Dynamic Determination of Mixed Model Covariance Structures. in Double-blind Clinical Trials. Matthew Davis - Omnicare Clinical Research

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1 PharmaSUG Paper SP12 Dynamic Determination of Mixed Model Covariance Structures in Double-blind Clinical Trials Matthew Davis - Omnicare Clinical Research Abstract With the computing power of SAS software, the use of repeated measures models in double-blind randomized clinical trials is possible. However, with the large size of various clinical trials and the blinded nature of the data, it can be difficult to determine the covariance structure that should be used in the mixed model analysis. While the unstructured covariance matrix is considered the best structure to use for a blinded clinical trial, cases arise where this covariance fails to converge. In order to ensure that appropriate analysis is still performed on the data, alternate methods must be pre-specified before unblinding that will determine the best covariance structure if the unstructured covariance fails to converge. This paper will demonstrate strategies, recommendations and a macro for a dynamically programmed determination of the covariance structure that will best fit the data, which allows for flexibility in mixed model design before treatment unblinding. In order to use the methods presented in this paper, both BASE SAS and SAS /STAT will be used. The code used is not platform dependent. Introduction In order to accurately model data in many types of clinical trials, mixed model specifications are written prior to unblinding in order to reduce bias in the conduct of the trial. One issue that arises from this determination is the specification of the covariance structure. From common practice, the best covariance structure to specify in the model is the unstructured covariance structure. This is able to provide a separate covariance for each patient in the trial and can best model the data. The unique issue that arises in modeling double-blind data is the random assignment of treatment groups. Consider the following model: PROC MIXED; CLASS treatment visit subject investigator MODEL response = visit treatment; REPEATED visit/ GROUP=subject(investigator) TYPE=un; QUIT; Typically in a clinical trial, all output is programmed and checked before unblinding of treatment codes. As a result, it is possible that the above model could fit the data prior to unblinding, and fail to fit the data after unblinding. If this case happens, when time is short, other methods need to be taken into account to dynamically choose the correct covariance structure in the case that the unstructured covariance structure fails to converge. The following will describe appropriate covariance structures to consider, methods to choose the best covariance structure and a macro that will dynamically choose the best covariance structure to fit the data. FDA Guidance The FDA provides very little guidance on the topic of mixed model covariance structures. One place where they mention the covariance structure is in their guidance on Statistical Approaches to Establishing Bioequivalence : In the Random statement, TYPE=FA0(2) could possibly be replaced by TYPE=CSH. This guidance recommends that TYPE=UN not be used, as it could result in an invalid (i.e., not nonnegative definite) estimated covariance matrix. (FDA, 34) The FDA is clearly concerned with reporting models where the covariance matrix is not estimable. However, arguments can be made the FA0(2) or CSH covariance matrices will result in models that are non-estimable. Thus,

2 according to some publications (Chow, 87), the TYPE=UN covariance matrix should be checked for a positive definite estimated covariance matrix. If this matrix is not found, other covariance matrices, such as FA02 or CSH, could be fit in its place. Therefore, the need arises for an efficient way to determine if the unstructured covariance matrix fits the data and, if it does not, to find the optimal covariance matrix to apply to the model. Choosing the Right Informational Criteria If desired, the -2 Res Log Likelihood (-2 times the log likelihood value for the model) can be used to determine the model of best fit. We can perform a likelihood ratio test between any two models to statistically show whether one fits significantly better than another. To perform the likelihood ratio test, consider the -2 Res Log Likelihood Scores from two separate models, each with a different number of covariance parameters. A Chi Square test statistic can be computed by subtracting the -2 Res Log Likelihood from the model with more parameters from the model with less. This Chi-Square test statistic will have degrees of freedom equal to the difference of the number of parameters of the covariance matrices. If this test statistic is significant, the model with the greater number of parameters in the covariance matrix fits significantly better than the model with less parameters (Stokes et. al). If the test statistic is not significant, there is no significant difference in the fit of the two models and the model with the lesser number of parameters should be used for simplicity and parsimony. This method, however, may not be a realistic option. There are three other information criterion produced for each model: AIC (Akaike s Information Criteria) AICC (Akaike s Information Criteria Corrected) BIC (Bayesian Information Criteria) Each of these information criterions are based off of the -2 Res Log Likelihood Score. The AIC is calculated by taking the -2 Res Log Likelihood and adding twice the number of parameters of the covariance matrix, thus penalizing the model for adding more parameters. The AICC corrects the AIC for small samples, but can also be used for large samples since it converges to AIC. The BIC uses Bayesian estimating methods to add to the -2 Res Log Likehood score based on the number of observations in the dataset and then number of covariance parameters in the model. Each of the three covariance structures has its strengths and weaknesses. The AIC tends to choose the more complex model over the BIC. Due to the definition of the AICC, if the model being fit is a small model, the AICC should be used instead of the AIC. The BIC is sometimes preferred over the AIC due to the loss of power that can result from the more complex model that the AIC chooses (Littell, 184). Any of these three informational criteria are valid for choosing a model as long as it is pre-specified before unblinding. Choosing the Best Covariance Structure In most cases, the best covariance structure to fit data in clinical trials will be the unstructured covariance structure. This covariance structure works well to model each patient differently and has the least amount of assumptions behind it. However, if this fails to converge, other covariance structures should be considered. Some of these covariance structures are the following: Autoregressive Toeplitz Compound Symmetry Variance Components In addition, some of these covariance structures have heterogeneous counterparts that also may accurately fit the data. In order to choose the best covariance structure, assuming the unstructured covariance does not fit the data, different information criterion should be considered. The lower the information criteria is, the better the model fit is considered to be. As one can imagine, this can be an arduous process of running through 8 possible models to find the model of best fit. In addition, if many models of this type exist, it can take a great deal of time to run 8 models for a number of parameters or subsets of data. The macro at the end of the paper will run the data through all possible models and go through the following process:

