# Conditional Regression Analysis for Recurrence Time Data

## Article excerpt

Recurrence time data can be regarded as a specific type of correlated survival data in which recurrent event times of a subject are stochastically ordered. Given the ordinal nature of recurrence times, this article focuses on conditional regression analysis. A semiparametric hazards model, including the structural and episode-specific parameters, is proposed for recurrence time data. In this model the order of episodes serves as the stratification variable. Estimation of the structural parameter can be constructed on the basis of all of the observed recurrence times. The structural parameter is estimated by the profile-likelihood approach. Although the structural parameter estimator is asymptotically normal, the episode-specific parameters may or may not be estimated consistently due to the sparseness of data for specific events. Examples are presented to illustrate the performance of the estimators of the structural and episode-specific parameters. An extension of the univariate recurrent events to the bivar iate events, which occur repeatedly and sequentially, is discussed with an example.

KEY WORDS: Correlated survival data; Counting processes; Martiagale; Partial likelihood; Profile likelihood.

1. INTRODUCTION

Recurrence time data are commonly encountered in longitudinal studies when failure events can occur repeatedly over time for each study subject. Such data can be identified in various scientific areas, including biomedical sciences, demographical studies, industrial research, and possibly other fields. Taking the disease of schizophrenia as an example from biomedical research, a patient may be repeatedly hospitalized due to a relapse of schizophrenic symptoms. Examples in other areas include multiple live births in a woman's lifetime, repeated breakdowns of an automobile, multiple opportunistic infections in studies of acquired immunodeficiency syndrome (AIDS), and multiple injuries in aging studies. When analyzing recurrent events of the same type, a natural failure time is the time between two successive events, because the occurrence of each event serves as a time origin for the next event.

In multivariate survival analysis, correlated survival data usually refer to failure time data sampled by clusters--for example, tumor occurrences in littermates. Under the sampling, correlation exists among failure times from each cluster (Cai and Prentice 1995; Lee, Wei, and Amato 1992; Liang, Self, and Chang 1993; Lin 1994; Wei, Lin, and Weissfeld 1989). Although recurrence time data share some common characteristics with standard correlated survival data, there are fundamental distinctions between these two types of multivariate survival data:

a. The occurrence times of recurrent events from a subject are stochastically ordered, but standard correlated survival data are not.

b. Whereas each component of a correlated survival time unit could be censored, for recurrent event data only the last recurrence time of a subject is subject to right censoring.

c. The number of recurrent events is informative for the recurrence time distributions, but the size of a cluster for standard correlated data is generally not.

This article studies regression models and methods that exploit the special features of recurrence time data. Specifically, the ordinal nature of recurrent events allows the development of a conditional regression analysis utilizing semiparametric hazards models (Cox 1972 and Prentice, Williams, and Peterson 1981). Given the nature that each recurrent event appropriately serves as a new time origin for the next event, the chosen failure time is defined to be the time between two successive events.

Define [T.sub.0] = 0. Let [T.sub.j] be the random variable representing the occurrence time of the jth recurrent event and let [Y.sub.j] = [T.sub.j] - [T.sub.j-1] be the recurrence time of interest, j = 1,2,.... Let z(u) be a vector of covariates at time u and let Z(t) = {z(u): u [less than or equal to] t} be the corresponding covariates history up to and including t. …