The Stratified Cox Procedure

1 5 TheStratifiedCoxProcedure1731745. The Stratified Cox ProcedureIntroductionWe begin with an example of the use of the Stratified Coxprocedure for a single predictor that does not satisfy the PHassumption. We then describe the general approach for fittinga Stratified cox model , including the form of the (partial) like-lihood function used to estimate model also describe the assumption of no interaction that istypically incorporated into most computer programs thatcarry out the Stratified Cox Procedure . We show how the no-interaction assumption can be tested, and what can be doneif interaction is conclude with a second example of the Stratified Cox pro-cedure in which more than one variable is outline below gives the user a preview of the material tobe covered by the presentation.

2 A detailed outline for reviewpurposes follows the (page 176) Example (pages 176 180) General Stratified Cox (SC) model (pages 180 181) No-Interaction Assumption and How to TestIt (pages 182 188) Second Example Involving Several StratificationVariables (pages 188 193) Graphical View of the Stratified Cox Approach(pages 193 194) (pages 195 196)Objectives175 ObjectivesUpon completing the chapter, the learner should be able to:1. Recognize a computer printout for a Stratified Cox State the hazard form of a Stratified cox model for a givensurvival analysis scenario and/or a given set of computerresults for such a Evaluate the effect of a predictor of interest based on com-puter results from a Stratified Cox For a given survival analysis scenario and/or a given setof computer results involving a Stratified cox model , state the no-interaction assumption for the given model ; describe and/or carry out a test of the no-interactionassumption.

3 Describe and/or carry out an analysis when the no-interaction assumption is not The Stratified Cox ProcedureI. PreviewStratified cox model : modification of Cox PH model Stratification of predictor notsatisfying PH includes predictors satisfyingPHFOCUSHow stratification iscarried out: computer results hazard function single predictor vs. 2 predictors no-interaction vs. interactionThe Stratified cox model is a modification of theCox proportional hazards (PH) model that allowsfor control by stratification of a predictor thatdoes not satisfy the PH assumption.

4 Predictorsthat are assumed to satisfy the PH assumption areincluded in the model , whereas the predictor be-ing Stratified is not this presentation, we focus on how stratificationis carried out by describing the analysis of com-puter results and the form of the hazard functionfor a Stratified cox model . We first consider strati-fying on a single predictor and then later considerstratifying on two or more predictors. Further, wedistinguish between the use of a no-interaction version of the Stratified cox model and an alterna-tive approach that allows An ExampleConsider the computer results shown here for aCox PH model containing the three variables, logWBC, treatment group (Rx), and SEX.

5 These re-sults derive from a clinical trial of 42 leukemiapatients, where the response of interest is days trial: 42 leukemia patients Response-days in remissionlog WBC (PH) log WBC and Rxsatisfy PH Sex does not satisfy PH(Same conclusions using graphical approaches) Stratified Cox (SC): control for sex ( Stratified ); simultaneously include log WBC and Rx in the modelFrom the printout, theP(PH) values for log WBCand treatment group are nonsignificant. However,theP(PH) value for SEX is significant below level.

6 These results indicate that log WBCand treatment group satisfy the PH assumption,whereas the SEX variable does not. The same con-clusions regarding the PH assumption about thesevariables would also be made using the graphicalprocedures described we have a situation where one of thepredictors does not satisfy the PH assumption,we carry out a Stratified Cox (SC) procedurefor the analysis. Using SC, we can control forthe SEX variable which does not satisfy thePH assumption by stratification while simulta-neously including in the model the log WBC andtreatment variables which do satisfy the PH An Example177 EXAMPLE (continued) STATA OUTPUT USING SC: Stratified Cox regression Analysis time _t: survtStratified Cox regression Analysis time _t: survtAppendix A illustrates SC procedures using Stata, SAS, and SPSS.

7 Log WBC and Rx are included in SC model . SC model is Stratified by of Rx adjusted for log WBC and SEX: Hazard ratio: = Interpretation: Placebo group (Rx= 1) has times the hazard as the treatment group (Rx= 0)95% CI for Rx ( , ) indicates considerable formula: exp( ) [95% Conf. Interval]log WBC Rxp> |z| of subjects = 42 Log likelihood = by [95% Conf. Interval]log WBC Rxp> |z| of subjects = 42 Log likelihood = by test: P = (two-tailed), significant at the computer results from a SC Procedure areshown here.

8 These results come from the Statapackage. (See the Computer Appendix for runninga SC Procedure in Stata, SAS, or SPSS).The computer results show that the log WBC andRxvariables are included in the model listing,whereas the SEX variable is not included; rather,the model stratifies on the SEX variable, as indi-cated at the bottom of the output. Note that theSEX variable is being adjusted by stratification,whereas log WBC is being adjusted by its inclu-sion in the model along the above output, we have also circled some keyinformation that can be used to assess the effectof theRxvariable adjusted for both log WBC andSEX.

9 In particular, we can see that the hazard ra-tio for the effect ofRxadjusted for log WBC andSEX is given by the value This value can beobtained by exponentiating the coefficient theRxvariable. The hazard ratio value can beinterpreted to mean that the placebo group (forwhichRx=1) has times the hazard for goingout of remission as the treatment group (for whichRx=0).Also, we can see from the output that a 95% con-fidence interval for the effect of theRxvariable isgiven by the limits to This is a fairlywide range, thus indicating considerable variabil-ity in the hazard ratio point estimate.

10 Notethat these confidence limits can be obtained by ex-ponentiating the quantity plus or minus the standard error the above output, a test for the significanceof theRxvariable adjusted for log WBC and SEX isgiven by the Wald statistic P value of This isa two-tailed P-value, and the test is just significantat the The Stratified Cox ProcedureLR test: Output for reduced modelEXAMPLE (continued) [95% Conf. Interval]log WBCp> |z| of subjects = 42 Log likelihood = by sexSC model for males and females:Females (g= 1):h1(t,X)=h01(t)exp[ 1Rx+ 2 log WBC]Males (g= 2):h2(t,X)=h02(t)exp[ 1Rx+ 2 log WBC]Rx and log WBC in the modelSexnot in the model ( Stratified )Hazard function for Stratified Coxmodel:hg(t,X)=h0g(t)exp[ 1Rx+ 2 log WBC]g= 1,2;g denotes stratum #.