Efficiency Lost by Analyzing Counts Rather than Event Times in Poisson and Overdispersed Poisson Regression Models

Abstract

Inference for point processes is most efficient if the event times for each individual are available. Sometimes, the study design is such that only aggregated data are collected, consisting of the number of events or recurrences for each individual over the observation period. This article discusses the loss in efficiency of an analysis of the aggregated counts versus an analysis of the actual event times. One particular case is exemplified—that in which the purpose of the experiment or trial is to compare the effects of treatments—and the loss in efficiency in the estimator of the treatment effect is computed. The specific point process considered here is the nonhomogeneous Poisson process, with a proportional intensity model for the treatment effects. Random-effects models are also considered, with estimation via a quasi-likelihood approach. The quasi-likelihood analysis proposed here is an extension of such techniques for the homogeneous Poisson process. The resulting estimating equations for the parameters in the random-effects models are simple and intuitive. The results show that for many usual situations, treatment effects are very efficiently estimated using aggregated data, but the underlying intensity function is not.

Key Words:

• Clinical trial
• Estimating equation
• Negative binomial regression
• Quasi-likelihood
• Random effect
• Recurrent event
• Unobserved heterogeneity.