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1. Yes There are some observations with missing values indicating that at the time these patients were enrolled the investigators had not yet started to record chlamydia antibody status Column 3 Gonnorhoea 0 No 1 Yes Column 4 Contracept The use of contraceptives 0 No 1 Yes Column 5 Sexpatr Multiple sex partners 0 No 1 Yes In order to run the Meanscore analysis the user must specify the response variable first stage variables and the predictor variables in the model Multiple first stage variables may be specified but they must all be categorical although the second stage variables can be continuous There is a facility to fit a separate coefficient for each level of a categorical predictor variable The implementation of these features is slightly different from software to software see sections 4 2 4 3 and 4 4 for the specific details for each software 15 4 2 Using the Meanscore package in R 4 2 1 Installation guide You need to have R installed in your computer Our program has been tested under R 1 2 0 so we advise you to first update your R if you still use an older version see section 3 1 to learn more about R You can download our program from the following sites http www r project org R website http www ucc ie ucc depts pubh programs programs html The zip file contains README packages file see Appendix B where you can find the instructions on how to
2. 1 MEANSCORE this function is called with the combined first and second stage data where the missing values in the incomplete covariate s are represented by NA the usual notation in Splus R 2 MS NPREV this function is called with the second stage i e complete data and the first stage sample sizes or prevalences if only prevalences are available then estimates are provided but no standard errors Prior to running this function the CODING function 3 should be run to see the order in which MS NPREV expects the first stage sample sizes or prevalences to be provided 3 CODING this function recodes multiple columns of first stage covariates into a single vector and displays the coding scheme Help on these functions and on the illustrative data sets provided can be viewed using help or or the HTML help file system This code has only been tested under R 1 2 0 for Windows and may need some modifications for use with other versions or other operating systems We would be happy to hear about any bugs that you find and to receive any comments or suggestions for improvements Marie Reilly PhD amp Agus Salim Dept of Epidemiology Dept of Statistics University College Cork University College Cork Ireland Ireland E mail marie reilly ucc ie agus stat ucc ie 71 2 Meanscore package in S PLUS The Mean Score method for missing and auxiliary covariate data is described in the paper by Reil
3. should be run to see the order in which MS NPREV expects the first stage sample sizes or prevalences to be provided 3 CODING this function recodes multiple columns of first stage covariates into a single vector and displays the coding scheme 72 Help on these functions and on the illustrative data sets provided can be viewed using help or This code has only been tested under S PLUS 4 0 for Windows and may need some modifications for use with other versions or other operating systems We would be happy to hear about any bugs that you find and to receive any comments or suggestions for improvements Marie Reilly amp Agus Salim Dept of Epidemiology Dept of Statistics University College Cork University College Cork Ireland Ireland E mail marie reilly ucc ie E mail agus stat ucc ie 3 Meanscore package in STATA SUBJECT Meanscore algorithm for missing covariate data in logistic regression models AUTHORS Marie Reilly Dept of Epidemiology amp Public Health and Agus Salim Dept of Statistics University College Cork UCC Cork Ireland SUPPORT marie reilly ucc ie INSTALLATION Stata version 6 Installing from internet please check 73 U 20 6 How do I install an addition R net Installing from floopy disk type net from a net install meanscor to install program net get meanscor to access illustrative datasets Installing fr
4. zlevel stratum and a list called se containing se the standard errors of estimates achieved by the optimal design budget Optimal sampling design for 2 stage studies with fixed budget Usage budget x x y y z z prev prev fctvar NULL var NULL b b cl c1 c2 c2 Arguments REQUIRED ARGUMENTS y response variable binary 0 1 x matrix of predictor variables z matrix of the surrogate or auxiliary variables can be more than one column prev the prevalence of each y z stratum where y z is the different levels of y and z var The name of the predictor variable whose coefficient is to be optimised If this is a factor variable please see DETAILS at the end of this section b the total budget available 44 cl the cost per first stage observation c2 the cost per second stage observation OPTIONAL ARGUMENTS fctvar the names of any factor variables in the predictor matrix SIDE EFFECTS The following lists will be returned n theoptimal number of observations first stage sample size design a list consisting of the following items ylevel the different levels of the response variable zlevel the different levels of first stage covariates z prev the prevalence of each ylevel zlevel stratum n2 the sample size of pilot observations for each ylevel zlevel stratum prop optimal 2nd stage sampling proportion for each ylevel zlevel stratum samp 2nd optimal 2nd st
5. 0 01584 0 00797 0 01584 0 00817 0 01563 0 00772 0 01533 0 00738 Age 0 04594 0 01194 0 04595 0 01371 0 04304 0 01248 0 04284 0 01247 Angina 0 13795 0 29977 0 13796 0 30433 0 15164 0 29085 0 15032 0 29080 Chf 0 36390 0 09456 0 36391 0 09486 0 40086 0 09380 0 40235 0 09355 LVDBP 0 02521 0 01172 0 02521 0 01137 0 02112 0 01107 0 02125 0 01107 Surg 1 04214 0 19520 1 04215 0 20183 0 99580 0 18658 1 00156 0 18634 Table A 4 Comparison of meanscore and other likelihood based methods using the Ectopic pregnancy data see section 6 3 Variable Meanscore WL PL ML Coef SE Coef SE Coef SE Coef SE Intercept 0 78892 0 21727 1 58107 0 28143 1 55017 0 28756 1 60172 0 28130 Chlam 0 90485 0 31592 0 85737 0 31862 0 89721 0 30574 0 89400 0 30467 Gonn 0 05172 0 30077 0 06865 0 33482 0 03353 0 35763 0 05117 0 32208 Contracept 2 36030 0 18715 2 25818 0 23860 2 25721 0 23934 2 26657 0 23400 sexpatr 0 74156 0 23526 0 85534 0 31331 0 81079 0 30252 0 87355 0 30779 67 Table A 5 Comparison of meanscore and other likelihood based methods using the NWTSG data with Institutional Histology as the first stage variables Variable Meanscore WL PL ML Coef SE Coef SE Coef SE Coef SE Intercept 3 2032 0 5708 3 2035 0 5648 3 2408 0 5518 3 2590 0 5470 Stage I 1 1781 0 2138 1 1
6. Dempster et al 1977 For simplicity of notation let Y denote the response variable Z the complete covariates which must be categorical and X the covariates of interest in the regression model where some components of X are missing The complete covariates Z may contain some auxiliary or surrogate variables that are informative about the missing components of X Interest is focused on estimating the parameters in the regression model fg Y X If the relationship between Z and X was fully known we could obtain the Maximum Likelihood Estimator MLE for the parameters of the regression model by using the EM algorithm which is equivalent to solving the score equation Y SsQ X Y ALS a Vi X 2 5 2 0 ieV jeV where Sg Yi Xi dfg Y X OP the usual score statistic Reilly amp Pepe 1995 V denotes the set of complete validation cases and V denotes the set of incomplete non validation cases Throughout this chapter we will use the same notation Because the exact relationship between X and Z is unknown Mean Score uses a non parametric estimate for the conditional expectation above Each incomplete case is assigned the average score of complete cases with matching Y and Z A little algebra shows that the Mean Score estimator is thus the solution to the score equation Zi Yi n dram S Fi Xi 0 ieV V Zi Zi Yi i where N denotes the total number of cases with Z Z and Y Y And N m denotes t
7. given by Holcroft and Spiegelman s algorithm is counter intuitive in the sense that their OPTIMAL design does not like to sample from rare cells Our algorithm conversely samples more from rare cells indicating that observations in rare cells tend to be more informative We have communicated our findings to the authors but at the time of this writing we have not had any response Our assessment to their program is that it is not user friendly not general enough it can only take single predictor and in its current form we could not recommend it 60 7 Rweb modules for optimal sampling development and configuration Rweb is a Web based interface to the R statistical package R is a freely distributed open source system for statistical computation and graphics Hornik 2000 R was initially written by Ross Ihaka and Robert Gentleman and is available for download from any CRAN Comprehensive R Archive Network mirror site See R home page at lt http www r project org gt for more information about downloading and installing R Rweb was developed by Banfield 1999 to provide an easy to use interface to all of R statistical and data management functions Rweb is available at http www math montana edu Rweb index html It comes in three versions e The basic Rweb code window will run on most browsers but requires knowledge of R programming e The JavaScript version provides a more sophisticated interface but requires a JavaScript
8. samp 2nd optimal 2nd stage sample size for each ylevel zlevel stratum var the variance of estimates achieved by the optimal design cost the minimum study cost coding combines two or more surrogate auxiliary variables into a vector Usage coding x x y y z z output F REQUIRED ARGUMENTS y response variable should be binary 0 1 x matrix of predictor variables for regression model z matrix of any surrogate or auxiliary variables OPTIONAL ARGUMENTS output logical value if it s TRUE T the original surrogate or auxiliary variables and the re coded auxiliary variables will be returned The default is False F SIDE EFFECTS This function does not return any values except if output T If used with only second stage 1 e complete data it will print the following ylevel the distinct values or levels of y 46 zl zi the distinct values of first stage variables zl zi new z recoded first stage variables Each value represents a unique combination of first stage variable values n2 second stage sample sizes in each ylevel new z stratum If used with combined first and second stage data i e with NA for missing values in addition to the above items the function will also print the following nl first stage sample sizes in each ylevel new z stratum DETAILS The response predictor and surrogate variables have to be numeric If you have multiple co
9. such as laboratory tests and radiological imaging might only be collected for some of the study subjects By appropriately choosing the total number of observations and the second stage sampling fractions such design can yield more efficient and cost effective estimates than simple random sampling The missing covariates setting referred to in Chapter 1 can also be viewed as a two stage design where the response variable Y and the complete variables Z are regarded as the first stage information and the incomplete components of X are regarded as the second stage information 2 1 Optimal Design and Meanscore In chapter 1 we noted that the variance of meanscore estimates depends on the total number of observations and the validation sampling fraction in each Z Y stratum Thus it is possible to minimise the variance using an appropriate study design In this chapter we will outline how one can derive optimal sampling designs for two stage studies for the following three scenarios 1 Where we already have the first stage data and would like to sample a specified number of observations at the second stage For example if we already have a database or registry and we wish to gather additional information on some subjects in order to address a research question 2 Where a fixed budget is available and we wish to design a study that will minimise the variance of an estimate subject to the budget constraint 3 Where a coefficient of in
10. 1 1 Introduction Missing data is one of the most common problems in data analysis Perhaps the most common approach when confronted with missing data is excluding the incomplete cases from analysis and proceeding to analyse the complete cases using standard methods While valid under certain assumptions regarding the missingness mechanism this approach results in a loss of precision due to the ignored observations In this report we are interested in the problem of missing covariates in regression models In the last two decades some analysis methods that accommodate all available cases have been developed Those methods include meanscore Reilly and Pepe 1995 pseudo likelihood Breslow and Cain 1988 weighted likelihood Flanders and Greenland 1991 and nonparametric maximum likelihood Breslow and Holubkhov 1998 The meanscore method is the subject of this report The meanscore method that incorporates information from all available cases into the regression model is a likelihood based method For completely random missingness this results in an improvement in efficiency over the analysis of complete cases only More importantly the method is applicable to a wide range of patterns of missingness known as MAR Missing at Random where missingness may depend on the completely observed variables but not on the unobserved value of the incompletely observed variable s 1 2 Meanscore The meanscore method is motivated by the EM algorithm