3 1) If the unstructured covariance structure fits, model the data with this covariance structure. 2) If the unstructured does not fit, run through the model with 7 different covariance structures (Topelitz, Heterogeneous Toeplitz, Variance Components, Compound Symmetry, Heterogeneous Compound Symmetry, Autoregressive and Heterogeneous Autoregressive). 3) Of all of these structures, the macro will choose the one with the least information criteria value. 4) The model will return a dataset with the specified LSMEANS, confidence intervals and p-values. Benefits of this Approach In many studies, a double-blind is applied so the treatment codes are not known before the study has completed. In order to return timely results, SAS programming needs to be completed ahead of database lock. The issue that can arise with this situation is, once all of the actual treatment codes have been applied, the covariance matrix of best fit can change. While this may not cause an issue for one or two models, studies that have a number of questionnaires or other efficacy endpoints that require numerous models will have delays in their results if a method like this is not followed. Following these ideas should decrease the amount of time needed to produce accurate results soon after hardlock. While the specific application to the unblinding of clinical trials is discussed in length, there are many other uses for this method. In general, any mixed model can be assessed using these criterions. Regardless of blinded or unblinded data, these rules can be used in general to assess the model of best fit and will be helpful in choosing the most appropriate model. Conclusion Choosing the correct covariance matrix for a mixed model is a vital part of an accurate model. However, in a double-blind clinical trial, the unblinding of the study can change the covariance matrix and affect the timely production of the results of the study. Employing the methods proposed of a dynamic programming for the covariance matrices will ensure that changing of the treatment codes will have no effect on the programming of the study; the SAS output will be able to handle the new data. Therefore, by following these parameters, the output produced will be more accurate and be produced in a more timely fashion. Macro for Covariance Determination **************************************************************************************; * Macro: optmixed.sas *; * Author: Mat Davis - Omnicare Clinical Research *; * Description: This macro will check a mixed model to determine if *; * the unstructured covariance matrix will fit. If it fits, then that *; * model should be used. If it does not fit, the model will check the *; * seven other covariance structures to determine the model of best fit *; * based on the AIC, BIC or AICC values *; * Date: March 2010 *; **************************************************************************************; *%optmixed(dsn=,outcome=,treatment=,baseline=n,visid=visid,interaction=n,fitcrit=bic)*; * DSN=The dataset to be used in analysis *; * OUTCOME=The outcome variable for the model *; * TREATMENT=The treatment variable for the model *; * BASELINE=N if no baseline is to be used. Otherwise, this should be the name of the*; * baseline variable (should be on each observation) *; * VISID=The visit variable for the study *; * INTERACTION=N if no interaction is to be used. Otherwise, this will be the *; * the interaction of the visit and the treatment *; * FITCRIT=The information criteria to be used for the study (AIC, BIC or AICC) *; * SUBJECT=The subject statement for the study *; * COVARIATES=Any extra covariates needed for the model should be listed here *; **************************************************************************************; %MACRO xmexist1(_mvxvar); %LOCAL _MVXdsid _MVXnum _MVXi _MVXval; %LET _MVXvar=%UPCASE(&_MVXvar); %* open the list of currently defined values;

