1 Introduction
Although the incorporation of random effects into Cox models has gained increasing attention in analyses of event history data, these models pose considerable theoretical difficulties in the development of estimation and inference procedures (Clayton 1991). Until recently, previous research in this area has focussed mainly on survival models with one level of random effects (Sastry 1997; Sargent 1998). The frequentist approaches to nested frailty survival models have usually been restricted to piecewise constant baseline hazard functions and specific random effects distributions (Sastry 1997). On the other hand, Bayesian approaches to nested random effects Cox models are computationally intensive, and the assessment of convergence of computational techniques such as the Gibbs sampler remains an area of debate (Glifford 1993; Smith and Roberts 1993; Sargent 1998). Flexible frailty models that can be fit with reasonable computational effort are therefore needed.
Considerable progress has been made in recent years in the area of random effects generalized linear models (Breslow and Clayton 1993; Lee and Nelder 1996; Ma 1999). The connection between the Cox and Poisson regression models has long been recognized (Whitehead 1980). In this paper, we show that random effects methods developed for use with generalized linear models can be applied by characterizing the random effects Cox model as a random effects Poisson regression model. Our approach deals with an unspecified baseline hazard function and a wide range of random effects distributions. Our approach can also handle ties and stratification in the same way as in the standard Cox model. Further, our explicit expressions for the random effects facilitate incorporation of relatively large numbers of random effects.
The organization of the paper is as follows. We introduce the random effects Cox model and its auxiliary random effects Poisson models in Sections 2 and 3, respectively. In Section 4, we discuss the estimation of the nested random effects Cox models based on the orthodox BLUP approach to the auxiliary random effects Poisson models. An illustrative example involving animal carcinogenesis data is presented in Section 5, and potential extensions of the models are discussed in Section 6.
2 Random Effects Cox Model
In this section, we consider a Cox model with two levels of random effects. Suppose that the cohort of interest is stratified on the basis of one or more relevant covariates. Let the hazard function for individual $(i, j, k)$ from stratum $s = 1, 2, \dots, a$ at time $t$ be denoted by $h_{ijk}^{(s)}(t)$. Given the random effects, we assume that the individual hazard functions are conditionally independent