11. 10 10 10 10 10 10 10 10 10 o Oo o Ho H Type coding for details prop samp 2nd 6181 0000 5107 1465 1061 0000 0000 0000 0000 0000 1 0000 1 0000 107 476 598 65 622 409 9 19 36 28 111 15 ylevel new z 49 se SE Intercept 1 181027372 sex 0 219738497 wt 0 006671847 age 0 014483091 angina 0 241016110 chf 0 074636521 lve 0 009852944 surg 0 175539344 5 4 Using the Optimal package in STATA 5 4 1 Installation guide We have written the package in STATA version 6 so you need to have STATA version 6 or later The optimal package can be installed directly from the STATA website by following these instructions From inside STATA type net cd stb net cd stb58 net describe sxd2 net install sxd2 net get sxd2 After executing the last command the optimal package is installed in your computer To test 1f you have installed all the components you can type help optfixnorhelp optbudorhelp optprec to try some of the examples IMPORTANT Since we submitted the program to the STB we have improved the calling syntax and the output format The most recent version is available at http www ucc ie depts ucc pubh programs programs html Note that this program is slightly different from the STB version The most noticeable change is that the functions now call the name of the variable to be optimised in the optvar option instead of the position of the variable in the
12. of multiple imputation It refers to a procedure in which each incomplete case is filled in i e imputed using several cases that are sampled randomly with replacement from the list of similar complete cases to form several imputed data sets The hotdeck estimate is the average of the standard estimates from all imputed data sets However the usual multiple imputation variance formula when applied to hotdeck multiple imputation underestimates the between imputation variance This is because simple hotdeck multiple imputation acts as if the distribution of non missing sample values was exactly the same as the population distribution of the values Little and Rubin 1987 The Approximate Bayesian bootstrap ABB method can be used to increase the between imputation variability so that hotdeck can be implemented using a two step procedure Mander and Clayton 2000 to yield unbiased variance estimate To investigate the behaviour of the various estimators and more importantly their variance estimates we generated 200 observations of a standard normal predictor variable X N 0 1 The response variable Y was then generated as a Bernoulli random variable with p exp x 1 exp x A dichotomous surrogate variable for X called Z was generated as follows Z 1 X gt 0 0 otherwise 56 We deleted 100 1 p 6 observations of the predictor variable X where p is the validation sampling fraction using both a random and a balanced sampli
13. supporting function named coding The fixed n function calculates the sampling fractions at the second stage given fixed first and second stage sample sizes which will minimise the variance of a specified co efficient in the regression model The budget function calculates the first stage sample size and the second stage sampling fractions that will maximise precision of a specified co efficient subject to a given budget The precision function calculates the first stage sample size and the second stage sampling fractions that will minimise cost subject to a given precision for a specified co efficient The supporting function coding combines two or more first stage covariates into a vector i e it recodes a matrix of categorical variables into a vector that takes a unique value for each combination Before running any of the three modules you should run the coding function to see in which order you must supply the vector of prevalences 2 Supply the dataset to be used in the analysis Small datasets can be copied and pasted into the text box near the bottom of the screen One can also type in a URL for a Web accessible dataset For testing purposes there are two built in datasets provided cassl or cass2 The data must be in text format where lines represent observations and columns represent variables are separated by spaces The first line should contain the variable names separated by spaces When the module and a dataset have bee
14. vecname is provided to optbud in the correct order For this reason we strongly suggest that any call to optbud is preceded by a call to coding For more details about the coding function see at section 4 4 Using the Meanscore package in STATA optprec Optimal sampling design for 2 stage studies with fixed precision Command line optprec depvar indepvars if exp in range first varlist prev vecname prec cl c2 optvar varname coding Options first varlist prev vecname specifies the first stage variables vector of prevalences for each stratum formed by different levels of dependent variable and first stage covariates 52 prec the variance we want to achieve while minimising cost cl cost per study subject at the first stage c2 cost per study subject at the second stage optvar varname the covariate for which we want to achieve a variance prec If the covariate is a factor variable you need to specify the level whose coefficient is to be optimised see ANALYSIS WITH CATEGORICAL VARIABLES coding a logical flag default of 0 FALSE means that prior to calling the optprec function you have run the coding function help coding for details to create the vector grp_yz containing the distinct groups strata formed by the different levels of response Y and first stage covariates Z If you have not run coding and you call the optprec function with coding 1 the grp yz vecto
15. with only second stage i e complete data it will print the following ylevel the distinct values or levels of y zl zi the distinct values of first stage variables zl zi new z recoded first stage variables Each value represents a unique combination of first stage variable values n2 second stage sample sizes in each ylevel new z stratum If used with combined first and second stage data i e with NA for missing values in addition to the above items the function will also print the following nl first stage sample sizes in each ylevel new z stratum 26 4 3 3 Examples 4 3 3 1 meanscore Here we again demonstrate the example we illustrated in section 4 2 3 The simNA data set see section 4 1 is stored in the simNA matrix The matrix is automatically available when you declare library meanscore This implementation is slightly different from R as in R you need to load the data matrix to make it available meanscore yssimNA 1 z simNA 2 x simNA 3 1 For calls to ms nprev input nl or prev in the following ylevel new z order ylevel z new z nl n2 1 0 0 1 310 150 2 0 1 2 166 85 3 10 1177 86 4 11 2 347 179 parameters est se Z pvalue Intercept 0 04939797 0 07155154 0 6903831 0 4899533 x 1 01885599 0 10187166 10 0013679 0 0000000 4 3 3 2 ms nprev The ms nprev command provides a way of doing Meanscore analysis if we only have the complete observations but we kno
16. y z stratum the sample size of pilot observations for each y z stratum optimal 2nd stage sampling proportion for each y z stratum optimal 2nd stage sample size for each y z stratum 38 coding combines two or more surrogate auxiliary variables into a vector Usage coding x x y y z z return F Arguments REQUIRED ARGUMENTS y response variable should be binary 0 1 x matrix of predictor variables for regression model z matrix of any surrogate or auxiliary variables OPTIONAL ARGUMENTS return logical value if it s TRUE T the original surrogate or auxiliary variables and the re coded auxilliary variables will be returned The default is False F Value This function does not return any values except if return T If used with only second stage i e complete data it will print the following ylevel the distinct values or levels of y zl zi the distinct values of first stage variables zl zi new z recoded first stage variables Each value represents a unique combination of first stage variable values n2 second stage sample sizes in each ylevel new z stratum If used with combined first and second stage data i e with NA for missing values in addition to the above items the function will also print the following nl first stage sample sizes in each ylevel new z stratum 39 DETAILS The response predictor and surrogate variables have to be num
17. 00 The version we used to develop our software is S PLUS version 4 0 3 3 STATA STATA is a statistical software package sold by STATA corporations It has attracted a lot of interest from biostatisticians epidemiologists and medical researchers for its ease of use and its flexibility In a single environment the user has access to a wide range of commands from simple tables to complex models in any sequential order The software has a large library of contributed functions written by users The documentation for these functions is published in the STATA Technical Bulletin STB and the code made available in the STATA website For example the programs we developed have been published in STB 58 November 2000 A powerful capability in STATA is the web compatibility the search command inside STATA allows the user to quickly and easily identify contributed functions and the net install command allows one to directly install from the web as part of STATA any function they request These functions can be installed directly from the internet if you already have STATA software installed in your computer The books by Rabe Hesketh and Everitt 2000 and Hamilton 1997 provide an introduction to STATA More detail about STATA can be found at http www stata com We developed the package using STATA 6 although STATA 7 has been released since then However all programs written in the older version of STATA can be run in the newer version by puttin
18. 1 1 0 2 1 2 0 3 1 3 Nn UT d W PN FP WD Ul A W NH 10 10 10 10 10 10 8 10 10 10 10 10 zlevel prev 1 0 0197823937 2 0 0544015826 3 0 1339020772 4 0 0503214639 5 0 6698813056 6 0 0467359050 1 0 0009891197 2 0 0022255193 3 0 0040801187 4 0 0032146390 5 0 0127349159 6 0 0017309594 1 181028836 219738725 006671856 014483114 1 Check sample sizes prevalences n2 10 10 10 10 10 10 10 10 10 10 10 o Oo Oo H Qo H 1 prop samp 2nd 6181 0000 5107 1465 1061 0000 0000 0000 0000 0000 1 0000 1 0000 107 476 598 65 622 409 9 ake 36 28 ITI 15 41 42 angina 0 241016357 chf 0 074636552 lve 0 009852953 surg 0 175539553 5 3 Using the Optimal package in S PLUS 5 3 1 Installation guide Our program has been tested under SPLUS 4 for windows Some modifications may be needed for other versions If you have S PLUS installed on your computer you can download our program from the following sites http lib stat cmu edu DOS S STATLIB website http www ucc ie ucc depts pubh programs programs html The zip file contains a README file where you can find the instructions on how to install the package Once the package has been installed you can use the functions by issuing the command library optimal in the S PLUS session window 5 3 2 Syntax and features fixed n Optimal second stage sampling fra
19. 1987 and hence it yields the constrained optimum solution In the fixed budget and fixed precision scenarios the problem is more complicated since in addition to the second stage sampling fractions we have to estimate the optimal study size total number of observations Note that the formula for this quantity involves the square root of 3 p I Var Sp Z YM Dye ZY In practical examples this term can be negative and thus there is no solution If this occurs we proposed to sample 100 from the stratum with maximum o I Var Sg Z Y 1 and optimally sample from 10 the remaining strata The step was done iteratively until J7 5 p u Var Ss Z YU Du gt 0 This ad ZY hoc method has also been shown to yield the constrained optimum solution Salim and Reilly 2000 11 3 Computer Packages We have written the Meanscore and optimal sampling algorithms in three programming languanges R S PLUS and STATA In this chapter we give a brief introduction to each of these packages Chapter 4 presents detailed instructions for installing and running the Meanscore function in R S PLUS and STATA with an introduction section explaining features that are common to all 3 systems Chapter 5 presents a similarly structured guide to the optimal sampling software The R version of the optimal package is also available as a web based module for users who do not have access to R S PLUS or STATA The module uses the R web Banfield 19
20. 2 0 2309 8 81 0 942 Hotdeck 1 04 0 2444 0 2149 12 07 0 928 1 041 0 2476 0 2134 13 81 0 906 200 0 25 3 Meanscore 1 064 0 3097 0 2931 5 36 0 940 1 057 0 2925 0 2884 1 40 0 960 ABB 1 096 0 3482 0 2924 16 03 0 932 1 087 0 32 0 302 5 63 0 936 Hotdeck 1 068 0 3146 0 2250 28 48 0 866 1 068 0 3035 0 2276 25 01 0 862 Comparison of Meanscore weighted likelihood WL pseudo likelihood PL and nonparametric maximum likelihood ML Table A 2 Comparison of meanscore and other likelihood based methods using 1000 2 stage observations from CASS data optimal with respect to age see section 6 3 Variable Meanscore WL PL ML Coefl SE Coef SE Coef SE Coef SE Intercept 7 75272 0 62692 7 75284 0 67666 7 59179 0 63523 7 59180 0 63485 Sex 0 59564 0 17130 0 59564 0 17340 0 61852 0 16995 0 61852 0 16486 Age 0 06801 0 01053 0 06801 0 01141 0 06515 0 01064 0 06515 0 01066 Table A 3 Comparison of meanscore and other likelihood based methods using 1000 2 stage observations from CASS data optimal with respect to left ventricular blood pressure see section 6 3 Variable Meanscore WL PL ML Coef SE Coefl SE Coef SE Coefl SE Intercept 6 09724 1 00602 6 09740 1 11720 5 91887 1 01303 5 93357 1 00116 Sex 0 21135 0 21497 021134 0 22433 0 27818 0 20438 0 29358 0 19507 Weight
21. 3 2 S PLUS AE A T E ET op A E TEE A OR 12 3 3 SEA dp LL P 12 4 MEANSCORE PACKAGE cceseeseeseeeeeeeee eee nnn nnn nnn nnn nnn nnn n nnn n nn nnna 13 4 1 Package Features p 13 4 2 Using the Meanscore package in R eese eee essent enses ense ta setatis sensns enne ense enne 15 4 2 1 Installation guide eet edet a s ben epi e es 15 4 2 2 Synt x and feat res dune e e RR eO E DERE AR Ee re eene 15 4 2 3 Examples ses RR RS UR OE E D USERS od Lee Ys 19 4 3 Using the Meanscore package in S PLUS eese eee ee seen senate sensns tnnt tn estu ne tnne 21 4 3 1 Installation guide ce aco eerte oq itedosttatatiad watilttgbdtetec streiten etes 21 4 3 2 Syntax and features a ecd eR et et E RR mete ete CHI Nh ege etta 22 4 3 3 Examples sata AN STE fu dent vato eds SO ANE pm ttis di ien ew eee 26 4 4 Using the Meanscore package in STATA eese eee eese eese tenete stent n tentes sense tuse tn senno 27 4 4 1 Installation Guide Rr ne ee Oe ae a RO 27 4 4 2 Syntax and Features scien ses cus nie cei RAR as He el Rane Th BRR AE 28 4 4 3 Examples 5 tte eee ok ive ee I tief entis 30 5 OPTIMAL PACKAGE ass sa 32 5 1 Package Features p 32 5 2 Using the Optimal package in R esee eee eese eese einen enne tn netus tassa se ass asc asses
