Neyman, Markov processes and survival analysis

J. Neyman used stochastic processes extensively in his applied work. One example is the Fix and Neyman (F–N) competing risks model (1951) that uses finite homogeneous Markov processes to analyse clinical trials with breast cancer patients. We revisit the F–N model, and compare it with the Kaplan–Meier (K–M) formulation for right censored data. The comparison offers a way to generalize the K–M formulation to include risks of recovery and relapses in the calculation of a patient’s survival probability. The generalization is to extend the F–N model to a nonhomogeneous Markov process. Closed-form solutions of the survival probability are available in special cases of the nonhomogeneous processes, like the popular multiple decrement model (including the K–M model) and Chiang’s staging model, but these models do not consider recovery and relapses while the F–N model does. An analysis of sero-epidemiology current status data with recurrent events is illustrated. Fix and Neyman used Neyman’s RBAN (regular best asymptotic normal) estimates for the risks, and provided a numerical example showing the importance of considering both the survival probability and the length of time of a patient living a normal life in the evaluation of clinical trials. The said extension would result in a complicated model and it is unlikely to find analytical closed-form solutions for survival analysis. With ever increasing computing power, numerical methods offer a viable way of investigating the problem.