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. 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.

2 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; 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.

3 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. 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.

4 However,theP(PH) value for SEX is significant below level. 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. Log WBC and Rx are included in SC model .

5 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. 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.

6 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. 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.]

7 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 #.LR and Wald give same Cox regressionAnalysis time _t: survtLR= ( 2 ) ( 2 ) = (P < ) HR for effect of Rx adjusted for log WBCand sex:e 1where 1 is the coefficient of alternative test involves a likelihood ratio (LR)statistic that compares the above model (fullmodel) with a reduced model that does not con-tain theRxvariable. The output for the reducedmodel is shown here. The log-likelihood statisticfor the reduced model is 2 times ,which is to be compared with the log-likelihoodstatistic of 2 times for the full is therefore , which equals Under H0, thisstatistic has a chi-square distribution with onedegree of freedom and is significant at the Thus, theLRand Wald tests lead to thesame far, we have illustrated the results from a strat-ified Cox Procedure without actually describingthe model form being used.

8 For the remissiondata example, we now present the hazard func-tion form for the Stratified cox model , as shownhere. This hazard function formula contains asubscriptgthat indicates thegth , in our remission data example, where wehave Stratified on SEX,gtakes on one of twovalues, so that we have a different baseline hazardfunction for males and that the hazard function formula containsthe variablesRxand log WBC, but does notcontain the variable SEX. SEX is not includedin the model because it doesn t satisfy the PHassumption. So, instead, the SEX variable iscontrolled by the variablesRxand log WBC areincluded in the model , we can estimate the effectof each variable adjusted for the other variableand the SEX variable using standard exponentialhazard ratio expressions. For example, the esti-mated hazard ratio for the effect ofRx, adjustedfor log WBC and SEX, is given byeto the 1 hat, where 1is the coefficient of An Example179 Cannot estimate HR for SEX variable(SEX doesn t satisfy PH).

9 Different baseline hazard functions:h01(t) for females and h02(t) for interaction assumption(see Section IV)Same coefficients 1and 2for bothfemale and male models. EXAMPLE (continued) Females and males:same 1and 2 sameHR s, , e 1 Differentbaselinesh01(t) Survival curve for femalesh02(t) Survival curve for males{Estimates of 1and 2:Maximize partial likelihood (L),whereL= L1 L2L1is the likelihood for females derivedfromh1(t),andL2 is the likelihood for males derivedfromh2(t).Nevertheless, because the SEX variable is notincluded in the model , it is not possible to obtaina hazard ratio value for the effect of SEX adjustedfor the other two variables. This is the price to bepaid for stratification on the SEX variable. Notethat a single value for the hazard ratio for SEXis not appropriate if SEX doesn t satisfy the PHassumption, because the hazard ratio must thenvary with also that the hazard functions for malesand females differ only insofar as they havedifferent baseline hazard functions, namely,h01(t) for females andh02(t) for males.}

10 However,the coefficients 1and 2are the same for bothfemale and male there are different baseline hazardfunctions, the fitted Stratified cox model will yielddifferent estimated survival curves for femalesand males. These curves will be described , however, that because the coefficients ofRxand log WBC are the same for females and males,estimates of hazard ratios, such aseto the 1 hat, are the same for both females and feature of the Stratified cox model is calledthe no-interaction assumption. It is possibleto evaluate whether this assumption is tenableand to modify the analysis if not tenable. We willdiscuss this assumption further in Section obtain estimates of 1and 2, a (partial)likelihood function (L) is formed from the modeland the data; this function is then maximizedusing computer iteration. The likelihood function(L) for the Stratified Cox (SC) model is differentfrom the nonstratified cox model .