Penn BSTA 653 - Regression models for survival data (21 pages)

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Regression models for survival data



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Regression models for survival data

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Pages:
21
School:
University of Pennsylvania
Course:
Bsta 653 - Survival Analysis
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Regression models for survival data Why not use standard models e g linear regression for failure time data 1 Typically don t restrict failure times to be positive How can one impose restriction 2 Use transformation especially log accelerated failure time model Can also apply transformation to expectation e g model log of the expectation of the failure time Regression models for failure times model failure time directly Accelerated failure time model Models for hazards Multiplicative hazard models Proportional hazards model most popular model Additive hazards models Other models Proportional odds model Will consider models with time varying covariates later 3 Accelerated failure time model linear model for log failure times Y ln T log failure time error distribution typically assumed to be i i d Look at model on natural scale T define baseline survival function 4 now that is provides mapping of survival functions between baseline function and function at any given covariate level If t0 5 X is median survival in group X median survival in baseline group is t0 5 Xexp X If t0 5 0 is median survival in baseline group median survival in group X is t0 5 0exp X informally covariates lengthen life by factor exp X 5 problem with accelerated failure time model estimation In practical work with standard software one must choose error distribution problems with error distributions considered before semiparametric models what is meant by term what are advantages and disadvantages compared to fully parametric models or nonparametric models 6 parametrize some of model distribution but not all for accelerated failure time model semiparametric version parameterize association of covariates X with mean or median outcome leave distribution of errors unspecified let f denote density of error function consider parameter for error distribution f write f 7 in parametric models the form of the model is known up to a finite dimensional parameter exponential is one dimensional hazard 8 Weibull can be 2 dimesional shape parameter and scale parameter semiparametric parametric nonparametric models is infinite dimensional in model error distribution is nonparametric remainder of model parametric estimation of regression parameters in semiparametric model worked out in principle Tsiatis 1990 in practice good implementations lacking will discuss further later one reason for relative unpopularity of accelerated failure time model is absence of easy semiparametric estimators 8 parametric versions of model widely implemented SAS PROC LIFEREG distribution options exponential Weibull log logistic lognormal generalized gamma Stata streg adds Gompertz 9 Hazard models models for multiplicative hazard models Where is a baseline hazard function most popular proportional hazards model Thus What is nice about this model form 10 Proportional hazards naturally restricts hazard to be nonnegative for other functions c one may need to apply constraint define proportional hazards relate to model 11 proportional hazards for all times t the hazards given different covariate levels X1 and X2 are proportional i e ratio of hazards is the same at any time t 12 again semiparametric model formulation is unspecified or equivalently unknown parameter in is infinite dimensional estimation of in semiparametric model worked out done routinely will discuss in more detail later 13 survival function in multiplicative hazards model shown for continuous survival times 14 log negative log plots under multiplicative hazard models regression functions of ln cumulative hazard on time are parallel for different covariate levels same vertical separation between curves at different times basis for log negative log plots used for model checking 15 Weibull hazards same as above follows proportional hazards model vertical separation of hazards or cumulative hazards constant over time most easily visualized with log transformed time straight line for Weibull 16 log logistic distribution as above 17 departures from proportional hazards easily seen for hazard log transformed time log cumulative hazard plot not as easily seen for natural scale time why might cumulative hazard plots be more useful than hazard plots 18 Estimation hazard estimation like density estimation not as easily done as cumulative hazard estimation will consider later in course 19 additive hazards models assumes effect of X constant over time on additive scale book presents more complicated model where coefficients can be functions of time in this model there is a constraint must be greater than 0 The constraint is not naturally imposed by constraining the baseline hazard to be greater than 0 20 proportional odds model t nonincreasing in t if t left otherwise unspecified semiparametric model related model widely used for ordinal outcome data not used much in practical work yet xx additional topics in model specification time varying covariates collapsibility analogous models for discrete time data will put off discussion until later 21


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