4 %LET _MVXdsid=%SYSFUNC(open(sashelp.vmacro)); %* get the position of the "name" in the dataset; %LET _MVXnum=%SYSFUNC(varnum(&_MVXdsid,name)); %* loop until all are tested or we find the one we are looking for; %LET _MVXval="."; %DO %UNTIL("&_MVXval" EQ "" OR %UPCASE("&_MVXval") EQ "&_MVXvar"); %LET _MVXi=%EVAL(&_MVXi+1); %LET _MVXval=%SYSFUNC(fetchobs(&_MVXdsid,&_MVXi)); %LET _MVXval=%SYSFUNC(getvarc(&_MVXdsid,&_MVXnum)); %* close the data set; %LET num=%sysfunc(close(&_mvxdsid)); %* if we found it, return true, otherwise return false; %IF "&_MVXval" EQ "&_MVXvar" %THEN 1; %ELSE 0; %MEND xmexist1; %MACRO _domixed(varx,reason=); /* reason=check will return values for AR, TOEP, CS, VC reason=uncheck will check for unstructured covariance reason=prod will produce the actual results for a specified covariance structure */ %GLOBAL unaccept; %IF &reason.=check OR &reason.=prod %THEN %DO; %IF &varx=%str(ar) %THEN %LET variance=ar(1); %ELSE %IF &varx=%str(arh) %THEN %LET variance=arh(1); %ELSE %LET variance=&varx.; %IF &reason.=check %THEN %DO; ODS LISTING CLOSE; %ELSE %IF "&varx." ne "un" %THEN %DO; %LET variance=&cvar.; %ELSE %IF "&varx."="un" %THEN %LET variance=un; PROC MIXED DATA=&dsn. SCORING=2; CLASS &treatment. &visid. &subject.; MODEL &outcome.= %IF "&baseline." NE "N" %THEN %DO; &baseline. &treatment. &visid. &covariates. %IF "&interaction." NE "N" %THEN %DO; &treatment.*&visid. %end; /ddfm=kenwardroger; repeated &visid./sub=&subject. type=&variance.; %IF "&interaction." ne "N" %THEN %DO; lsmeans &treatment.*&visid./cl diff; LSMEANS &treatment./cl diff; ODS OUTPUT covparms=covc; ODS OUTPUT fitstatistics=fitc; ODS OUTPUT tests3=test3c; ODS OUTPUT lsmeans=lsmeansc; ODS OUTPUT diffs=diffsc; ODS OUTPUT convergencestatus=convergec; QUIT; %GLOBAL reasonxc; DATA _NULL_; SET convergec; reasonxc=reason; CALL SYMPUT('reasonxc',reasonxc); %PUT REASONXC &reasonxc.; %IF "&reason." EQ "check" AND "&reasonxc." EQ "Convergence criteria met." %THEN %DO; ODS LISTING; %GLOBAL aic&varx.;

5 DATA fitc; SET fitc; WHERE descr=&criteria.; aic&varx.=value; CALL SYMPUT ("aic&varx.",aic&varx.); %PUT "aic&varx. = &&&aic&varx."; ***IF THE UNSTRUCTURED MODEL DID NOT FIT, RUN THROUGH THE OTHER SEVEN MODELS***; %IF &reason.=uncheck %THEN %DO; %IF "%UNQUOTE(&reasonxc.)" NE "Convergence criteria met." %THEN %DO; %LET unaccept=no; %_domixed(ar,reason=check) %_domixed(toep,reason=check); %_domixed(vc,reason=check); %_domixed(cs,reason=check); %_domixed(arh,reason=check) %_domixed(toeph,reason=check); %_domixed(csh,reason=check); %ELSE %DO; DATA _NULL_; unaccept='yes'; CALL SYMPUT ('unaccept',unaccept); PUT "**************************************************************"; PUT "THE UNSTRUCTURED COVARIANCE MATRIX FITS"; PUT "**************************************************************"; run; %end; %end; %IF "&reason." EQ "check" AND "%UNQUOTE(&reasonxc.)" EQ "Convergence criteria met." %THEN %DO; ODS LISTING; %GLOBAL aic&varx.; DATA fitc; SET fitc; WHERE descr=&criteria.; aic&varx.=value; CALL SYMPUT ("aic&varx.",aic&varx.); %PUT "aic&varx. = &&&aic&varx."; %MEND; ***MACRO TO GET THE BEST COVARIANCE STRUCTURE AND DO THE ANALYSIS WITH THE COVARIANCE STRUCTURE SELECTED***; %MACRO _getbest; ODS LISTING; %IF "&unaccept." ne "YES" %THEN %DO; %GLOBAL rerun; ***DETERMINE WHICH OF THE SEVEN MODELS FITS BEST***; %IF %xmexist1(aicar) EQ 1 OR %xmexist1(aictoep) EQ 1 OR %xmexist1(aicvc) EQ 1 OR %xmexist1(aiccs) EQ 1 OR %xmexist1(aicarh) EQ 1 OR %xmexist1(aictoeph) EQ 1 OR %xmexist1(aiccsh) EQ 1 %THEN %DO;