22. 787 0 2116 1 0150 0 2085 1 0172 0 2087 Stage II 0 3742 0 2150 0 3742 0 2129 0 2635 0 2064 0 2647 0 2065 Stage III 0 1461 0 2152 0 1461 0 2110 0 1400 0 2067 0 1407 0 2067 Central Hist 1 8020 0 1522 1 8022 0 1496 1 7876 0 1526 1 8099 0 1324 Age 0 0109 0 0022 0 0109 0 0022 0 0117 0 0021 0 0117 0 0021 Study 0 2088 0 1394 0 2088 0 1383 0 2270 0 1332 0 2282 0 1331 Table A 6 Comparison of meanscore and other likelihood based methods using the NWTSG data with Institutional Histology and Stage of Tumor as the first stage variables Variable Meanscore WL PL ML Coefl SE Coef SE Coefl SE Coef SE Intercept 3 2813 0 5495 3 2815 0 5398 3 2565 0 5300 3 2283 0 5282 Stage I 1 1330 0 1718 1 1335 0 1686 0 9525 0 1659 1 0476 0 1502 Stage II 0 3572 0 1646 0 3572 0 1610 0 2448 0 1578 0 3210 0 1408 Stage III 0 3088 0 1627 0 3088 0 1566 0 2824 0 1575 0 3340 0 1384 Central Hist 1 8554 0 1530 1 8555 0 1481 1 7844 0 1498 1 8527 0 1293 Age 0 0113 0 0021 0 0114 0 0021 0 0119 0 0020 0 0115 0 0021 Study 0 2053 0 1392 0 2053 0 1369 0 2241 0 1316 0 2299 0 1326 69 Appendix B INSTALLATION GUIDELINES 1l Meanscote package in R The Mean Score method for missing and auxiliary covariate data is described in the paper by Reilly amp Pepe in Biometrika 1995 This likelihood based method is asymptotically equivalent to hot deck multiple imputa
23. 99 environment The last two chapters of this report chapter 7 and chapter 8 provide more details about this module 3 1 R language R is a language and environment for statistical computing and graphics It was chosen because it is an open programmable extendable package with most statistical functions already available is freely available and has a large user base of academic statisticians committed to constant improvement and update of the package R is similar to the S language and environment which was developed at Bell Laboratories by John Chambers and colleagues and in fact R can be viewed as an implementation of S There are some important differences but much code written for S runs unaltered under R R is available as Free Software under the terms of the Free Software Foundation s GNU General Public License in source code form see http www gnu org It runs on many operating systems including Windows 9x NT UNIX and Mac R can be downloaded at the R project website at http www r project org or other mirror sites around the world The latest version available at the time of writing is R 1 2 3 Once installed the software includes a reference manual and help files from which one can learn The r help mailing list publishes announcements about the development of R the availability of new code questions and answers about problems and solutions using R and so on One can subscribe to this list by sending sub
24. A second stage sample of 1142 observations were drawn using a balanced sampling scheme The term balanced here 1s used as the investigators attempted to sample all relapsed cases and those with UH tumors predicted by institutional pathologists The sampling fraction for controls with FH tumor was chosen so that the number of controls and cases in the dataset are the same 5 National Wilms Tumor data with Institutional Histology and Stage of tumor as the first stage variable Breslow and Chatterjee 1999 This dataset contain the same observations as dataset 4 above In addition to Institutional Histology we also have Stage of Tumor as a first stage variable Each dataset was analysed using 4 different methods meanscore weighted likelihood pseudo likelihood and maximum likelihood The results are presented in Table A 2 Table A 6 From those tables we note the similar performance of all methods In particular it is encouraging to see that the performance of Meanscore is comparable to the full maximum likelihood method even when the 2 stage sampling fraction 1s small As expected maximum likelihood is slightly more efficient yielding smaller standard errors However this small gain is achieved at the expense of a more complex algorithm and lack of robustness when the model being fit is wrong Breslow and Chatterjee 1999 59 6 4 Optimal Two stage Validation Studies The paper by Holcroft and Spiegelman 1999 derives a met
25. Appl Statist 48 457 68 Dempster A P Laird N M Rubin D B 1977 Maximum likelihood from incomplete data via the EM algorithm J R Statist Soc B 39 1 38 Flanders W D and Greenland S 1991 Analytic methods for two stage case control studies and other stratified designs Statist Med 10 739 47 Fletcher R 1987 Practical methods of optimization Chichester Wiley Hamilton L C 1997 Statistics with STATA 5 Brooks Cole Pub Co Holcroft C and Spiegelman D 1999 Design of validation studies for estimating the odd ratio of exposure disease relationships when exposure is misclassified Biometrics 55 1193 1201 Hornik K 2000 The R FAQ http www ci tuwien ac at hornik R Little R J and Rubin D B 1987 Statistical analysis with missing data New York Wiley Mander A and Clayton D 1999 sg116 Hotdeck imputation STATA Technical Bulletin 54 32 4 Rabe Hesketh S and Everitt B 2000 4 handbook of statistical analyses using Stata Second Edition London CRC Reilly M and Pepe MS 1995 A mean score method for missing and auxiliary outcome covariate data in regression models Biometrika 82 299 314 Reilly M 1996 Optimal sampling strategies for two stage studies Amer J Epidemiol 143 92 100 Reilly M and Pepe MS 1997 The relationship between hot deck multiple imputation and weighted likelihood Statist Med 16 5 19 Rweb Statistical analysis on the web http www mat
26. In the following section we give a brief description of each dataset simNA Simulated dataset for illustrating the meanscore function DESCRIPTION A simulated data set of 1000 observations with 500 missing values In STATA this dataset is called sim miss There are 3 variables in the dataset Y 1s the response variable It was generated as a Bernoulli random variable with P Y 1 exp x 1 exp x where X is the true covariate generated as a standard normal variable N 0 1 Finally Z is the surrogate for the true covariate X and was generated using the following rule Z 0 x lt 0 1 otherwise 14 ectopic The ectopic pregnancy dataset DESCRIPTION This dataset which was analysed in Table 3 of Reilly and Pepe 1995 is from a case control study of the association between ectopic pregnancy and sexually transmitted diseases STDs The total sample size 1s 979 consisting of 264 cases and 715 controls One year after the study began the investigators started collecting serum samples for determining chlamydia antibody status in all cases and in a 50 percent subsample of controls As a result only 327 out of the 979 patients have measurements for chlamydia antibody The dataset has 979 observations with 5 variables arranged in the following columns Column 1 Pregnancy The ectopic pregnancy status of patients at the time of interview 0 No 1 Yes Column 2 Chlamydia The chlamydia antibody status of patients 0 No
27. R S PLUS and STATA R S PLUS STATA Features cass cass pilotcas Pilot observations from CASS study Vliestra et al 1980 The response variable is mortality the first stage variable is sex and the second stage variable is age There are 25 observations from each stratum formed by the different levels of mortality and sex cass2 cass2 wtpilot Pilot observations from CASS study Vliestra et al 1980 The response variable is mortality there are two first stage variables sex and categorical weight The second stage variables are weight continuous age CHF angina LVDBP Ive and urgency of surgery surg There are 10 observations from each stratum 5 2 Using the Optimal package in R 5 2 1 Installation guide You need to have R installed on your computer see section 4 2 1 for details You can download the optimal package from the following sites http www r project org R website http www ucc ie ucc depts pubh programs programs html The zip file contains a README packages file see Appendix B where you can find the procedures on how to install the package Once the package has been installed you can use the functions by issuing the command library optimal in the R session window The command help package optimal will open a window where you can read more details about the package 5 2 2 Syntax and features fixed n Optimal second stage sampling fractions
28. S PLUS installed on your computer you can download our program from the following sites http lib stat cmu edu DOS S STATLIB website http www ucc ie ucc depts pubh programs programs html The zip file contains a README file see Appendix B where you can find the instructions on how to install the package Once the package has been installed you can make the meanscore package available by issuing the command library meanscore in the S PLUS session window 22 4 3 2 Syntax and features meanscore Mean Score Method for Missing Covariate Data in Logistic Regression Models USAGE meanscore x x y y z z fctvar NULL print all F REQUIRED ARGUMENTS y response variable binary 0 1 x matrix of predictor variables one column of which contains some missing values NA z matrix of the surrogate or auxiliary variables which must be categorical OPTIONAL ARGUMENTS fctvar optional factor variables if the columns of the matrix of predictor variables have names supply these names otherwise supply the column numbers MEANSCORE will fit separate coefficients for each level of the factor variables SIDE EFFECTS A list called parameters containing the following will be returned est the vector of estimates of the regression coefficients se the vector of standard errors of the estimates z Wald statistic for each coefficient pvalue 2 sided p value Ho est 0 when print all T it will also return th
29. SOFTWARE TOOLS FOR OPTIMAL TWO STAGE SAMPLING A USER GUIDE Marie Reilly PhD Agus Salim BSc Salaheddin Mahmud MD MSc Department of Epidemiology and Public Health and Department of Statistics University College Cork and National Cancer Registry Ireland Funded by The Health Research Board Ireland TABLE OF CONTENTS LIST OF TABLES iii en Rie ERE CE ar Taa pa r aaa a reei Ee aa Daaa aeaaaee Daaa raana Coina Re ME 3 1 MISSING COVARIATE DATA eeeeeeeeeeeeeeeennnn nnne nnn nnn nnn nnn nnns 4 1 1 Introduction serie ee three hereto Yee eee DEOR SEEN EYE Tso dacoas e Eu Dee Dew e sobncsesessnuesteeseeoseseentessee 4 1 2 M anscorc urine eoe ren event uve po deco oes e ees pt E e EES ET SETE EEE EE E iE EESTO 4 2 OPTIMAL SAMPLING DESIGN FOR TWO STAGE STUDIES 6 2 1 Optimal Design and Meanscore sscccccsscssccssessscssesssesssessesssssssssssssssesssesessssesesesessssssseesees 6 2 2 Optimal Design Derivation sscsscsssssccssessscssscesessssssscsssssssesesssssscsssnsesssssssensesssssssesesseees 7 2 2 1 Fixed second stage sample size sse ener enne enne 7 2 2 2 Fixed budget iere I eee de eee eos 7 2 2 3 Fixed precision 5 een caves sey sei eite de e e HE EC E ea 8 2 3 Computational IsSSUeS 9 3 COMPUTER PACKAGES aasssssssnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn nnmnnn nnne 11 3 1 R language TESE A E E EE E E 11
30. The default 1s False F fctvar factor variables if the columns of the matrix of predictor variables have names supply these names otherwise supply the column numbers MS NPREV will fit separate coefficients for each level of the factor variables SIDE EFFECTS If called with prev will return only A list called table containing the following ylevel the distinct values or levels of y zlevel the distinct values or levels of z prev the prevalences for each y z stratum n2 the sample sizes at the second stage in each stratum defined by y z and a list called parameters containing est the Mean score estimates of the coefficients in the logistic regression model 24 If called with n1 it will return A list called table containing ylevel the distinct values or levels of y zlevel the distinct values or levels of z nl the sample size at the first stage in each y z stratum n2 the sample sizes at the second stage in each stratum defined by y z and a list called parameters containing est the Mean score estimates of the coefficients in the logistic regression model se the standard errors of the Mean Score estimates z Wald statistic for each coefficient pvalue 2 sided p value HO est 0 If print all T the following lists will also be returned wzy the weight matrix used by the mean score algorithm for each Y Z stratum this will be in the same order as nl and prev varsi the variance of
31. age cost of 0 5 unit the code below will calculate the sampling strategy that optimises the precision of the coefficient for Surgery surg see output below 40 data cass2 y cass2 1 response variable z cass2 c 2 3 auxiliary variable x cass2 c 2 4 9 predictor variables run CODING function to see in which order we should enter prevalences coding x x y y z z 1 For calls requiring n1 or prev as input use the following order ylevel sex wtcat new z n2 1 0 0 1 1 10 2 0 1 1 2 10 3 0 0 2 3 10 4 0 1 2 4 10 5 0 0 3 5 10 6 0 1 3 6 10 7 1 0 1 1 8 8 1 1 1 2 10 95 1 0 2 3 10 10 1 1 2 4 10 11 1 0 3 5 10 12 1 1 3 6 10 supplying the prevalence from Table 5 Reilly 1996 prev c 0197823937 0 0544015826 0 1339020772 0 0503214639 0 6698813056 0 0467359050 0 0009891197 0 0022255193 0 0040801187 0 0032146390 0 0127349159 0 0017309594 optimise surg coefficient budget x x y y Z Z var Surg prev prev b 10000 c1 1 c2 0 5 1 please run coding function to see the order in which you 1 must supply the first stage sample sizes or prevalences 1 Type coding for details 1 For calls requiring nl or prev as input use the following order ylevel sex wtcat new z n2 Oo Oo Oo O O O H sn 1 8752 Sdesign ylevel Oo Oo Oo oo se Intercept sex weight age 0 1 1 1 0 2 1 2 0 3 1 3 0 1
