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 161Regression models for survival dataRegression models for failure timesAccelerated failure-time modelModels for hazardsMultiplicative hazard modelsProportional hazards model: most popular modelAdditive hazards modelsOther modelsProportional odds modelWill consider models with time-varying covariates later2Accelerated 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 function3now, that is, provides mapping of survival functions betweenbaseline function and function at any given covariatelevelIf t0.5,X is median survival in group X, median survival inbaseline group is t0.5,Xexp(!X$)If t0.5,0 is median survival in baseline group, mediansurvival in group X is t0.5,0exp(X$)informally, covariates “lengthen” life by factor exp(X$)4problem with accelerated failure-time model: estimationIn practical work with standard software, one mustchoose error distributionproblems with error distributions considered beforesemiparametric models: what is meant by term?what are advantages and disadvantages compared to fullyparametric models or nonparametric models?5parametrize some of model/distribution, but not allfor accelerated failure-time model, semiparametricversionparametrize association of covariates X with mean ormedian outcomeleave distribution of errors , unspecifiedlet f(,) denote density of error function; considerparameter ( for error distribution f(,); write f(,;()in parametric models, the form of the model is known upto a finite dimensional parameter (exponential: ( is one-dimensional (hazard 8)Weibull: ( can be 2-dimesional (shape parameter andscale parameter)6semiparametric (parametric-nonparametric models): ( isinfinite dimensionalin model error distribution is nonparametric; remainder ofmodel parametricestimation of regression parameters $ in semiparametricmodel worked out in principle (Tsiatis, 1990); in practice,good implementations lacking; will discuss further laterone reason for relative unpopularity of acceleratedfailure-time model is absence of easy semiparametricestimators7parametric versions of model widely implementedSAS: PROC LIFEREG distribution options: exponentialWeibulllog logisticlognormalgeneralized gammaStata:streg distribution optionsexponentialWeibullloglogisticlognormalgeneralized gammaGompertz8Hazard models:models for multiplicative hazard modelsWhere is a baseline hazard functionmost popular: proportional hazards modelThus, Proportional hazards naturally restricts hazard to benonnegativefor other functions c(@) one may need to applyconstraint9proportional hazards: for all times t, the hazards givendifferent covariate levels X1 and X2 are proportional:i.e., ratio of hazards is the same at any time tagain, semiparametric model formulation is unspecified; or, equivalently, unknown parameter( in is infinite dimensionalestimation of $ in semiparametric model worked out;done routinely; will discuss in more detail later10survival function in multiplicative hazards model; shownfor continuous survival times11log- negative log plotsunder multiplicative hazard models, regression functionsof ln(cumulative hazard) on time are parallel for differentcovariate levelssame vertical separation between curves at different timesbasis for log-negative log plots, used for model checking12Weibull hazards (same as above); follows proportionalhazards modelvertical separation of hazards or cumulative hazardsconstant over timemost easily visualized with log-transformed time-straightline for Weibull13log-logistic distribution (as above)departures from proportional hazards easily seen forhazard, log-transformed time log cumulative hazard plotnot as easily seen for natural scale timewhy might cumulative hazard plots be more useful thanhazard plots?14Estimation: hazard estimation, like density estimation,not as easily done as cumulative hazard estimationwill consider later in course15additive hazards modelsassumes effect of X constant over time on additive scalebook presents more complicated model where coefficientscan be functions of timein this model, there is a constraint: must be greaterthan 0. The constraint is not naturally imposed byconstraining the baseline hazard to be greater than 0.16proportional odds model"(t) nonincreasing in tif "(t) left otherwise unspecified, semiparametric modelrelated model widely used for ordinal outcome datanot used much in practical work yetadditional topics in model specification: time-varyingcovariates, collapsibility, analogous models for discrete-time datawill put off discussion until


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Penn BSTA 653 - Regression models for survival data

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