Estimation in a Semi-Markov Transformation Model

Semi-Markov and modulated renewal processes provide a large class of multi-state models which can be used for analysis of longitudinal failure time data. In biomedical applications, models of this kind are often used to describe evolution of a disease and assume that patient may move among a finite number of states representing different phases in the disease progression. Several authors proposed extensions of the proportional hazard model for regression analysis of these processes. In this paper, we consider a general class of censored semi-Markov and modulated renewal processes and propose use of transformation models for their analysis. Special cases include modulated renewal processes with interarrival times specified using transformation models, and semi-Markov processes with with one-step transition probabilities defined using copula-transformation models. We discuss estimation of finite and infinite dimensional parameters and develop an extension of the Gaussian multiplier method for setting confidence bands for transition probabilities and related parameters. A transplant outcome data set from the Center for International Blood and Marrow Transplant Research is used for illustrative purposes.