Saddlepoint approximation for semi-Markov processes with application to a cardiovascular randomised study

Semi-Markov processes have gained popularity as multistate models of disease progression, but have rarely been used in randomised clinical trials (RCTs) when many endpoints are collected over time. Reasons for this include the additional complexity created by censoring, the complexity of the model usually characterised by transition intensities, and difficulties in estimating clinically relevant quantities. In this paper, we show that an indirect approach can address these issues, while accommodating censored data and permitting accurate estimation of survivor functions, hazard ratio and net reductions in risk. The technique combines saddlepoint approximation and likelihood theory to fit a semi-Markov model, specified in terms of its transition probabilities and moment-generating functions (MGFs). A key feature of this approach is that the proportional hazard (PH) assumption commonly assumed for all transition time distributions is no longer needed. Inference for clinically relevant quantities is obtained through bootstrapping or randomisation tests; a simple goodness of fit procedure exploiting the link with the cumulative incidence function for competing risks is also introduced. As illustration, an illness–death model (with transition times between randomisation, non-fatal stroke, and death) for a cardiovascular trial (LIPID) is provided. Evidence of a cumulative benefit of continued treatment, not identified by standard analysis methods in earlier published work, is presented. The technique is flexible enough to be applied extensively in clinical trials.