32. age sample size for each ylevel zlevel stratum se the standard error of estimates achieved by the optimal design precision Optimal sampling design for 2 stage studies with fixed precision Usage precision x x y y z z prev prev fctvar NULL var NULL pre pre c 1 c 1 c2 c2 Arguments REQUIRED ARGUMENTS y response variable binary 0 1 x matrix of predictor variables z matrix ofthe surrogate or auxiliary variables can be more than one column prev the prevalence of each y z stratum where y z is the different levels of y and z var The name of the predictor variable whose coefficient is to be optimised If this is a factor variable please see DETAILS at the end of this section pre the fixed variance of var coefficient cl the cost per first stage observation c2 the cost per second stage observation 45 OPTIONAL ARGUMENTS fctvar the names of any factor variables in the predictor matrix SIDE EFFECTS The following lists will be returned n the optimal number of observations first stage sample size design a list consisting of the following items ylevel the different levels of response variable zlevel _ the different levels of first stage covariates z prev the prevalence of each ylevel zlevel stratum n2 the sample size of pilot observations for each ylevel zlevel stratum prop optimal 2nd stage sampling proportion for each ylevel zlevel stratum
33. at we have a fixed budget B available and the first stage and second stage cost per observation are known to be c and c respectively Usin the fact that ZY 52 n note that we can write the constraint as g p p ZY V Z Y ZY f V Z Y B zn ci e V p hp where n is the study size and p and p d 2 7 are the prevalence probability and the second stage sampling fraction for the Z Y stratum Again using a Lagrange multiplier our task is to minimise V A nci ne V p p B Z Y where Vw is the k k element of the variance covariance matrix V The optimal study size and second stage sampling fractions can be obtained by taking the derivatives of the V Z Y function above with respect to n and p setting these to zero and solving them simultaneously It can be shown see Reilly and Pepe 1995 for more details of the theoretical derivation that the optimal study size is given by Vee X p JI Var SB Z Ys B Z Y fr J M p r Var Ss Z NI Je Z Y And the optimal second stage sampling fraction for Z Y stratum is given by pen B nc I Var Ss Z Y 1 Ju no V p Jr Var Sg Z Y Ju Z Y n B oc 2 2 3 Fixed precision In this case we would like to achieve a fixed variance estimate say 6 for the k component of the regression coefficient vector B while minimising the study cost Assume again that the first stage cost is cy per observation and second
34. capable browser like Netscape Navigator or Microsoft Internet Explorer e Rweb modules these are point and click interfaces that allow the user to perform standard statistical analyses on built in or user supplied datasets Rweb currently has modules for summary statistics two way tables ANOVA and linear regression We developed four additional modules for calculating the optimal two stage sampling strategies in Reilly 1996 The following section briefly describes the implementation of these modules The rest of the document will explain how to install and configure them 7 1 Implementation of the optimal sampling modules Rweb modules are implemented as dynamically created HTML forms Behind the scenes the scripts collect user choices and convert them to R code Rweb then runs R in batch mode with the submitted code and returns the output printed and graphical in standard HTML format The optimal sampling modules were developed using this same approach Each module is implemented in two script files The setup script builds the HTML form where users select the options they want The run script analyzes and validates user input and builds a text file that contains the appropriate R commands for the selected options For our application we created a cut down version of Rweb with new improved opening and help pages 61 This version excludes the code window in order to minimise the security risks associated with giving direct acce
35. ctions subject to fixed sample sizes at the first and second stage Usage M fixed n x x y y z z n2 n2 fctvar NULL var var nl option prev option frac option Arguments REQUIRED ARGUMENTS y response variable binary 0 1 x matrix of predictor variables z matrix of the surrogate or auxiliary variables can be more than one column n2 size of second stage sample var The name ofthe predictor variable whose coefficient is to be optimised Ifthis is a factor variable please see DETAILS at the end of this section 43 and one of the following nl vector of the first stage sample sizes for each y z stratum OR prev vector of the estimated prevalences for each y z stratum AND frac the second stage sampling fraction i e the ratio of second stage sample size to first stage sample size NOTE if prev is given frac will also be required OPTIONAL ARGUMENTS fctvar the names of any factor variables in the predictor matrix SIDE EFFECTS A list called design consisting of the following items ylevel the different levels of response variable zlevel the different levels of first stage variables z ni the first stage sample size for each ylevel zlevel stratum n2 the sample size of pilot observations for each ylevel zlevel stratum prop optimal 2nd stage sampling proportion for each ylevel zlevel stratum samp 2nd optimal 2nd stage sample size for each ylevel
36. der For this reason we strongly suggest that any call to optfixn 1s preceded by a call to coding For more details about the coding function see section 4 4 Using the Meanscore package in STATA 51 optbud Optimal sampling design for 2 stage studies with fixed budget Command line optbud depvar indepvars if exp in range first varlist prev vecname b cl c2 optvar varname coding Options first varlist specifies the first stage variables prev vecname vector of prevalences for each stratum formed by different levels of dependent variable and first stage covariates b available budget cl cost per study subject at the first stage c2 cost per study subject at the second stage optvar varname the covariate whose variance estimate is to be minimised i e optimised If the covariate is a factor variable you need to specify the level whose coefficient is to be optimised see ANALYSIS WITH CATEGORICAL VARIABLES coding a logical flag default of 0 FALSE means that prior to calling the optbud function you have run the coding function help coding for details to create the vector grp yz containing the distinct groups strata formed by the different levels of response Y and first stage covariates Z If you have not run coding and you call the optbud function with coding 1 the grp yz vector will be created within the optbud function but it is imperative that the vector
37. e analysis using the ms nprev command for the simulated data above Notice that we have to run the coding function first to see the order in which we have to enter the vector of first stage sample sizes data simNA extract the complete cases only complete simNA is na simNA 3 run the coding function coding x simNA 3 y simNA 1 z simNA 2 1 For calls to ms nprev input nl or prev in the following order ylevel z new z nl n2 1 0 0 0 310 150 2 05 4 1166 85 3 10 O 177 86 4 11 1 347 179 supply the first stage sample sizes in the correct order nl c 310 166 177 347 ms nprev x complete 3 z complete 2 y complete 1 nl n1 1 please run coding function to see the order in which you 21 1 must supply the first stage sample sizes or prevalences 1 Type coding for details 1 For calls to ms nprev input n1 or prev in the following order ylevel z new z n2 1 0 0 0 150 2 01 1 85 3 10 0 86 4 11 1 179 1 Check sample sizes prevalences Stable ylevel zlevel n1 n2 1 0 0 310 150 2 0 1 166 85 3 1 O0 177 86 4 1 1 347 179 parameters est se Z pvalue Intercept 0 0493998 0 07155138 0 6904103 0 4899362 x 1 0188437 0 10187094 10 0013188 0 0000000 4 3 Using the Meanscore package in S PLUS 4 3 1 Installation guide Our program has been tested under SPLUS 4 for windows Some modifications may be needed for other versions If you have
38. e coding function a variable called grp_yz is created It contains the distinct groups formed by different levels of the dependent variable Y and first stage covariates Z A list is printed indicating the definition of each 30 stratum For calls to msnprev optfixn optbud and optprec the first stage sample sizes or prevalences must be entered following the order of grp yz 4 4 3 Examples 4 4 3 1 meanscor Again the example illustrated in section 4 2 3 is presented The data set is called sim miss see section 4 1 The following code in STATA will give the same results as illustrated in section 4 2 3 and 4 3 3 for R and S PLUS use sim miss meanscor y x first z second x odd 0 meanscore estimates Coef Std Err Zz P gt z 95 Conf Interval SEC PES H 91 cons 0494025 0715522 0 690 0 490 0908373 1896424 x 1 01891 1018749 10 002 0 000 8192386 1 218581 4 4 3 2 msnprev The ms nprev command provides a way of doing Meanscore analysis if we only have the complete observations but we know the first stage sample size in each stratum The following lines will do the Meanscore analysis using the msnprev command for the simulated data above Notice that we have to run the coding function first to see the order in which we have to enter the vector of first stage sample sizes use sim miss coding y z keep the second stage sample only keep if
39. e following lists That the Fisher information matrix varsi variance of the score for each ylevel zlevel stratum ms nprev Logistic regression of two stage data using second stage sample and first stage sample sizes or proportions prevalences as input BACKGROUND This algorithm will analyse the second stage data from a two stage design incorporating as appropriate weights the first stage sample sizes in each of the strata defined by the first stage variables If the first stage 23 sample sizes are unknown you can still get estimates but not standard errors using estimated relative frequencies prevalences of the strata To ensure that the sample sizes or prevalences are provided in the correct order it is advisable to first run the coding function USAGE ms nprev y y x x z z nl option prev option fctvar NULL print all F REQUIRED ARGUMENTS y response variable should be binary 0 1 x matrix of predictor variables for regression model z matrix of any surrogate or auxiliary variables and one of the following nl vector of the first stage sample sizes for each y z stratum must be provided in the correct order see coding function OR prev vector of the first stage or population proportions prevalences for each y z stratum must be provided in the correct order see coding function OPTIONAL ARGUMENTS print all logical value determining all output to be printed
40. eric If you have multiple columns of z say z1 z2 zn these will be recoded into a single vector new z These new z values are reported as new z in the output see value For example zl z2 73 new z 000 1 100 2 010 3 110 4 001 5 101 6 011 7 111 8 If some of the value combinations do not exist in your data the function will adjust accordingly For example if the combination 0 1 1 is absent then 1 1 1 will be coded as 7 If you wish to optimise the coefficient of a factor variable you need to specify which level of the variable to optimise For example if weight is a factor variable with 3 categories 1 2 and 3 then var Weight2 will optimise the estimation of the coefficient which measures the difference between weight 2 and the baseline weight 1 By default the baseline is always the category with the smallest value 5 2 3 Examples We give an example using the pilot subsample from the CASS data discussed in Reilly 1996 and described briefly in Table 5 1 The data are in the cass2 matrix which can be loaded using data cass2 and a description of the data set can be seen using help cass2 Our model is I RID Bo BISEX Dzweighti Dsagei DACHF Ps ANGINA BoLVEi BiSurgeryi where the response variable is operative mortality In our examples below we use sex and categorical weight as auxiliary variables Given an available budget of 10 000 a first stage cost of 1 unit and second st
41. eta atn annnn 60 7 2 Software needed to run Optimal Sampling Software on Rweb eese 61 8 USING THE RWEB OPTIMAL SAMPLING MODULES 63 APPENDIX Cr MM 65 xc spam 69 BIBLIOGRAPHY sapata p RR pP RA d RR Cu coU DOR GRE e hc T9 List of Tables Table 5 1 Illustrative data sets for optimal package in R S PLUS and STATA sees 34 Table A 1 Simulation Studies to Compare Meanscore and Hotdeck Multiple Imputation 65 Table A 2 Comparison of meanscore and other likelihood based methods using 1000 2 stage observations from CASS data optimal with respect to age see section 6 3 66 Table A 3 Comparison of meanscore and other likelihood based methods using 1000 2 stage observations from CASS data optimal with respect to left ventricular blood pressure see section 6 3 66 Table A 4 Comparison of meanscore and other likelihood based methods using the Ectopic pregnancy data see section 6 3 cs steve e etc esed ies t eee et eee eur E ae IEEE eu YES REUS 67 Table A 5 Comparison of meanscore and other likelihood based methods using the NWTSG data with Institutional Histology as the first stage variables essen 67 Table A 6 Comparison of meanscore and other likelihood based methods using the NWTSG data with Institutional Histology and Stage of Tumor as the first stage variables sess 68 1 Missing covariate data
42. evel zlevel stratum samp 2nd optimal 2nd stage sample size for each ylevel zlevel stratum and a list called se containing se the standard errors of estimates achieved by the optimal design budget Optimal sampling design for 2 stage studies with fixed budget Usage 36 budget x x y y z z prev prev factor NULL var NULL b b c1 c1 c2 c2 Arguments REQUIRED ARGUMENTS y response variable binary 0 1 x matrix of predictor variables z matrix of the surrogate or auxiliary variables can be more than one column prev the prevalence of each y z stratum where y z is the different levels of y and z var The name of the predictor variable whose coefficient is to be optimised If this is a factor variable see DETAILS at the end of this section b the total budget available cl the cost per first stage observation c2 the cost per second stage observation OPTIONAL ARGUMENTS factor the names of any factor variables in the predictor matrix Value The following lists will be returned n the optimal number of observations first stage sample size design a list consisting of the following items ylevel the different levels of the response variable zlevel the different levels of first stage covariates z prev the prevalence of each ylevel zlevel stratum n2 the sample size of pilot observations for each ylevel zlevel stratum prop optimal 2nd stage sampling proportio
