Regression to the mean for overdispersed count data

In repeated measurements, regression to the mean (RTM) is a tendency of subjects with observed extreme values to move closer to the mean when measured a second time. Not accounting for RTM could lead to incorrect decisions such as when observed natural variation is incorrectly attributed to the effect of a treatment/intervention. A strategy for addressing RTM is to decompose the total effect, the expected difference in paired random variables conditional on the first being in the tail of its distribution, into regression to the mean and unbiased treatment effects. The unbiased treatment effect can then be estimated by subtraction. Formulae are available in the literature to quantify RTM for Poisson distributed data which are constrained by mean–variance equivalence, although there are many real life examples of overdispersed count data that are not well approximated by the Poisson. The negative binomial can be considered an explicit overdispersed Poisson process where the Poisson intensity is chosen from a gamma distribution. In this study, the truncated bivariate negative binomial distribution is used to decompose the total effect formulae into RTM and treatment effects. Maximum likelihood estimators (MLE) and method of moments estimators are developed for the total, RTM, and treatment effects. A simulation study is carried out to investigate the properties of the estimators and compare them with those developed under the assumption of the Poisson process. Data on the incidence of dengue cases reported from 2007 to 2017 are used to estimate the total, RTM, and treatment effects.