Mixed-Poisson point process with partially observed covariates: ecological momentary assessment of smoking

Ecological momentary assessment is an emerging method of data collection in behavioral research that may be used to capture the times of repeated behavioral events on electronic devices and information on subjects' psychological states through the electronic administration of questionnaires at times selected from a probability-based design as well as the event times. A method for fitting a mixed-Poisson point-process model is proposed for the impact of partially observed, time-varying covariates on the timing of repeated behavioral events. A random frailty is included in the point-process intensity to describe the variation in baseline rates of event occurrence among subjects. Covariate coefficients are estimated using estimating equations constructed by replacing the integrated intensity in the Poisson score equations with a design-unbiased estimator. An estimator is also proposed for the variance of the random frailties. Our estimators are robust in the sense that no model assumptions are made regarding the distribution of the time-varying covariates or the distribution of the random effects. However, subject effects are estimated under gamma frailties using an approximate hierarchical likelihood. The proposed approach is illustrated using smoking data.