43. for other versions or operating systems We would be happy to hear about any bugs that you find and to receive any comments or suggestions for improvements Marie Reilly amp Agus Salim Dept of Epidemiology Dept of Statistics University College Cork University College Cork Ireland Ireland E mail marie reilly ucc ie E mail agus astat ucc ie 76 S Optimal package in S PLUS This library contains functions for calculating the optimal two stage sampling strategies in Reilly 1996 Briefly the methods are applicable in studies where some categorical covariates Z and a dichotomous outcome variable Y are to be measured at the first stage and additional covariates X which may be continuous are to be gathered on a subsample at the second stage Logistic regression analysis of all the data will then proceed using the Mean Score method Reilly and Pepe 1995 In addition to the total sample size the variance of the Mean Score estimate depends on the second stage sampling fractions in each of the Y Z strata Hence the study size and or the second stage sampling fractions can be optimised The three functions here provide the optimal sampling strategies under different constraints fixed n calculates the sampling fractions at the second stage given fixed first and second stage sample sizes which will minimise the variance of a specified coefficient in the regression model budget calculates the fir
44. g the information about the older version in the beginning of the program For example our program uses the version 6 0 command to inform STATA the version we used to write it 13 4 Meanscore package 4 1 Package Features There are many similarities between our packages in R S PLUS and STATA In this section we highlight common features shared by the packages and advise you to read this section regardless of which software you intend work with Sections 4 2 4 3 and 4 4 deal with some minor differences in the package for the different software environments The Meanscore package contains functions to implement the Meanscore method Reilly and Pepe 1995 for estimating the coefficients in a logistic regression model from two stage data There are 3 functions in the package 1 MEANSCORE is called with the combined first and second stage data where the missing covariate values are represented by NA in R and S PLUS and in STATA missing values are represented by a dot 2 MS NPREV is called with the second stage i e complete data and the first stage sample sizes or prevalences Prior to running this function the CODING function 3 should be run to see the order in which MS NPREV expects the first stage sample sizes or prevalences to be provided 3 CODING which recodes multiple columns of first stage covariates into a single column and displays the coding scheme Two illustrative data sets are also provided with the package
45. h montana edu Rweb Salim A and Reily M 2000 Practical problems arising in computing optimal sampling designs for two stage studies Presentation at the Research student conference RSC 2000 Cardiff Sherman K J et al 1990 Sexually transmitted diseases and tubal pregnancy Sex Transm Dis 7 115 21 SPLUS http www mathsoft com splus STATA http www stata com 80 The R Project for Statistical Computing http www r project org Venables W N and Ripley B D 1999 Modern Applied Statistics with S PLUS Third Edition Springer Venables W N and Ripley B D 2000 S Programming Springer Vliestra R E Frye R L Kromnal R A et al 1980 Risk factors and angiographic coronary artery disease a report from the Coronary Artery Surgery Study CASS Circulation 62 254 61
46. he number of complete validation cases with Z Z and Y Y The meanscore estimator is unbiased and has asymptotic variance given by Reilly amp Pepe 1995 pm S la Q7 IO n where I the observed Fisher information matrix n V Z Y Q Y gt valse X Z Y Z E V Z Y Z Y n n total number of observations study size V Z Y n The term zy is referred as the validation sampling fraction or second stage sampling fraction for n the Z Y stratum It can be seen from the form of the variance formula that the variance of the estimates is a function of the number of observations and the validation sampling fraction in each Z Y stratum Thus it is possible to minimise the variance using an appropriate study design We will develop this idea further in Chapter 2 2 Optimal Sampling Design for two stage studies We define a two stage study as a study where a response variable and some predictor variables are measured at the first stage for all study subjects and one or more predictor variables are collected only for a subset of the study subjects at the second stage The second stage subjects are selected using stratified random sampling within the strata defined by the different levels of response and first stage variables This type of study is popular in epidemiology where researchers usually collect information on some cheap variables from all study subjects while expensive variables
47. his function calculates the total number of study observations and the second stage sampling fractions that will minimise the study cost subject to a fixed variance for a specified coefficient In addition to specifying the required variance the user must also supply 1 the unit cost of observations at the first and second stage 11 the vector of prevalences in each of the strata defined by different levels of dependent variable and first stage covariates and iii the name of the variable to be optimised Before running this function you should run the coding function to see the order in which you must g y g y supply the vector of prevalences In STATA this function is called optprec while R and S PLUS both use the same name precision coding combines two or more surrogate auxiliary variables into a vector The coding function is very important because it must be run prior to running any of the optimal sampling functions in order to see the order in which you should enter the vector of prevalences or first stage sample sizes see syntax and features under the different software This function has already been described in section 4 2 2 4 3 2 and 4 4 2 Illustrative datasets There are two illustrative data sets provided as examples with the package The data sets have slightly different names from software to software see Table 5 1 34 Table 5 1 Illustrative data sets for optimal package in
48. hod for two stage validation sampling for estimation of the odd ratio OR in a logistic regression analysis essentially the same problem we addressed These authors offer a FORTRAN programme for calculating the optimal design We obtained the program from Dr Spiegelman stdls channing harvard edu The program uses the FFSQP routine by Dr Andre Tits andre isr umd edu which is free for non commercial purposes After some email correspondence we managed to get a copy of the routine by linking it with the main program we could run the optimal design calculations This program only accommodates a single predictor and one surrogate variable in the analysis In addition the user must supply the prevalence of the outcome P Y 1 the sensitivity and specificity of the surrogate variable the prevalence of the surrogate variable P V 1 the odd ratio estimates and the 2 stage sampling fraction Unfortunately the program failed to run for many settings even using the same parameters the authors considered in the paper For one setting for which the program successfully ran it returned the optimal sampling design for different sampling schemes OPTIMAL BALANCED RANDOM CASE CONTROL and PROSPECTIVE The OPTIMAL sampling scheme here is a hybrid design which is the same as what we considered in our algorithm Hence we compared the OPTIMAL sampling scheme with our results using the same set of parameters However it appears that the results
49. ing designs using the Meanscore algorithm AUTHORS Marie Reilly Dept of Epidemiology amp Public Health and Agus Salim Dept of Statistics University College Cork UCC Cork Ireland SUPPORT marie reilly ucc ie 78 INSTALLATION Stata version 6 Installing from internet please check U 20 6 How do I install an addition R net Installing from floopy disk type net from a net install optimal to install program net get optimal to access illustrative datasets Installing from C drive type net from c dirname net install optimal to install program net get optimal to access illustrative datasets where dirname is the name of the directory where you put the source files HELP After installation from inside Stata online help is available by typing help coding help optfixn help optbud help optprec 79 Bibliography Banfield J 1999 Rweb web based statistical analysis Journal of Statistical Software 4 1 15 Breslow N E and Cain K C 1988 Logistic regression for two stage case control data Biometrika 75 11 20 Breslow N E and Holubkhov R 1997 Maximum likelihood estimation of logistic regression parameters under two phase outcome dependent sampling J R Statist Soc B 59 447 61 Breslow N E and Chatterjee N 1999 Design and analysis of two phase studies with binary outcome applied to Wilms tumor prognosis
50. install the package Once the package has been installed you can make the meanscore package available by issuing the command ibrary meanscore in the R session window The command help package meanscore will open a window where you can read more details about the package 4 2 2 Syntax and features meanscore Mean Score Method for Missing Covariate Data in Logistic Regression Models Usage meanscore y y x x z z factor NULL print all F Arguments y response variable binary 0 1 x matrix of predictor variables one column of which contains some missing values NA z matrix ofthe surrogate or auxiliary variables which must be categorical factor optional factor variables if the columns of the matrix of predictor variables have names supply these names otherwise supply the column numbers MEANSCORE will fit separate coefficients for each level of the factor variables Value A list called parameters containing the following will be returned est the vector of estimates of the regression coefficients se the vector of standard errors of the estimates z Wald statistic for each coefficient 16 pvalue 2 sided p value Ho est 0 when print all T it will also return the following lists That the Fisher information matrix varsi variance of the score for each ylevel zlevel stratum ms nprev Logistic regression of two stage data using second stage sample and first stage sample sizes or p
51. ist called table containing ylevel the distinct values or levels of y zlevel the distinct values or levels of z nl the sample size at the first stage in each y z stratum n2 the sample sizes at the second stage in each stratum defined by y z and a list called parameters containing est the Mean score estimates of the coefficients in the logistic regression model se the standard errors of the Mean Score estimates z Wald statistic for each coefficient pvalue 2 sided p value Ho est 0 If print all T the following lists will also be returned Wzy the weight matrix used by the mean score algorithm for each Y Z stratum this will be in the same order as nl and prev varsi the variance of the score in each Y Z stratum That the Fisher information matrix coding combines two or more surrogate auxiliary variables into a vector DESCRIPTION recodes a matrix of categorical variables into a vector which takes a unique value for each combination BACKGROUND From the matrix Z of first stage covariates this function creates a vector which takes a unique value for each combination as follows zl z2 73 new z 000 1 100 2 010 3 18 110 4 001 5 101 6 011 7 111 8 If some of the combinations do not exist the function will adjust accordingly for example if the combination 0 1 1 is absent above then 1 1 1 will be coded as 7 The values of this new z are reported as new z in the prin
52. l 3 6 0 1 1 1 1 2 0 2 3 1 2 4 0 3 5 1 3 6 response variable dauxiliary variables predictor variables to see in which order we should enter prevalences ms nprev input nl or prev in the following n2 10 10 10 10 10 10 8 10 10 10 10 10 the prevalences from Table 5 Reilly 1996 prev c 0197823937 0 0544015826 0 1339020772 0 0503214639 0 6698813056 0 0467359050 0 0009891197 0 0022255193 0 0040801187 0 0032146390 0 0127349159 0 0017309594 optimise the surg coefficient budget x x y y Z Z var Surg prev prev b 10000 c1 1 c2 0 5 1 please run coding function first 1 you have to supply the first sample sizes 1 For calls to ms nprev input nl or prev in the following order ylevel 1 0 2 0 3 0 4 0 5 0 6 0 7 1 8 1 9 1 10 1 12 1 1 Check n 1 8752 design ylevel al 0 2 0 3 0 4 0 5 0 6 0 7 1 8 1 9 1 10 1 TE 1 12 1 zl z2 new z n2 0 1 1 1 0 2 l 2 0 3 l 3 Oo 1 1 1 0 2 l 2 O 3 l3 10 10 10 10 10 10 8 10 10 10 10 Nn Ul d W N FP WD Ul d W N Hn 10 48 it will give you idea on which order sample sizes prevalences zlevel ON oO FF W PN FP WD Ul dg W NH Oo O oo O O O O O O O QO 003214 012734 prev 0197823937 0544015826 1339020772 0503214639 6698813056 0467359050 0009891197 0022255193 0040801187 16390 19159 0017309594 n2 10 10
53. lumns of z say z1 z2 zn these will be recoded into a single vector new z These new z values are reported as new z in the output see value zl z2 z3 new z 000 1 10 e o o o a O O 2 3 4 5 6 7 8 e js If some of the value combinations do not exist in your data the function will adjust accordingly For example if the combination 0 1 1 is absent then 1 1 1 will be coded as 7 If you wish to optimise the coefficient of a factor variable you need to specify which level of the variable to optimise For example if weight is a factor variable with 3 categories 1 2 and 3 then var Weight2 will optimise the estimation of the coefficient which measures the difference between weight 2 and the baseline weight 1 By default the baseline is always the category with the smallest value 5 3 3 Examples Here we show the same example as in section 5 2 3 above We assume as before that we have a 10 000 47 budget and the first stage and second stage cost per observations are 1 and 0 5 respectively Suppose we would like to optimise the precision of the urgency of surgery surg coefficient y cass2 1 z cass2 c 2 3 x cass2 c 2 4 9 run CODING function coding x x y y z z 1 For ylevel new z ylevel O o 0 JN O Ul FF W N B H 11 12 enter 0 0 0 0 0 0 calls to order zl z2 new z O0 1 1 loc d 2 0 2 3 1 2 4 0 3 5
