Edward Rotan Visiting Professor

Wesley O. Johnson, Ph.D.

Professor of Statistics

University of California, Davis

Semiparametric AFT Models for Survival Data

The semiparametric proportional hazards (PH) model is ubiquitous in survival literature.

It is flexible and easily fit using standard packages, at least for right censored data.

However, the assumption of proportional hazard functions may be violated and we may

seek a proper alternative semiparametric model. One such model is the accelerated

failure time (AFT) model. Whereas the PH model assumes the covariates act multiplic-

atively on a baseline hazard function, the AFT model assumes that the covariates act

multiplicatively on the argument of the baseline survival distribution.

Early approaches to the semiparametric AFT model were considered by Miller (1976),

Buckley and James (1978), Kalbfleisch (1978), Koul, Susarla and Van Ryzin (1981), and

Christensen and Johnson (1988). All these approaches are difficult or sub-adequate in

one way or another. As an example, Johnson and Christensen (1989) demonstrated that

the mathematics involved for a fully Bayesian solution to even the non-censored problem

in this context is horrendous at best.

Recently, the analytic intractability of Bayesian semiparametric inference for the AFT

model for right censored data was overcome by utilizing Markov chain Monte Carlo

(MCMC) methods. See Doss (1994), Kuo and Mallick (1997), Walker and Mallick (1999)

and Kottas and Gelfand (2000). Here, we consider two alternative approaches that build

on these works. We model the baseline survival distribution either with a mixture of Polya

tree (MPT) processes or a mixture of Dirichlet processes (MDP), given a standard para-

metric family of base measures. Our models allow for a complete Bayesian solution

using MCMC methods and are direct nonparametric extensions of existing parametric

models. Additionally, existing prior information for all parametric model parameters can

be incorporated immediately into the more general nonparametric approach. The MDP

model presented is for interval censored data and is a straightforward generalization of

the approach taken by Christensen and Johnson (1988), and thus provides a numerically

tractable and practically implementable, complete Bayesian approach to a useful general-

ization of the problem that they considered to be analytically intractable. The MPT model

builds on the work of Walker and Mallick. This work is joint with Tim Hanson at the

University of New Mexico.