title: Neyman, Markov processes and survival analysis.
abstract: 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.
authors: Department of Mathematics, University of Maryland, College Park, MD 20742, USA. gly@math.umd.edu
authors: Grace
authors: G
authors: Yang
mesh_terms: D056808:Biostatistics; D001943:Breast Neoplasms / Q000401:mortality / Q000628:therapy; D003430:Cross-Sectional Studies; D005260:Female; D035922:Hepatitis A Antibodies / Q000097:blood; D006801:Humans; D053208:Kaplan-Meier Estimate; D008390:Markov Chains; D015233:Models, Statistical; D012306:Risk; D013269:Stochastic Processes; D016019:Survival Analysis
chemical_list: D035922:Hepatitis A Antibodies