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Penn BSTA 653 - Regression models for survival data

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Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 17Page 18Page 19Page 20Page 211Regression models for survival dataWhy not use standard models (e.g., linear regression) for failure-time data?2Typically don’t restrict failure times to be positiveHow can one impose restriction?3Use transformation (especially log); accelerated failure-time modelCan also apply transformation to expectation (e.g., model log of theexpectation of the failure-time)Regression models for failure times (model failure-time directly)Accelerated failure-time modelModels for hazardsMultiplicative hazard modelsProportional hazards model: most popular modelAdditive hazards modelsOther modelsProportional odds modelWill consider models with time-varying covariates later4Accelerated failure-time modellinear model for log failure-timesY / ln(T): log failure-time, / error distribution; typically assumed to be i.i.d.Look at model on natural scale (T)define baseline survival function5now, that is, provides mapping of survival functions between baseline function andfunction at any given covariate level0.5,XIf t is median survival in group X, median survival in baseline group is0.5,Xt exp(!X$)0.5,0If t is median survival in baseline group, median survival in group X is0.5,0t exp(X$)informally, covariates “lengthen” life by factor exp(X$)6problem with accelerated failure-time model: estimationIn practical work with standard software, one must choose error distributionproblems with error distributions considered beforesemiparametric models: what is meant by term?what are advantages and disadvantages compared to fully parametric models ornonparametric models?7parametrize some of model/distribution, but not allfor accelerated failure-time model, semiparametric versionparameterize association of covariates X with mean or median outcomeleave distribution of errors , unspecifiedlet f(,) denote density of error function; consider parameter ( for errordistribution f(,); write f(,;()8in parametric models, the form of the model is known up to a finite dimensionalparameter (exponential: ( is one-dimensional (hazard 8)Weibull: ( can be 2-dimesional (shape parameter and scale parameter)semiparametric (parametric-nonparametric models): ( is infinite dimensionalin model error distribution is nonparametric; remainder of model parametricestimation of regression parameters $ in semiparametric model worked out inprinciple (Tsiatis, 1990); in practice, good implementations lacking; will discussfurther laterone reason for relative unpopularity of accelerated failure-time model is absenceof easy semiparametric estimators9parametric versions of model widely implementedSAS: PROC LIFEREG distribution options: exponentialWeibulllog logisticlognormalgeneralized gammaStata:streg adds Gompertz10Hazard models:models for multiplicative hazard modelsWhere is a baseline hazard functionmost popular: proportional hazards modelThus, What is nice about this model form?11Proportional hazards naturally restricts hazard to be nonnegativefor other functions c(@) one may need to apply constraintdefine proportional hazards, relate to model12proportional hazards: for all times t, the hazards given different covariate levels12X and X are proportional:i.e., ratio of hazards is the same at any time t13again, semiparametric model formulation is unspecified; or, equivalently, unknown parameter ( in is infinitedimensionalestimation of $ in semiparametric model worked out; done routinely; willdiscuss in more detail later14survival function in multiplicative hazards model; shown for continuous survivaltimes15log- negative log plotsunder multiplicative hazard models, regression functions of ln(cumulativehazard) on time are parallel for different covariate levelssame vertical separation between curves at different timesbasis for log-negative log plots, used for model checking16Weibull hazards (same as above); follows proportional hazards modelvertical separation of hazards or cumulative hazards constant over timemost easily visualized with log-transformed time-straight line for Weibull17log-logistic distribution (as above)18departures from proportional hazards easily seen for hazard, log-transformedtime log cumulative hazard plotnot as easily seen for natural scale timewhy might cumulative hazard plots be more useful than hazard plots?19Estimation: hazard estimation, like density estimation, not as easily done ascumulative hazard estimationwill consider later in course20additive hazards modelsassumes effect of X constant over time on additive scalebook presents more complicated model where coefficients can be functions oftimein this model, there is a constraint: must be greater than 0. The constraintis not naturally imposed by constraining the baseline hazard to be greater than 0.21proportional odds model"(t) nonincreasing in tif "(t) left otherwise unspecified, semiparametric modelrelated model widely used for ordinal outcome datanot used much in practical work yetxxadditional topics in model specification: time-varying covariates, collapsibility,analogous models for discrete-time datawill put off discussion until


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