOVA Enter a value for alpha a the significance level for the analysis of covariance Input the number of groups that are to be studied the variance of the means and the common standard deviation within groups Specify the number of covariates and the R squared with covariates then specify for power or sample size and nTerim will compute the other Test significance level a Alpha is the probability of rejecting the null hypothesis of equal means when it is true the probability of a Type error Input Advice Enter 0 05 a frequent standard Entry Options 0 001 to 0 20 Clear Compute Transfer References Zanni ye 3 Compute Effect Size Assistant a Specify Multiple Factors jug Output jus Help ja Notes 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 Onc
130. p Sizes Covariance Matrix u Specify Multiple Factors ui Output Figure 4 6 12 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 14 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 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size MANOVA 1 ey ee ee ee 4 3 orleveltable ts iii Cy oe woe a Treen te Number of variables p x wij Factor Level Table Means Matrix Group Sizes ui Covariance Matrix ui Specify Multiple Factors u Output Figure 4 6 13 Enter Number of Response variables 9 Once the number of levels for each factor has been specified click the Fill button at the bottom right corner of the Factor Level Table as shown in Figure 4 6 14 10 The word Filled will now be displayed in the main table in the Factor Level Table row telling the user that the Factor Level table
131. played in Figure 4 4 10 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size ANOVA 1 Ee 1 2 3 4 Test significance level a 0 05 0 05 0 05 Number of groups G 3 3 3 Variance of means V 11 76 11 76 11 76 Common standard deviation o 6 8 Effect size A V a 0 32667 0 735 0 18375 Power 94 82 99 98 75 13 N as multiple of n1 Sri Sni ni 25 2 5 2 5 Total sample size N 50 50 50 Cost per Bac unit 85 85 85 gt Calculate attainable power with the given sample sizes v _ All columns Figure 4 4 10 Re run calculations for Column 3 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 107 108 Metin 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
132. qual 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 1 or 2 sided test Number of levels M Between level correlation p Group 1 Proportions m1 Group 2 Proportions tt2 Odds Ratio Y m2 1 m1 m1 1 12 2 Test Significance Level a 3 Group 1 size N1 04 Group 2 size N2 0 45 0 55 Ratio N2 N1 0 39 0 51 Power 4 Cost per sample unit 0 05 uz Specify Multiple Factors lua 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 View Assistants New Fixed Term Test New Interim Test Plot Tools Window Help Plot Power vs Sample Size _ RM Two Proportions 1 Passi Kareena 2 3 4 Test significance level a 10 05 0 05 0 05 0 05 ior 2 sided test 2 x 2 x 2 x 2 x umber of levels M 3 3 3 3 Between level correlation p 04 0 4 0 4 0 4 Group 1 proportion p1 0 4
133. s 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 40 File Edit View Assistants Plot Tools Window Help Plot Power vs Sample Size GST Two Proportions 1 Pl eel 2 i 3 4 Sa a aS 10 05 E APARANA 2 iy 1 1 ae x New Fixed Term Test New Interim Test Group 1 proportions n1 0 41 Group 2 proportions n2 0 465 Odds ratio Y n2 1 n1 ni 1 n2 1 25074 Group 1 size n1 1400 Group 2 size n2 1400 Ratio n2 n1 A 1 i 1 Continuity correction off off off off a Power 82 17 Cost per sample unit 180 Total study cost so4000 Number of looks 5 5 5 5 Information Times Equally Spaced Equally Spaced Equally Spaced 7 Equally Spaced x Max times 1 1 1 1 Determine bounds Spending Function iy Spending Function x Spending Function x Spending Function Spending function Power Family x O Brien Fleming O Brien Fleming x O Brien Fleming x P
134. s 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 77 IM 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 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 4 Design Goal No of Groups Fixed Term Means Interim Proportions C Survival 0 Agreement C Regression Study Goal And Design es Analysis Method Test C Confidence Interval 0 Equivalence Two sample t test Student s t test equal variances f 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
