Piecewise Exponential Models for Survival Data with Covariates

Abstract

A general class of models for analysis of censored survival data with covariates is considered. If $n$ individuals are observed over a time period divided into $I(n)$ intervals, it is assumed that $\lambda_j(t)$, the hazard rate function of the time to failure of the individual $j$, is constant and equal to $\lambda_{ij} > 0$ on the $i$th interval, and that the vector $\ell = \{\log \lambda_{ij}: j = 1, \ldots, n; i = 1, \ldots, I(n)\}$ lies in a linear subspace. The maximum likelihood estimate $\hat{\ell}$ of $\ell$ provides a simultaneous estimate of the underlying hazard rate function, and of the effects of the covariates. Maximum likelihood equations and conditions for existence of $\hat{\ell}$ are given. The asymptotic properties of linear functionals of $\hat{\ell}$ are studied in the general case where the true hazard rate function $\lambda_0(t)$ is not a step function, and $I(n)$ increases without bound as the maximum interval length decreases. In comparison with recent work on regression analysis of survival data, the asymptotic results are obtained under more relaxed conditions on the regression variables.