6 DATA answer; %IF %xmexist1(aicar) EQ 1 %THEN %DO; col1=&aicar.;%end; %IF %xmexist1(aictoep) EQ 1 %THEN %DO;col2=&aictoep.;%end; %IF %xmexist1(aicvc) EQ 1 %THEN %DO;col3=&aicvc.;%end; %IF %xmexist1(aiccs) EQ 1 %THEN %DO;col4=&aiccs.;%end; %IF %xmexist1(aicarh) EQ 1 %THEN %DO; col5=&aicarh.;%end; %IF %xmexist1(aictoeph) EQ 1 %THEN %DO; col6=&aictoeph.;%end; %IF %xmexist1(aiccsh) EQ 1 %THEN %DO;col7=&aiccsh.;%end; if col1=. AND col2=. AND col3=. AND col4=. AND col5=. AND col6=. AND col7=. THEN DO; PUT "*********************************************************"; PUT "NO COVARIANCE STRUCTURES FIT YOUR DATA WITH THIS MODEL"; PUT "*********************************************************"; END; DATA answer; LENGTH cvar $5.; SET answer; winner=min(col1,col2,col3,col4,col5,col6,col7); IF winner=col1 THEN cvar="ar"; ELSE IF winner=col2 THEN cvar="toep"; ELSE IF winner=col3 THEN cvar="vc"; ELSE IF winner=col4 THEN cvar="cs"; ELSE IF winner=col5 THEN cvar="arh"; ELSE IF winner=col6 THEN cvar="toeph"; ELSE IF winner=col7 THEN cvar="csh"; CALL SYMPUT ('cvar',cvar); PUT "****************************************************************"; PUT "Winner is " cvar " with an %upcase(&fitcrit.) value of " winner; PUT "****************************************************************"; %_domixed(&cvar.,reason=prod ); %MEND; %MACRO optmixed(dsn=,outcome=,treatment=,baseline=n,interaction=n,visid=visid,fitcrit=bic,subject=subjid,covariates=); %GLOBAL criteria; %IF "%UPCASE(&fitcrit.)"="AIC" %THEN %LET criteria='aic (smaller is better)'; %ELSE %IF "%UPCASE(&fitcrit.)"="BIC" %THEN %LET criteria='bic (smaller is better)'; %ELSE %IF "%UPCASE(&fitcrit.)"="AICC" %THEN %LET criteria='aicc (smaller is better)'; ***MACRO TO DO THE MIXED PROCEDURE AND DETERMINE THE APPROPRIATE COVARIANCE STRUCTURE***; %_domixed(un,reason=uncheck); ***GETS INFO FROM THE BEST COVARIANCE STRUCTURE***; %_getbest; %MEND;

7 References Average, Population, and Individual Approaches to Establishing Bioequivalence. Guidance for Industry. Washington DC: US Department of Health and Human Services Food and Drug Administration, Print. Stokes, Maura E., Charles S. Davis, and Gary G. Koch. Categorical Data Analysis Using the SAS(R) System. 2nd ed. Cary, NC: SAS Institute and Wiley, Print. Contact Information Your comments and questions are valued and encouraged. Contact the author at: Name: Mat Davis Enterprise: Omnicare Clinical Research Address: 630 Allendale Rd City, State ZIP: King of Prussia, PA Work Phone: (484) SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. Other brand and product names are trademarks of their respective companies.

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