Zero inflated Poisson distributions and regression to the mean

Regression to the mean (RTM) is a statistical phenomenon in which extreme obser vations or measurements tend to move closer to the population average upon subsequent remeasurement. In pre-post studies subjects are often selected based on specific base line criteria or thresholds. The total effect, which is the observed mean difference of the pre-post observations, includes both RTM and the average true intervention effects. This often leads to incorrect conclusions about the effectiveness of the intervention. To avoid inaccurate conclusion, it is essential to account for the RTM effect when estimating an intervention effect. In the literature, formulas for quantifying RTM have been developed for various forms of the bivariate Poisson distributions. However, these formulations often fail to adequately address the excess zeros commonly found in correlated bivariate count data. This article focuses on the Type I Bivariate Zero-Inflated Poisson Distribution (TI-BZIPD), which is particularly valuable for modeling bivariate count data with a significant number of zero observations, and the Type I Bivariate Zero-Inflated Generalized Poisson Distribution (TI-BZIGPD), which is effective for modeling both a substantial number of zero observations and varying correlations and dispersion. These distributions are highly relevant for various applications. Expressions are formulated to represent the total effect, which is then divided into the RTM and intervention effects. Maximum likelihood estimators for these effects are derived, and their properties are validated through extensive simulations. Data on Lacistema aggregatum and Protium guianense plant counts, together with health care utilization data from the Australian Health Survey (1977–1978), are used to estimate RTM and intervention effects for the Type I Bivariate Zero-Inflated Poisson Distribution and the Zero-Inflated Generalized Poisson Distribution, respectively.