Discrete-time nonparametric estimation for semi-Markov models of chain-of-events data subject to interval censoring and truncation

Chain-of-events data are longitudinal observations on a succession of event s that can only occur in a prescribed order. One goal in an analysis of thi s type of data is to determine the distribution of times between the succes sive events. This is difficult when individuals are observed periodically r ather than continuously because the event times are then interval censored. Chain-of-events data may also be subject to truncation when individuals ca n only be observed if a certain event in the chain (e.g., the final event) has occurred. We provide a nonparametric approach to estimate the distribut ions of times between successive events in discrete time for data such as t hese under the semi-Markov assumption that the times between events are ind ependent. This method uses a self-consistency algorithm that extends Turnbu ll's algorithm (1976, Journal of the Royal Statistical Society, Series B 38 , 290-295). The quantities required to carry out the algorithm can be calcu lated recursively for improved computational efficiency. Two examples using data from studies involving HIV disease are used to illustrate our methods .