54. ly amp Pepe Biometrika 1995 This likelihood based method is asymptotically equivalent to hot deck multiple imputation Reilly amp Pepe Stats in Medicine 1997 Missingness may depend on the available response and covariate values but not on the unobserved covariate values i e MAR Missing At Random and the method is applicable to cohort or case control designs The subsample of subjects on whom the incomplete covariate is available is referred to as the validation sample or the second stage sample and the remaining subjects are the non validation sample or the first stage sample The code provided here implements a Mean Score analysis for a logistic regression model where the incomplete covariate s may be continuous but the first stage covariates and or auxiliary variables must be categorical After extracting the ZIP file in the library subfolder of your SPlus directory the command gt library meanscore makes available the following 3 functions 1 MEANSCORE this function is called with the combined first and second stage data where the missing values in the incomplete covariate s are represented by NA the usual notation in Splus R 2 MS NPREV this function is called with the second stage i e complete data and the first stage sample sizes or prevalences if only prevalences are available then estimates are provided but no standard errors Prior to running this function the CODING function 3
55. method for logistic regression models Users need to provide the first stage sample size for each stratum the second stage sample size required and the name of the predictor variable to be optimised Optimality is with respect to the standard error of a coefficient of interest Before running the fixedn function you should run the coding function to see the order in which you g y g y must supply the vector of first stage sample size In STATA this function is called optfixn while R and S PLUS both use the name fixed n budget Optimal sampling design for 2 stage studies with fixed budget Description This function calculates the total number of study observations and the second stage sampling fractions that will maximise precision subject to an available budget In addition to specifying the budget the user must also supply 1 the unit cost of observations at the first and second stage 11 the vector of prevalences in each of the strata defined by the different levels of response variable and first stage variables and iii the name of the variable to be optimised 33 Before running the budget function you should run the coding function to see the order in which you must supply the vector of prevalences In STATA this function is called optbud while R and S PLUS both use the same name budget precision Optimal sampling design for 2 stage studies with fixed precision Description T
56. missing x input the first stage sample sizes for each stratum matrix samp 310 166 177 347 msnprev y x first z sample samp odd 0 the second stage sample sizes lxi quem ec group y z Freq uui E E quce ees 1 150 2 85 3 86 4 179 sage sea FIRE en le A grp yz y z grp z ni n2 1 0 0 i 310 150 2 0 1 2 166 85 3 1 0 1 177 86 4 1 1 2 347 179 meanscore estimates Coef Std Err Zz P z 95 Conf Interval n e re cons 0494025 0715522 0 690 0 490 0908373 1896424 x 1 01891 1018749 10 002 0 000 8192386 1 218581 32 5 Optimal package 5 1 Package Features The optimal package derives optimal sampling designs for 3 different scenarios as outlined in Chapter 2 To run any of the three functions you need to have a pilot sample typically consisting of a few observations from each stratum defined by the different levels of response variable and first stage variables The names of the functions are slightly different for the different software but they implement the same algorithm We summarise the three functions below fixed n Optimal second stage sampling fractions subject to fixed sample sizes at the first and second stage Description This function computes the optimal second stage sampling fractions and sample sizes for each stratum defined by the different levels of response variable and first stage variables using the mean score
57. n for each ylevel zlevel stratum samp 2nd optimal 2nd stage sample size for each ylevel zlevel stratum se the standard error of estimates achieved by the optimal design precision Optimal sampling design for 2 stage studies with fixed precision Usage 37 precision x x y y z z prev prev factor NULL var NULL pre pre c 1 c 1 c2 c2 Arguments REQUIRED ARGUMENTS y response variable binary 0 1 x matrix of predictor variables z matrix of the surrogate or auxiliary variables can be more than one column prev the prevalence of each y z stratum where y z is the different levels of y and z var The name of the predictor variable whose coefficient is to be optimised See DETAILS at the end of this section if this is a factor variable pre the required variance of the var coefficient cl the cost per first stage observation c2 the cost per second stage observation OPTIONAL ARGUMENTS factor Value the names of any factor variables in the predictor matrix The following lists will be returned n the optimal number of observations first stage sample size var the variance of estimates achieved by the optimal design cost the minimum study cost and a list called design consisting of the following items ylevel zlevel prev n2 prop samp 2nd the different levels of response variable the different levels of first stage covariates z the prevalence of each
58. n selected the user clicks the submit button The system will attempt to open the dataset and report any errors encountered in the process If no errors were detected you will be presented with the analysis page The options in this page will depend on the module you selected in the previous step You simply follow the instructions on the screen and click submit when you have provide d all the parameters required for the analysis APPENDIX Appendix A Results of Comparative Study of Meanscore and Hotdeck Table A 1 Simulation Studies to Compare Meanscore and Hotdeck Multiple Imputation 65 N p No Imp Method Random Sampling Balanced Samping Mean D Monte Estimated Bias 95 Mean B Monte Estimated Bias 96 95 Carlo SE coverage Carlo SE coverage SE SE 200 0 5 10 Meanscore 1 032 0 2335 0 2268 2 87 0 936 1 046 0 2426 0 2283 5 89 0 948 ABB 1 044 0 2420 0 2282 5 70 0 942 1 055 0 2468 0 2286 7 37 0 938 Hotdeck 1 037 0 2370 0 2101 11 35 0 912 1 05 0 2462 0 2131 13 44 0 918 200 0 25 10 Meanscore 1 023 0 3166 0 2896 8 53 0 934 1 046 0 3084 0 2866 7 07 0 948 ABB 1 052 0 3332 0 2934 11 94 0 924 1 072 0 3222 0 2854 11 42 0 942 Hotdeck 1 029 0 3195 0 2186 31 58 0 850 1 053 0 3113 0 2218 28 75 0 844 200 0 5 3 Meanscore 1 037 0 2388 0 2284 4 36 0 938 1 037 0 243 0 2279 6 21 0 944 ABB 1 048 0 2520 0 2336 7 30 0 936 1 045 0 253
59. ng scheme We used two different sampling fraction p 0 25 and p 0 5 Under each setting 3 and 10 imputations were used to compare the effect on the estimates of the increasing number of imputations We performed 500 simulations to study the variability of the estimates In every simulation the data are analysed using 3 different methods simple hotdeck hotdeck using the ABB method and the meanscore method We developed R functions for simple hotdeck and hotdeck using the ABB method The function for hotdeck using the ABB method was built using the same procedure suggested by Mander and Clayton 2000 Several statistics were calculated and compared Mean p the average of the estimate from all 500 simulations Monte Carlo SE the true standard error of the estimate computed from 500 estimates as square Usa dio z root of 2 E where B is the estimate from the i simulation and B so i l is the average of the estimate as listed in Mean B column Estimated SE represents the average over 500 simulations of the estimated standard error using the appropriate variance formula for the methods This was computed as 500 the square root of m2 var 3 where var Q is the variance estimate from i l the i simulation Bias measures the bias of the standard error estimates of each method it was computed as the difference between the Monte Carlo SE and estimated standard error relative to the Monte Carlo standard error 95 coverage the
60. om C drive type net from c dirname net install meanscor to install program net get meanscor to access illustrative datasets where dirname is the name of the directory where you put the source files HELP After installation from inside Stata online help is available by typing help meanscor help msnprev help coding 74 4 Optimal package in R This library contains functions for calculating the optimal two stage sampling strategies in Reilly 1996 Briefly the methods are applicable in studies where some categorical covariates Z and a dichotomous outcome variable Y are to be measured at the first stage and additional covariates X which may be continuous are to be gathered on a subsample at the second stage Logistic regression analysis of all the data will then proceed using the Mean Score method Reilly and Pepe 1995 In addition to the total sample size the variance of the Mean Score estimate depends on the second stage sampling fractions in each of the Y Z strata Hence the study size and or the second stage sampling fractions can be optimised The three functions here provide the optimal sampling strategies under different constraints fixed n calculates the sampling fractions at the second stage given fixed first and second stage sample sizes which will minimise the variance of a specified coefficient in the regression model budget calculates the first stage sam
61. ond varlist specifies the incomplete covariates i e measured at the second stage odd 0 reports regression co efficients default 1 reports odds ratios 29 msnprev Meanscore method for missing covariate data in logistic regression models using validation second stage data and first stage sample sizes or prevalances Command line msnprev depvar indepvars if exp in range first varlist prev vecname sample vecname odd 0 Options first varlist specifies the first stage covariates odd 0 reports regression co efficients default odds ratios and one of the following sample vecname vector of the first stage sample sizes for each stratum OR prev vecname vector of the prevalences for each stratum If prevalences are provided no standard errors are estimated NOTE you have to run the coding function see below prior to using this function in order to know the order in which to enter the prev or sample vector coding orders the strata formed by different levels of dependent variable and first stage covariates Command line coding depvar first stage variables Description The coding function orders the strata formed by different levels of the dependent variable and first stage covariates This is the order in which the vector of first stage sample sizes or prevalences must be entered before calling msnprev or any of the optimal sampling functions described in section 5 4 Within th
62. ple size and the second Stage sampling fractions that will maximise precision of a specified coefficient subject to a given budget precision calculates the first stage sample size and the second Stage sampling fractions that will minimise cost subject to a given precision for a specified coefficient Each of the functions requires pilot data on Z Y X as input this would typically be a small number of X observations in each of the Y Z strata Knowledge is also required of the prevalences of these strata in the population which can be provided as estimates or can be computed from the first stage data if available 75 INSTALLATION GUIDE The simplest installation can be done by unzipping the BINARY package directly to R HOME library For users familiar with building R packages the following command can be used to install the package make BUILD option pkg optimal executed from R HOME src gnuwin32 after you have unzipped this package to R HOME src library Alternatively you can use the UNIX command Remd install optimal executed from R HOME src library After the package has been installed the command library optimal will make the functions available Detailed help on each function and on the illustrative data sets cassl cass2 can then be viewed by using help or or the HTML help file system This code has been tested under R1 2 0 for windows some changes may be needed
63. predictor matrix 50 Since the syntax and features section below are written based on the version in our website there may be some minor differences from the STB help files 5 4 2 Syntax and features optfixn optimal sampling design for 2 stage study with fixed second stage sample size Command line optfixn depvar indepvars if exp in range first varlist nl vecname n2 optvar varname coding Options first varlist nl vecname n2 optvar coding specifies the first stage variables vector of first stage sample sizes for each stratum formed by different levels of dependent variable and first stage covariates second stage sample sizes the covariate whose variance estimate is to be minimised i e optimised If the covariate is a factor categorical variable you need to specify the level whose coefficient is to be optimised see ANALYSIS WITH CATEGORICAL VARIABLES a logical flag default of 0 FALSE means that prior to calling the optfixn function you have run the coding function help coding for details to create the vector grp yz containing the distinct groups strata formed by the different levels of response Y and first stage covariates Z If you have not run coding and you call the optfixn function with coding 1 the grp yz vector will be created within the optfixn function but it is imperative that the vector vecname is provided to optfixn in the correct or
