Bayesian Joint Estimation of Binary Outcome and Time-to-event Data: Effects of Leaf Quality on Pupal Survival and Time-to-Emergence in the Winter Moth

Abstract

Plant–herbivore interactions are complex and affect herbivore fitness components and life history traits in many different ways. In this paper, we present results from an experiment studying the effects of leaf quality on pupal survival and duration of pupation (as measured by time-to-emergence) in the winter moth. Because only surviving pupae are at risk of emerging, analysis of time-to-emergence should exclude the dead pupae. However, due to right censoring, the survival status could not be determined for each individual. This failure to determine the group of moths at risk of emerging a priori motivated the development of a joint model of both survival probability and time-to-emergence. We formulate the model in a Bayesian framework and apply Monte Carlo Markov Chain (MCMC) to obtain posterior distributions. Time-to-emergence is modeled by a Cox Proportional Hazards (CPH) model where only the surviving pupae are at risk of emergence. Probability of pupal survival was modeled by a Generalized Linear Mixed Model (GLMM). The censored individuals were included in the analysis as a missing value in the GLMM. The GLMM then generated prior distributions of survival probabilties—and thus of the probability of being at risk of emergence—for these 19 individuals, conditional on the model parameters. The CPH model was formulated as a count process and the binary frailty was incorporated as a zero-inflated Poisson model. Zeros in this model represent the non-survivors. Leaf quality did not appear to influence time-to-emergence. Pupal survival was affected in a complex and unexpected way showing opposite effects in males and females. We also explored the robustness of our model against increased levels of censoring. While the degree of censoring was low in our study (< 1%), we artificially increased it to 67%. Although further study is required to study the generality of these results in a theoretical framework, our explorations suggest that the newly proposed technique may be widely applicable in a variety of situations where the identification of the at risk population cannot be done in a straightforward way.