Journal of Agricultural, Biological, and Environmental StatisticsA Likelihood Framework for Multi-Stage Interval-Censored Processes

Abstract We propose a rigorous likelihood-based joint framework for modeling stage-specific transition times in multi- stage biological processes subjected to discrete inspection schemes and environmental forcing. The proposed methodology simultaneously resolves three intertwined observational and ecological challenges: (i) interval- censored stage entry and exit times, (ii) right censoring arising from both administrative follow-up limits and disease-induced premature failures (e.g., pathogenic attacks), and (iii) a partially ordered dual-event structure wherein a stage transition and a disease attack occur within the same inspection interval. The transition hazard is modeled as a function of cumulative time-varying environmental and climatic covariates integrated over calendar time, capturing how accumulated weather exposure affects instantaneous developmental risks between latent stage entry and exit. This non-proportional hazard formulation precludes standard partial likelihood methods and necessitates full likelihood inference. Because the observed likelihood involves high-dimensional intractable integrals over unobserved entry and exit dates, parameter estimation is conducted via a Monte Carlo Expectation- Maximization (MCEM) algorithm. The E-step draws exact conditional latent trajectories via probability integral transform rejection sampling, while the M-step updates piecewise-constant baseline hazard parameters in closed form and regression coeffcients via bounded L-BFGS-B optimization. Statistical uncertainty on environmental effects is rigorously quantified using Louis’s observed information principle, combining complete-data Hessian expectations with Monte Carlo score-variance corrections to account for missing information due to interval censoring. Motivated by the multi-stage fruit development of cocoa (Theobroma cacao) under phytosanitary pressure, the framework provides a versatile tool for longitudinal agro-environmental event-time data subject to periodic monitoring.