# Cox Regression in a Markov Renewal Model: An Application to the Analysis of Bone Marrow Transplant Data

By Dabrowska, Dorota M.; Sun, Guo-wen et al. | Journal of the American Statistical Association, September 1994 | Go to article overview

# Cox Regression in a Markov Renewal Model: An Application to the Analysis of Bone Marrow Transplant Data

Dabrowska, Dorota M., Sun, Guo-wen, Horowitz, Mary M., Journal of the American Statistical Association

1. INTRODUCTION

We consider estimation and prediction in a Cox proportional hazard model where observations on each individual under study form a Markov renewal process with a finite state space, say {1, . . . , r}. In the absence of covariates, for each subject we observe a (possibly censored) process (T, X) = {([T.sub.n], [X.sub.n]): n [greater than or equal to] 0}, where 0 = [T.sub.0] < [T.sub.1] < [T.sub.2]... are consecutive times of entrances into the states [X.sub.0], [X.sub.1], [X.sub.2], . .. , [X.sub.n] E {1, . . . , r}. Here the sequence X = {[X.sub.n],: n [greater than or equal to] 0} of states visited forms an embedded Markov chain and, given X, the sojourn times [T.sub.1], [T.sub.2] - [T.sub.1], . . . are independent with distributions depending on the adjoining states only. More precisely, (T, X) is a Markov renewal process if the distribution of the sojourn times [T.sub.n+1] - [T.sub.n], n [greater than or equal to] 0 satisfies

(1) Pr([T.sub.n+1] - [T.sub.n] [less than or equal to]X,[X.sub.n+1] =j|[X.sub.0], [T.sub.0],[X.sub.1],[T.sub.1],. . . [X.sub.n], [T.sub.n])

= Pr([T.sub.n+1] - ([T.sub.n] [less than or equal to][chi],[X.sub.n+1] =j|[X.sub.n]).

In medical applications, such a process can be used to model the patient's experience following a treatment in cases when the outcome of the treatment is affected by internal covariates whose values carry information about the patient's survival status. The state space {1, . . . , r} may correspond then to specific stages of the disease or to levels of covariates such as temperature or blood count. By way of (1), the Markov renewal model puts emphasis on the duration times of the consecutive episodes in the patient's post-treatment experience. Conditionally on the sequence of states visited X, these duration times are independent, whereas the intradependence among the times of the occurrence of the successive events is determined by the Markov chain assumption on the X process. Extensions of this model to the Cox regression setting fall into the framework of modulated renewal processes (Cox 1973; Kalbfleisch and Prentice 1980) and assume that the one-step transitions are determined by hazard functions of the form [[lambda].sub.ij](t, Z(t)) = [[alpha].sub.0,ij](L(t))exp{[[beta].sup.T][Z.sub.ij](t)}, where L(t) is the backwards recurrence time (i.e., the time elapsed between t and the last event prior to t) and Z(t) = {[Z.sub.ij](t): i, j [less than or equal to] r} is a vector of transition-specific time-dependent covanates. This model is more parsimonious than the usual Cox gegression model in that it accommodates both the calendar time t and the nonchronological duration time scale corresponding to the backwards recurrence time L(t). Along with Markov renewal processes, special cases of this model include homogeneous Markov chains, Cox regression for multiple decrement models (Aalen 1978), and variants of the Fix and Neyman (1951) model for recovery, relapse, and death due to cancer (Keiding 1990; McKeague and Utikal 1990). The dependence of the baseline hazard functions on the backwards recurrence time L(t) makes this model more difficult to analyze in that the nonchronological duration time clock used leads outside the framework of Aalen's multiplicative intensity models and invalidates the usual martingale methods. Gill (1980), Voelkel and Crowley (1984), Oakes and Cui (1992), and Andersen, Borgan, Gill, and Keiding (1993) have discussed this point in more detail. Here we assume that the covariates Z(t) depend on the backwards recurrence time only. In this case, estimation can be based on Jacod's (1975) likelihood for counting processes and leads to a partial likelihood that functionally is the same as in the case of ordinary Cox regression. In Section 3, a modification of Andersen and Gill's (1982) results is given to justify the asymptotic structure of the estimates. We further consider prediction of the realization of a process for a new subject. …

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