135. s Plot Tools Window Help E New Fixed Term Test E New Interim Test W Plot Power vs Sample Size Lu Open Manual Statistical Solutions Support 3 a 4 1 or 2 sided test 1 y 1 iy Group 1 proportionniattimet 0 5 Group 2 proportionn2attimet 0 4 Hazard ratio h In n1 In n2 0 756 Survival time assumption Proportional Hazard Exponential Survival Exponential Survival Exponential Survival Total sample size N 1000 Power 85 32 Number of events 550 Cost per sample unit 1100 Total study cost 100000 Number of Looks 5 5 5 5 Information times User Input Equally Spaced x Equally Spaced Equally Spaced z Max Times 1 1 1 1 Determine bounds Spending Function Spending Function Spending Function Spending Function Spending function Pocock x O Brien Fleming gt O Brien Fleming O Brien Fleming gt Truncate bounds No x No x No x No 7 Futility boundaries x Don t Calculate x Don t Calculate z Don t Calculate x Spending function A 7 O Brien Fleming O Brien Fleming O Brien Fleming 7 i 2 n 7a D Calculate attainable power with the given sample sizes X All columns Figure 3 3 10 Complete Survival Table with Pocock Spending Function 8 Once all the values have been entered select Calculate the attainable power with the given sample sizes from the drop down menu and click Run 9 The
136. s 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 in to q Where q Levels Levels Levels For example if 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 4 columns A B A B2 A2B 42B2 M p kM H12 H13 H14 4 6 1 P2 H21 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 o o p op 2 2 2 fea 2 oo Se ae 4 6 2 o p op o Where is a p x p matrix 123 124 In order to calculate power a value
137. s 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 for 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 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 Rencher A C 1998
138. st 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 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 6 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 7 Select Run and the sample size will be re calculated as shown in Figure 3 1 8 below File Edit View Assistants 3 __ New Fixed Term Test i New Interim Test Plot Tools Window Help Plot Power vs Sample Size GST Two Means 1 4 1 Test significance level a 0 05 0 05 1 or 2 sided test 2 2 Group 1 mean p1 220 220 Group 2 mean p2 200 200 Difference in means pi p2 20 20 Group 1 standard deviation o1 30 30 Group 2 standard devi
139. t 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 Columns 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 97 Metin 98 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 Power vs Sample Size Power vs Sample Size 4090 e Column 1 e Column 2 Column 3 Column 4 2090 3090 4090 5090 Power 82 68 S
140. t the matrix has been completed In this example we have also entered the sample size for each group Therefore the Group size n row displays the average group sample size and the Total sample size N is also provided 16 The next step in this MANOVA process is to generate the Covariance Matrix This is done by entering values for the Common Standard Deviation and Between Level Correlation where nTerim will automatically calculate the Covariance Matrix and display it in the Covariance Matrix window as shown in Figure 4 6 7 131 IM File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size fia All columns Means Matrix Group Sizes x l E E S a S a ay ee ey ee 1 1 1 2 1 2 3 4 2 n 7 8 7 4 5 6 4 4 ome ZE Lr u Factor Level Table u Means Matrix Group Sizes Covariance Matrix ui Specify Multiple Factors ui Output Figure 4 6 7 Completed MANOVA Design Table 17 In this example we know from similar studies that the common standard deviation is equal to 2 and the between level correlation is 0 5 To generate the Covariance Matrix simply enter 2 in the Common standard deviation row and 0 5 in the Between level correlation row as shown in Figure 4 6 7 To view the generated covariance matrix click on the Covariance Matrix tab at the bottom of the assistants table wij Factor Level
141. tates 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 means 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 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 a 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 c
142. ted measures analysis of variance Greenhouse Geisser 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 59 60 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size RM Contrast 1 z j N w ui Number of levels M Contrast C Fci pi Scale D SQRT Sci Standard deviation at each level o h Effect size A C D o SQRT 1 p Power Cost per sample unit Total study cost 3 p i ma Calculate required sample sizes for given power All columns Figure 4 1 2 One way Repeated Measures Contrast Test Table Compute Effect Size Assistant x cipi D SQRT ZcP wi Compute Effect Size Assistant aa Specify Multiple Factors ij Output Figure 4 1 3 Compute Effect Size Assistant Table 6 Once you enter a value for the number of levels M the Compute Effect Siz
