Stratification in the Cox model - MyWeb

1 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsStratification in the Cox modelPatrick BrehenyNovember 17 Patrick BrehenySurvival Data Analysis (BIOS 7210)1/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsIntroductionToday s topic is the use of Stratification in Cox regressionThere are two main purposes of Stratification :It is useful as a diagnostic for checking the proportionalhazards assumptionIt offers a way of extending the Cox model to allow fornon-proportionality with respect to some covariatesPatrick BrehenySurvival Data Analysis (BIOS 7210)2/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsVA Lung Cancer dataTo illustrate these concepts, we will look at a classic survivaldata set, the VA lung cancer data (veteranin thesurvivalpackage)The data comes from a clinical trial carried out by theVeterans Administration on male veterans with advanced,inoperable lung cancerIn the trial, patients were randomized to receive either astandard chemotherapy or an experimental chemotherapy, andthe primary endpoint was the time until deathPatrick BrehenySurvival Data Analysis (BIOS 7210)

2 3/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsCovariatesA number of covariates which potentially affect survival were alsorecorded:karno: The Karnofsky score, a way of quantifying thepatient s overall baseline status, with 70denoting that thepatient is able to care for themselves,40 60meaning thatthe patient requires assistance and regular medical care, and10 30meaning that the patient is hospitalizeddiagtime: Time in months from diagnosis to randomizationage: Age in years at randomizationprior: Indicator for whether the patient had received priortherapycelltype: Type of tumor (small cell, large cell, squamous, adenocarcinoma )Patrick BrehenySurvival Data Analysis (BIOS 7210)4/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox (years)SurvivalStandardTestPatrick BrehenySurvival Data Analysis (BIOS 7210)5/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsCox resultsllllllllHazard ratio< < < v squamousSmall v squamousLarge v squamousTreatmentPriorDiag.

3 BrehenySurvival Data Analysis (BIOS 7210)6/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsDiagnostics for proportional hazardsConsider the following as a way to assess the proportionalhazards assumption: rather than including a term in themodel as a covariate, we will estimate separate baselinehazards 01, 02, .. , for each level of the covariateIf the baseline hazards appear proportional, then it isreasonable to model the term in the regular mannerPatrick BrehenySurvival Data Analysis (BIOS 7210)7/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsDiagnostic plot typesBecause proportionality is difficult to assess by visualinspection, it is common to plotlog 0: i(t) = 0(t) exp( i)= log i(t) = log 0(t) + iAn alternative, known as theAndersen plot, is to plot 01versus 02.

4 Under proportional hazards this should be astraight line with slopeexp( )Patrick BrehenySurvival Data Analysis (BIOS 7210)8/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsTreatment (Version 1) 4 3 2 1012 Time (years)log( (t))StandardTestPatrick BrehenySurvival Data Analysis (BIOS 7210)9/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsTreatment (Version 2) ( = ) (years)log 1(t) log 2(t)Patrick BrehenySurvival Data Analysis (BIOS 7210)10/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsTreatment (Version 3, the Andersen plot)012340123456 1(t) 2(t)Patrick BrehenySurvival Data Analysis (BIOS 7210)11/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsCell 4 3 2 1012 Time (years)log( (t))squamoussmallcelladenolargePatrick BrehenySurvival Data Analysis (BIOS 7210)12/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox 4 3 2 101 Time (years)log( (t))(0,30](30,60](60,100]Patrick BrehenySurvival Data Analysis (BIOS 7210)13/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsRemarksTreatment appears broadly proportional except for veryshort-term survivalProportional hazards appears questionable with respect to celltypeKarnofsky status also appears non-proportional, with thevariable losing relevance over time (which makes sense)Patrick BrehenySurvival Data Analysis (BIOS 7210))))

5 14/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsThe stratified Cox modelWhat should we do in the presence of variables withnon-proportional effects?One remedy is to allow for different baseline hazards for eachlevel of the variable: ij(t) = 0j(t) exp(xTi ),where ij(t)is the hazard function for theith subject, whobelongs to thejth stratumThe model may seem complex, but is entirely straightforwardin the likelihood framework, as we can simply combinelikelihoods across strata:L( ) = jLj( )Patrick BrehenySurvival Data Analysis (BIOS 7210)15/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsStratified Cox model : DetailsFurthermore,`( ) = j`j( )u( ) = juj( )I( ) = jIj( ),so estimation, the Newton-Raphson algorithm, and inference are allstraightforward as well.

6 We simply have to sum the contributionsfrom each stratumPatrick BrehenySurvival Data Analysis (BIOS 7210)16/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsRcodeThesurvivalpackage makes it easy to fit stratified Coxmodels through the use of thestratafunction:fit <- coxph(S ~ trt + karno + .. + strata(celltype))summary(fit)will then provide a summary for all theparametric terms (trt,karno, .. ), but notcelltypesurvfit(fit)will estimateKdifferent baseline hazardfunctions, one for each stratum (here,K= 4)Patrick BrehenySurvival Data Analysis (BIOS 7210)17/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsPredictionsStandard treatment, wait 12 months, age 40, no prior (years)Survivalsqu: krn=80squ: krn=60lar: krn=60 Patrick BrehenySurvival Data Analysis (BIOS 7210)18/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsFinal remarksStratified Cox models are a useful extension of the standardCox models to allow for covariates with non-proportionalhazardsA minor drawback is that stratifying unnecessarily ( , eventhough the PH assumption is met) reduces estimationefficiency, although the loss is typically very smallA larger limitation of Stratification is that it becomes messywith continuous variables and with multiple stratificationvariables, as there is no way to impose an additive structurePatrick BrehenySurvival Data Analysis (BIOS 7210)19/20 IntroductionChecking the proportional hazards assumptionFitting stratified Cox modelsFinal remarks (cont d)

7 The other primary limitation of stratified models is that thereis no way to carry out inference for the Stratification variablesFor example, Stratification is commonly used to aggregateresults across multi-center studies, because comparing thesesites is typically not of interestStratification is less useful in dealing with non-proportionalitywith respect to treatment we are definitely interested inestimating the effect of treatment, and although we canobtain descriptive measures by estimating baselinecoefficients, confidence intervals and tests are lackingIn such cases, a more satisfying approach is to directly modelthe changing effect of the predictor over time, a topic we willcover in a future lecturePatrick BrehenySurvival Data Analysis (BIOS 7210)20/20