Asymptotic theory for the Cox semi-Markov illness-death model

Abstract

Irreversible illness-death models are used to model disease processes and in cancer studies to model disease recovery. In most applications, a Markov model is assumed for the multistate model. When there are covariates, a Cox (1972, J Roy Stat Soc Ser B 34:187-220) model is used to model the effect of covariates on each transition intensity. Andersen et al. (2000, Stat Med 19:587-599) proposed a Cox semi-Markov model for this problem. In this paper, we study the large sample theory for that model and provide the asymptotic variances of various probabilities of interest. A Monte Carlo study is conducted to investigate the robustness and efficiency of Markov/Semi-Markov estimators. A real data example from the PROVA (1991, Hepatology 14:1016-1024) trial is used to illustrate the theory.

Semiparametric estimation of quality adjusted lifetime distribution in semi-Markov illness–death model

“…15 is essentially a sum of n independent and identically distributed zero-mean random variables. Using the arguments of Shu et al (2007), we conclude that…”

“…It may be noted that the semi-Markov model does not fit readily into the multiplicative intensity framework because of its renewal nature (see Voelkel and Crowley 1984;Shu et al 2007). This can be dealt with by introducing timeshifted multivariate counting process over a fixed interval, say [0, τ ], given by…”

“…Let θ ∈ [0, τ ). Following Shu et al (2007) and under the regularity conditions 1-3 therein, we have the following theorems. The proofs of the theorems are given in the Appendix.…”

“…Let us assume the regularity conditions 1-3 of Shu et al (2007). Theorem 1 is Lemma 1(ii) of their work.…”

Estimation and Asymptotic Theory for Transition Probabilities in Markov Renewal Multi-State Models

“…Moreover, in the case of time-independent covariates, thanks to the renewal nature of the process, it was possible to estimate the transition probabilities and obtain confidence intervals by bootstrapping. For the special case of the three state irreversible illness-death model, Shu et al (2007) give explicit formulas for the transition probabilities and associated variances.…”