64. pregnancy data set Sherman et al 1990 This dataset which was analysed in Table 3 of Reilly and Pepe 1995 is from a case control study of the association between ectopic pregnancy and sexually transmitted diseases STDs The total sample size is 979 consisting of 264 cases and 715 controls The variables collected from the beginning of the study included gonnorhoea contraceptive use and sexual partners One year after the study began the investigators started collecting serum samples for determining chlamydia antibody status in all cases and in a 50 percent subsample of controls As a result only 327 out of the 979 patients have measurements for chlamydia antibody This dataset is described briefly in section 4 1 4 National Wilms Tumor dataset with Institutional Histology as the first stage variable Breslow and Chatterjee 1999 This dataset comes from the National Wilms Tumor Study Group NWTSG There are 3 variables in the dataset the treatment outcome relapse or not the type of tumor favourable histology FH or unfavourable histology UH measured at the NWTSG pathology centre Central Histology and the type of tumor predicted by pathologists at the participating institutions Institutional Histology There are 4088 patients in the original dataset In this dataset the treatment outcome is the response variable the Institutional Histology is the first stage variable and the Central Histology is the second stage variable
65. proportion of simulations where the nominal 95 confidence interval for B covers the true B 1 The results are presented Table A 1 As expected all methods give unbiased estimates for B 1 with the Meanscore giving the most stable estimate least variability Not surprisingly the largest departure from B 1 occurs when the validation sampling fraction is small p 0 25 and there are small number of imputations 3 imputations 57 By comparing the estimated SE and Monte Carlo SE columns under each sampling scheme we see that simple hotdeck consistently gives standard error estimates which are biased downward for all sampling fractions and number of imputations Examining the interval estimates nominal 95 confidence intervals for the simple hotdeck gives a low coverage which is not suprising in view of the bias discussed above This effect is more obvious for smaller validation sampling fraction where the coverage of the nominal 95 confidence interval never exceeded 87 The Mean Score and ABB hotdeck meanwhile give good coverage for all settings We conclude that the Meanscore method works as well as the hotdeck multiple imputation using Approximate Bayesian Boostrap This conclusion is supported by Reilly and Pepe 1997 who proved that the meanscore estimate has the same asympotic distribution as the hotdeck estimate with infinite number of imputations But meanscore has an advantage since it can produce the estimates in onl
66. pstopnm ppmtogif pnmcrop and pnmflip This is used by Rweb to convert R images to GIF images It can be downloaded from ftp wuarchive wustl edu graphics graphics packages NetPBM 8 The Rweb package version 1 03 62 Downloadable from http www math montana edu Rweb Resources html Read the included Readme file for information on how to install and configure Rweb 9 The optimal sampling R package Section 5 2 This library contains the optimal two stage sampling functions Consult your R documentation for information on how to install R add on packages The package is available from lt http www ucc ie depts ucc pubh programs programs html gt 10 The optimal sampling Rweb modules This module moves the CGI scripts to the AnalysisModules subdirectory in the Rweb CGI directory It also moves the data files and their associated description files to the DataSets directory 63 8 Using the Rweb optimal sampling modules The opening HTML page introduces the three RWeb modules for calculating the optimal two stage sampling designs as explained above Rweb is a Web based interface to R a statistical analysis package that takes the user submitted code runs R on the code in batch mode and returns the output printed and graphical Clicking on a link at the bottom of this opening page brings the user to the main screen This screen allows the user to 1 Select a module to use There are three modules and one
67. ptimal sampling fraction sample size for grp yz 10 1 18 the optimal sampling fraction sample size for grp yz 11 1 26 the optimal sampling fraction sample size for grp yz 12 1 18 the optimal number of obs 8756 the minimum variance for surg 0311946 total budget spent 10001 Note The optimal study size and second stage sampling fractions are slightly different from those obtained using R and S PLUS but this is simply because we rounded the prevalence vector 55 6 Comparative studies 6 1 Motivation The purpose of this chapter 1s to compare our methods with other approaches to the analysis of incomplete data and optimal two stage sampling Meanscore is only one of many approaches to the analysis of incomplete data There are many other methods such as multiple imputation and likelihood based methods such as pseudo likelihood Cain and Breslow 1988 and maximum likelihood Breslow and Holubkhov 1997 In section 6 2 we compare meanscore with hotdeck multiple imputation while in section 6 3 we compare meanscore to other likelihood based methods Our optimal sampling methodology relies on the Meanscore method in particular the variance formula We looked for published work and associated software which adopted an alternative approach so that we could compare the performance of our method This is described in section 6 4 6 2 Meanscore and Hotdeck multiple imputation Hotdeck is a non parametric version
68. r will be created within the optprec function but it is imperative that the vector vecname is provided to optprec in the correct order For this reason we strongly suggest that any call to optprec is preceded by a call to coding For more details about the coding function see section 4 4 Using the Meanscore package in STATA ANALYSIS WITH CATEGORICAL VARIABLES When we have categorical predictor variables we usually fit separate coefficients for each category The xi command prefix is a standard STATA command which can be used with optfixn optbud and optprec to accommodate this need STATA will create some new variables with names l varname level For example variable Isex_1 is a categorical variable for variable SEX with level 1 If you want to optimise the variance estimates for this variable you should set optvar Isex_1 in the command syntax xi optfixn mort i sex age first sex nl fstsamp n2 1000 optvar Isex 1 5 4 3 Examples We illustrate the same example as discussed in sections 5 2 3 and 5 3 3 We assume as before that we have a 10 000 budget and the first stage and second stage cost per observation are 1 and 0 5 respectively The first stage variables are sex and categorical weight Suppose we would like to optimise the precision of the urgency of the surgery surg coefficient The following STATA commands will run the analysis use wtpilot coding mort sex wtcat enter the prevalences in the order
69. riable Y in the 1 column dichotomous surrogate variable for X called Z in the 2 column and continuous predictor variable X in the 3 column A randomly selected 500 of the X values have been deleted i e are missing We would like to use all the data to estimate the coefficient of X in a logistic regression model P Yiz1 pp ED Bot Biki e 1 P Y 1 data simNA meanscore y simNA 1 z simNA 2 x simNA 3 OUTPUT 1 For calls to ms nprev input nl or prev in the following order ylevel z new z nl n2 1 0 0 0 310 150 2 O xL 1166 85 3 l 0 0 177 86 4 11 1 347 179 parameters est se Z pvalue Intercept 0 0493998 0 07155138 0 6904103 0 4899362 x 1 0188437 0 10187094 10 0013188 0 0000000 20 We can extract the complete cases and do analysis based on those cases only as follows complete simNA is na simNA 3 summary glm complete 1 complete 3 family binomial OUTPUT Coefficients Estimate Std Error z value Pr z Intercept 0 05258 0 09879 0 532 0 595 complete 3 1 01942 0 12050 8 460 2e 16 Notice that the standard error produced by Meanscore is smaller reflecting the additional information we gained by using the available cases 4 2 3 2 ms nprev The ms nprev command provides a way of doing Meanscore analysis if we only have the complete observations but we know the first stage sample size in each stratum The following lines will do the Meanscor
70. roportions prevalences as input Usage ms nprev y y x x z z n1 option prev option factor NULL print all F Arguments REQUIRED ARGUMENTS y response variable should be binary 0 1 x matrix of predictor variables for regression model z matrix of any surrogate or auxiliary variables and one of the following nl vector of the first stage sample sizes for each y z stratum must be provided in the correct order see coding function OR prev vector of the first stage or population proportions prevalences for each y z stratum must be provided in the correct order see coding function OPTIONAL ARGUMENTS print all logical value determining all output to be printed The default is False F factor factor variables if the columns of the matrix of predictor variables have names supply these names otherwise supply the column numbers MS NPREV will fit separate coefficients for each level of the factor variables Value If called with prev will return only A list called table containing the following ylevel the distinct values or levels of y zlevel the distinct values or levels of z prev the prevalences for each y z stratum 17 n2 the sample sizes at the second stage in each stratum defined by y z and a list called parameters containing p g est the Mean score estimates of the coefficients in the logistic regression model If called with n1 it will return A l
71. s sna 34 5 2 1 Installation guide n o RU e RR RS eri SE NIRE ROTEN a te Sage 34 5 2 2 Syntax and features 5 Asse en ee cud ERO D RN eid tree thas 34 5 2 3 Examples iss tuts esti ee en nente t en Ue E eR Needed 39 5 3 Using the Optimal package in S PLUS 4 eres eee eee essen seen sensns sensns enne tuse sn setae 42 5 3 1 Installation g ide etre o PE e ER DAUERTE UC det eres 42 5 3 2 Syntax and feats n nnee ir qat ie tee ie n e atia d ee e ede e e ent 42 5 3 3 Examples ute eu ie tide e e e Redes etit 47 5 4 Using the Optimal package in STATA csscsssssssssscsssesssesssessssessnssssssseseneseneeseessseseeseessoees 49 5 4 1 Installation guide dior ies E Ge Ora RUSO paa ei ein veta 49 5 4 2 Syntax and features ena ntpote n ea erbe E IE us 50 5 4 3 Examples Fasc too onn E D AL Rt a s a p S DUE 52 6 COMPARATIVE STUDIES 25 san nana dia cue Ete 55 6 1 Motivalions x 55 6 2 Meanscore and Hotdeck multiple imputation eee eee eee eene eene en setenta attain etn ennnn 55 6 3 Meanscore and other likelihood based method eeeeceee eerte eren eren ette 57 6 4 Optimal Two stage Validation Studies eee eee eee eee eee tenente nete seta seta ann 59 7 RWEB MODULES FOR OPTIMAL SAMPLING DEVELOPMENT AND CONFIGURATION sans cc eee eed 60 7 1 Implementation of the optimal sampling modules eere eee eerie eene tenete n
72. scribe in the body of an e mail not in the subject to r help request lists R project org There are many contributed packages that can be downloaded from the R website and this kind of contribution is the strength of R because it enables the software to seamlessly include many useful statistical functions designed by a large expert user base Some introductory manuals for R can be found on the R website The Venables and Ripley 1999 book on S PLUS can also be used as an introduction to R provided it is accompanied by its R Complements http www stats ox ac uk pub MASS3 which 12 describe how to use the book with R The newer book by Venables and Ripley 2000 gives a deeper introduction to programming issues and also discusses some major differences between S and R 3 2 S PLUS The S PLUS software was based on the S language originally designed by John Chambers and colleagues at Bell laboratories S PLUS is sold by Mathsoft Inc More details about S PLUS can be found on the S PLUS website http www mathsoft com splus A good introduction to the application of S PLUS in many statistical areas can be found in Venables and Ripley 1999 There is a huge amount of user contributed code for S available at the http lib stat cmu edu DOS S at Carnegie Melon University Discussion about the main differences between S S PLUS and R is available at the R FAQ section of the R website and in the recent book by Venables and Ripley 20
73. sion 6 so you need to have STATA 6 or later to be able to use the package Because STATA Technical Bulletin require package s name to be no longer than 7 characters the meanscore package in STATA is called meanscor The meanscor package can be installed directly from the STATA website by following these instructions 28 From inside STATA type net cd stb net cd stb58 net describe sg156 net install sg156 net get sg156 After executing the last command the meanscor package is installed in your computer To test 1f you have installed all the components you can type help meanscor or help msnprev and try some of the examples IMPORTANT Since we submitted the program to the STB we have improved the calling syntax and the output format of the functions The most recent version is available on our website at http www ucc ie ucc depts pubh programs programs html This program is slightly different from the one in the STB website Therefore the syntax and features section below are written based on the version on our website and there may be some minor differences from the STB help files 4 4 2 Syntax and Features meanscor Meanscore method for missing covariate data in logistic regression models Command line meanscor depvar indepvars 1f exp in range first varlist second varlist odd Options first varlist specifies the complete covariates i e measured at the first stage sec