143. termine bounds Spending Function x Spending Function x Spending Function x Spending Function x Spending function Pocock 7 O Brien Fleming 7 O Brien Fleming O Brien Fleming x Truncate bounds No No No e No Ly Truncate at Futility boundaries Don t Calculate z Don t Calculate Don t Calculate Don t Calculate x Spending function O Brien Fleming x O Brien Fleming O Brien Fleming u O Brien Fleming 7 Phi Figure 3 2 3 Completed Two Proportions Test Table 9 The boundaries calculated are shown in Figure 3 2 4 0 4 0 6 0 8 8 00000 8 00000 8 00000 8 00000 Upper bound 2 17621 2 14371 2 11322 2 08952 2 07091 Nominal alpha 0 01477 0 01603 0 01729 0 01833 0 01918 Incremental alpha 0 01477 0 01139 0 00927 0 00782 0 00676 Cumulative alpha 0 01477 0 02616 0 03543 0 04324 0 05000 Exit probability 22 98 25 92 20 00 13 21 8 00 i Cumulative exit probability 22 98 48 90 68 90 8211 90 12 3 Looks 3 Specify Multiple Factors a Output Figure 3 2 4 Boundary Table for Pocock Spending Function 38 10 Finally the boundaries calculated in the table in Figure 3 2 4 are automatically plotted as illustrated in Figure 3 2 5 Soundaes Graph _ a Pocock Boundaries with Alpha 0 05 2 3 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 ca
144. 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 93 IM 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 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 Study Goal And Design m Design Goal No of Groups Analysis Method Fixed Term Means One Test C Interim Proportions Two 0 Confidence Interval Survival gt Two Equivalence _ Agreement Regression Chi squared test to compare two proportions Compute power or sample size k Compute one or two proportions g Chi squared test continuity corrected f j Compute power or sample size a Compute one or two proportions Fisher s exact test fee Two group Chi square test comparing proportions in C categories Mantel Haenszel Cochran test Mantel H
145. the default setting in nTerim as well as a Ratio value of 1 for the group sizes 73 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 PEON 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 est SK w jeve 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 n1 j Group 2 size n2 Ratio n2 n1 fi 1 1 1 Power Cost per sample unit Total study cost T e o 7 a a eee E E L E OO _ o ee DES o o E o o o Calculate required sample sizes for given power v All columns Figure 4 2 2 Repeated Measures for Two Means Test Table Teer 6 We also know that the between level correlation is 0 5 so enter 0 5 into the Between level correlation row nQuery nTerim 2 File Edit View Assistants Plot Tools Window Help New Fixed Term Test _ New Interim Test Plot Power vs Sample Size RM Two Means 1 a 4 2 3 4 Test significance level a 0 05 1 or 2 sided test 2 y 2 2 2 Number of levels M 4 Differe
146. 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 4 112 Aner nie nQuery ALLE Alc File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size ANCOVA 1 ce lavel a Number of groups G r3 Figure 4 5 2 Analysis of Covariance Test Table E Compute Effect Size Assistant ij Specify Multiple Factors ui 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 4 113 114 Metin 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 Lg seem File Edit View Assistants Plot To
147. tions 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 www statistical solutions software com and iii you can email us for technical help at support statsol ie 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 17 18 Chapter 3 Group Sequential Interim Design 3 1 Two Means 3 1 1 Introduction nTerim 2 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 b
148. ttom 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 3 3 Survival 3 3 1 Introduction nTerim 2 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 Period designs in that the data from the trial is analyzed at one or more
149. uation Instead a search algorithm is used This search algorithm calculates power at various sample sizes until the desired power is reached 101 IM 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 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 5mmHg 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 Study Goal And Design oem Design Goal No of Groups Analysis Method Fixed Term Means Test C Interim Proportions C Confidence Interval Survival C Equivalence Agreement C Regression One way analysis of variance Single one way contrast Single one way contrast Unequal n s Two way analysis of variance Multivariate analysis of variance MANOVA
150. uired 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 p RGY ER o File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size ANCOVA 1 ance level a Number of groups G Variance of means V Common standard deviation o Number of covariates c R Squared with covariates R Cost per sample unit Total study cost Calculate required sample sizes for given power v All columns Figure 4 5 8 Analysis of Covariance Test Table Compute Effect Size Assistant OS ETN Total sample size N N as multiple of n1 Sri Sni ni Tastes wij Compute Effect Size Assistant W Specify Multiple Factors 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 117 118 Metin 8 For the Mean values for each group enter 31 for group 1 41 for group 2 and 45