74. ss to R and operating system commands 7 2 Software needed to run Optimal Sampling Software on Rweb Unix Linux Operating System Rweb was originally developed on a Sun workstation We installed and tested Rweb on an Intel Pentium machine running RedHat Linux version 6 2 without any modifications Linux is a free open source operating system that is widely used to power Internet servers For more information on downloading and installing Linux visit www redhat com 2 Unix Web server e g the Apache Web Server We used Apache to test Rweb in our setup Rweb does not use any specific feature of the Apache server so it should run on any Unix Web server Apache is a free open source web server and is available for download from many web sites including www apche org 3 R version 1 1 1 or greater R is downloadable from http ww w r project org 4 Perl version 5 004 or greater Perl is downloadable from http www perl com pace pub perldocs latest html 5 The following Perl modules e LWP Perl module for accessing URL s through Perl The LWP module is part of libwww and is available at http www perl com CPAN local CPAN html www e CGI Perl module for uploading local files and formatting some of the html output It can be downloaded from lt http stein cshl org WW W software CGI gt 6 Ghostscript Ghostscript is available at http www cs wisc edu ghost aladdin get5 10 html 7 The NetPBM libaray
75. st stage sample size and the second stage sampling fractions that will maximise precision of a specified coefficient subject to a given budget precision calculates the first stage sample size and the second stage sampling fractions that will minimise cost subject to a given precision for a specified coefficient Each of the functions requires pilot data on Z Y X as input this would typically be a small number of X observations in each of the Y Z Strata Knowledge is also required of the prevalences of these strata in the population which can be provided as estimates or can be computed from the first stage data if available 77 After extracting the ZIP file in the library subfolder of your SPLUS directory the command library optimal will make the functions available Detailed help on each function and on the illustrative data sets cassl cass2 can then be viewed by issuing the help or command The library has only been tested under S PLUS 4 for windows so some changes may be needed for other versions or operating systems We would be happy to hear about any bugs that you find and to receive any comments or suggestions for improvements Marie Reilly amp Agus Salim Dept of Epidemiology Dept of Statistics University College Cork University College Cork Ireland Ireland E mail marie reilly ucc ie E mail agus stat ucc ie 6 Optimal package in STATA SUBJECT Optimal sampl
76. stage cost is c per observation Note that as in the fixed budget case above we can write the total study cost as n ci cz2 gt p p Using Z Y a Lagrange multiplier we now wish to minimise the following function V Z Y ZY nci nci gt p 2 92 AVe 8 Z Y The optimal solution can be obtained by taking the first derivative of this function with respect to n and NP setting them to zero and solving the resultant equations simultaneously p It can be shown Reilly amp Pepe 1995 that the optimal study size is given by Pan p Wa ne Z Y E e by ERR ME gt p Wa and the optimal second stage sampling fraction for the Z Y stratum is given by pen Ci Wer Ju C2 E b3 p Wa Z Y Where Wz I Var Ss Z Y 2 3 Computational Issues The derivation of the optimal designs above was carried out without constraining the second stage sampling fractions to be less than or equal to 1 p lt 1 As a result the optimal second stage sampling fractions computed with these formulae can be greater than 1 Reilly and Pepe 1995 proposed the following ad hoc method to overcome this problem sample 100 from the stratum with the largest p gt 1 and optimally sample from the remaining strata This step was done iteratively until all p 1 In more recent work Salim and Reilly 2000 this ad hoc method was shown to be equivalent to the active set method discussed by Fletcher
77. subject to fixed sample sizes at the first and second stage Usage M W n fixed n x x y y z z n2 n2 factor NULL var var n1 option prev option frac option Based on Kurt Hornik s suggestion the optimal package on the R website has been renamed twostage 35 Arguments REQUIRED ARGUMENTS y response variable binary 0 1 x matrix of predictor variables z matrix of the surrogate or auxiliary variables can be more than one column n2 size of second stage sample var The name of the predictor variable whose coefficient is to be optimised See DETAILS if this is a factor variable and one of the following nl vector of the first stage sample sizes for each y z stratum OR prev vector of the estimated prevalences for each y z stratum AND frac the second stage sampling fraction i e the ratio of second stage sample size to first stage sample size NOTE if prev is given frac will also be required OPTIONAL ARGUMENTS factor the names of any factor variables in the predictor matrix Value A list called design consisting of the following items ylevel the different levels of response variable zlevel the different levels of first stage variables z ni the first stage sample size for each ylevel zlevel stratum n2 the sample size of pilot observations for each ylevel zlevelN stratum prop optimal 2nd stage sampling proportion for each yl
78. suggested by coding function NOTE the transpose operator is essential matrix prev 0 02 134 670 054 05 047 001 004 013 002 003 002 optimise the surg coefficient optbud mort sex surg first sex wtcat prev prev optvar surg b 10000 c1 1 c2 0 5 OUTPUT the second stage sample sizes ps pi do Oar ptg utu AE nes group mor t sex wtcat Freq PEIA Soit su der 10 2 10 3 10 4 10 5 10 6 10 7 8 8 10 9 10 10 10 11 10 12 10 accu tont E please check the sample sizes grp yz mort sex wtcat grp z prev n2 pilot 1 0 0 1 1 02 10 2 0 0 2 2 134 10 3 0 0 3 3 67 10 4 0 1 1 4 054 10 5 0 1 2 5 2 05 10 6 0 1 3 6 047 10 7 1 0 1 1 001 8 8 1 0 2 2 004 10 9 1 0 3 3 013 10 54 10 1 1 4 002 10 11 1 1 5 003 10 12 1 1 6 002 10 the optimal sampling fraction sample size for grp yz 1 604 106 the optimal sampling fraction sample size for grp yz 2 504 591 the optimal sampling fraction sample size for grp yz 3 106 624 the optimal sampling fraction sample size for grp yz 4 1 473 the optimal sampling fraction sample size for grp yz 5 146 64 the optimal sampling fraction sample size for grp yz 6 1 412 the optimal sampling fraction sample size for grp yz 7 1 9 the optimal sampling fraction sample size for grp yz 8 1 35 the optimal sampling fraction sample size for grp yz 9 1 114 the o
79. ted output see Value below This function should be run on second stage data prior to using the ms nprev function as it illustrates the order in which the call to ms nprev expects the first stage sample sizes to be provided Usage coding x x y y z z return F Arguments REQUIRED ARGUMENTS y response variable should be binary 0 1 x matrix of predictor variables for regression model z matrix of any surrogate or auxiliary variables OPTIONAL ARGUMENTS return logical value if it s TRUE T the original surrogate or auxiliary variables and the re coded auxilliary variables will be returned The default is False F Value This function does not return any values except if return T If used with only second stage i e complete data it will print the following ylevel the distinct values or levels of y 19 zl zi the distinct values of first stage variables zl zi new z recoded first stage variables Each value represents a unique combination of first stage variable values n2 second stage sample sizes in each ylevel new z stratum If used with combined first and second stage data 1 e with NA for missing values in addition to the above items the function will also print the following nl first stage sample sizes in each ylevel new z stratum 4 2 3 Examples 4 2 3 1 meanscore The simulated dataset simNA see section 4 1 has 1000 observations with dichotomous response va
80. terest 1s to be estimated with a specified precision and we wish to design a study that will achieve this for a minimum cost 2 2 Optimal Design Derivation 2 2 1 Fixed second stage sample size Suppose we wish to select n subjects at the second stage in such a way as to minimise the variance of the i component of the regression coefficient B This is equivalent to minimising the k k element in the variance covariance matrix V E Dos Intl Va gt Ia t TOF Ju n with the constraint that the second stage sample size n is fixed Note that we can write the constraint as V ZY zy Ph ppa 2 7 Y where pe is the prevalence probability of the Z Y stratum pe is the second stage sampling fraction for the Z Y stratum and n is the total number of observations A Lagrange multiplier can be used to accommodate the constraint so that in this case we would minimise Via A gt To No d nn 2 7 V Z Y Taking the first derivative of this function with respect to p and setting it to zero yields the optimal second stage sampling fractions After some algebraic manipulation it can be shown that the optimal V Z Y second stage sampling fraction in the Z Y stratum p are given by U Var Ss Z YI a n Y o JU Var Ss Z Y Z Y V Z Y _ 2 2 2 Fixed budget Assume now that we wish to minimise the variance of the k component of the regression coefficient B given th
81. the score in each Y Z stratum That the Fisher information matrix coding combines two or more surrogate auxiliary variables into a vector DESCRIPTION recodes a matrix of categorical variables into a vector which takes a unique value for each combination BACKGROUND From the matrix Z of first stage covariates this function creates a vector which takes a unique value for each combination as follows zl z2 z3 new z 000 1 1 0 O re O o o 2 3 4 5 6 7 O ec oo m 25 If some of the combinations do not exist the function will adjust accordingly for example if the combination 0 1 1 is absent above then 1 1 1 will be coded as 7 The values of this new z are reported as new z in the printed output see SIDE EFFECTS below This function should be run on second stage data prior to using the ms nprev function as it illustrates the order in which the call to ms nprev expects the first stage sample sizes to be provided USAGE coding x x y y z z output F REQUIRED ARGUMENTS y response variable should be binary 0 1 x matrix of predictor variables for regression model z matrix of any surrogate or auxiliary variables OPTIONAL ARGUMENTS output logical value if it s TRUE T the original surrogate or auxiliary variables and the re coded auxilliary variables will be returned The default is False F SIDE EFFECTS This function does not return any values except if output T If used
82. tion Reilly amp Pepe 1997 Missingness may depend on the available response and covariate values but not on the unobserved covariate values i e MAR Missing At Random and the method is applicable to cohort or case control designs The subsample of subjects on whom the incomplete covariate is available is referred to as the validation sample or the second stage sample and the remaining subjects are the non validation sample or the first stage sample The code provided here implements a Mean Score analysis for a logistic regression model where the incomplete covariate s may be continuous but the first stage covariates and or auxiliary variables must be categorical INSTALLATION GUIDE The simplest installation can be done by unzipping the BINARY package directly to R HOME library For users familiar with building R packages the full features of the R help systems can be made available by using the standard command make BUILD option pkg meanscore executed from R_HOME src gnuwin32 after you have unzipped this package to R_HOME src library Alternatively you can use the UNIX type command Remd install meanscore executed from R HOME src library 70 With the simpler installation the function will still work properly but some nice features of the help systems may be lost After you have installed the package the command library meanscore makes available the following 3 functions
83. w the first stage sample size in each stratum The following lines will do the Meanscore analysis using the ms nprev command for the simulated data above Notice that we have to run the coding function first to see the order in which we have to enter the vector of first stage sample sizes extract the complete cases only complete_simNA is na simNA 3 run the coding function coding x simNA 3 y simNA 1 z simNA 2 1 For calls to ms nprev input nl or prev in the following ylevel new z order 27 ylevel z new z nil n2 1 0 0 1 310 150 2 01 2 166 85 3 10 1 177 86 4 qt 2 347 179 supply the first stage sample sizes in the correct order nl c 310 166 177 347 ms nprev x complete 3 z complete 2 y complete 1 n1 n1 1 please run coding function first it will give you idea on which order 1 you have to supply the first sample sizes Type coding for details 1 For calls to ms nprev input nl or prev in the following ylevel new z order ylevel z new z n2 1 0 0 1 150 2 01 2 B5 3 10 1 86 4 1 1 2 179 1 Check sample sizes prevalences Stable ylevel zlevel n1 n2 3 2 0 1 310 150 2 0 2 166 85 3 1 1 177 86 4 1 2 347 179 parameters est se Zz pvalue Intercept 0 04939797 0 07155154 0 6903831 0 4899533 x 1 01885599 0 10187166 10 0013679 0 0000000 4 4 Using the Meanscore package in STATA 4 4 1 Installation Guide The program has been written in STATA ver
84. y one pass through the data whereas hotdeck is more demanding on computer time especially if the data set is large 6 3 Meanscore and other likelihood based method We compare Meanscore with three other methods of estimation for missing covariate data Those methods are pseudo likelihood Breslow and Cain 1988 weighted likelihood Flanders and Greenland 1991 and maximum likelihood Breslow and Holubkhov 1997 The S PLUS functions for implementing the last three methods were developed by Breslow and Chatterjee The programs can be downloaded from http www biostat washington edu norm software html Readers interested in the theoretical details of those methods should consult the original articles We used the following 5 data sets to compare the performance of the methods 1 Simulated dataset of 1000 2 stage observations from the CASS dataset The cass1 pilot dataset see Table 5 1 p 34 for more details about this dataset was used to obtain this 2 stage sample optimal with respect to age We used the first stage sample size from Table 3 of Reilly 1996 2 Simulated dataset of 1000 2 stage observations from the CASS dataset The cass2 pilot dataset see Table 5 1 p 34 for more details about this dataset was used to obtain this 2 stage sample optimal with respect to left ventricular diastolic blood pressure LVDBP We used the first stage sample size from Table 5 of Reilly 1996 58 3 Ectopic

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