151. umber of groups G 3 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 1 1 1 1 R Squared with covariates R 05 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 9600 9600 9600 9600 vE 77 m gt Calculate attainable power with the given sample sizes X All columns Figure 4 5 14 Completed multiple design ANCOVA Table 120 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 a Power vs Sample Size Column 1 r Column 2 Column 3 Column 4 50 100 150 200 250 300
152. ure 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 define the interim design such as number of looks spending function futility 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 File Edit View Assistants New Fixed Term Test New Interim Test Plot Tools Window Help Plot Power vs Sample
153. ure 4 6 5 Means Matrix Group Sizes Assistants Table 12 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 13 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 1 2 3 4 23 4 2 5 M 2 1 2 1 4 12 1 422 1 1 4 n 68 7 4 5 64 4 4 4 5 4 14 Enter this matrix in the Means Matrix Assistant table as illustrated in Figure 4 6 6 and then click the Fill button at the bottom right corner of the Means Matrix assistant table 130 Ce Wie gO a yQuery nierim 2 File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test Plot Power vs Sample Size MANOVA 1 Filled Filled Total study cost Calculate group size using Wilks lambda m jonan Means Matrix Group Sizes x Fe 4 5 6 7 8 1 1 2 1 6 en gt Nm ear W Factor Level Table u Means Matrix Group Sizes u Covariance Matrix i Specify Multiple Factors ui Output Figure 4 6 6 Completed Means Matrix Assistant Table 15 Once the user clicks on Fill the Means Matrix row in the main table displays Filled to signify tha
154. vies E AN EE 5 22 TOWNS VU AAO WY drs ss soe cteon te se haus coteaen A denteauseabesecsereed anes 5 2o MEM MN aes es ees reece cece ete i ces E eee nc scare eee at peeeacter Rac acest 6 2 4 Opening a New Design w cccisncrsacednecsasunndtarssndessacdvianseiededencnadnnvensesateeansssieieladadenescaisunstassaetenneens 8 2 5 Selecting an nQuery Advisor Design Table through nTerim cccssscccssseeeeeseeeeeeseeees 12 2 6 Using the Assistant Tables s cearciscscasenseaxecccssauesnuoctasssccsunserspartesseaatennecteamdcsnmrerasanteenecens 13 Dl POO A A E E ese 14 2B HeD ANG SUDON aaan E T E T 17 CRAPO ae am A EA E E EEE 18 Group Sequential Interim Design ccccccssssssssssssssssssssssesssesssesessseeesssssesessseeeeeeseeeens 18 se NO AVC QING rae ac racconsses cage E T acages toeraptencie sees meaeuaneavecuectee ss E OEE 19 de es ATO CUE ION o cette ce idee scour neacote E EE E E E S 19 Oa Methodology SECTION ie EAEE cose aeeade 21 PE ENPE a a NE E E EAE E E EE E ENA 23 DZ WOP OPON ee E A E 31 S2 LMU O E EE E EE EE 31 B22 VIET GOO SY ereenn enirn R EE E E ET 31 Ao EAI ONC r EE A ree vepsecnatan eeantetneeass 36 SN E E cee titae sate y snd sree atone arena E E E E E E cues 43 Di MOG UIC OM E S E E N E EE anes 43 Doe MONOCDE Y onioni oenina ena N EN EEE E NAE 43 Soo R ea 6 E A eee T A E ee N 47 CHD E Aana E EA EE E EEE eeeanes ee 56 Fed Term DESE onnan E A A TA 56 4 1 One Way Repeated Measures Contrast Constant Correlation
155. y 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 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 ON nTerim Power and Sample Size for Group File Edit View Assistants Plot Tools Window Help New Fixed Term Test New Interim Test W Plot Power vs Sample Size LL Open Manual a Statistical Solu
156. ypothesis to be rejected the type 1 error and the power 1 the drift parameter can be obtained using algorithms and procedures outlined by Reboussin et al 1992 and Jennison amp Turnbull 2000 HR log 3 3 1 S1 For the Exponential Survival Curve this is defined by the expression below 0 3 3 2 This can be solved for d the required number of events using the equation below d cl 3 3 3 log HR B Then to calculate the Proportional Hazards Curve Equation 3 3 4 is employed _ 1 HRydk 3 3 4 1 HR This can be solved for d the required number of events using Equation 3 3 5 2 1 HR O dy ae 3 3 5 1 HR 45 46 To calculate the sample size N the following formula is used 2d N 2 S1 So Calculate Attainable Power with the given Sample Size 3 3 6 Given a N group survival proportions s4 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 S1 S gt log HR 8 N 2 s4 s2 1 HR 2 1 HR 3 3 7 3 3 8 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

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