Quantification and Estimation of Regression to The Mean for Bivariate Distributions

Abstract

Regression to the mean (RTM) occurs when relatively high or low observations upon re-measurement are found closer to the population mean. When an intervention is applied to subjects selected in the tail of a distribution, an observed mean difference of the pre-post variables is called the total effect. The total effect is the sum of RTM and intervention/treatment effects, and estimation of RTM helps to accurately estimate the intervention/treatment effect.

The first study considers the bivariate Poisson distribution. Formulae for the total effect are derived and decomposed into RTM and intervention effects. The behaviour of RTM is demonstrated for homogeneous and inhomogeneous Poisson processes. Maximum likelihood estimators (MLE) for the total, RTM, and intervention effects are derived and their asymptotic properties are theoretically studied and verified through simulations. Using NSW data on road fatalities, the total, RTM, and intervention effects are estimated.

The second study considers the bivariate binomial distribution. Due to the dependence structure of the true and error components, subtracting RTM from the total effect does not give an unbiased estimator for the intervention effect. The correlation coefficient can take values in its full range, and RTM inflates comparatively more for negative correlation coefficient values. The Poisson and normal approximations to the binomial distribution underestimate the RTM effect. The MLE of the total, RTM and intervention effects are derived and their asymptotic properties are studied theoretically and verified through simulations. Data on obese individuals and cardboard cans are used to estimate the total, RTM and intervention effects.

Finally, we derive general formulae for the total, RTM and intervention effects under any bivariate distribution, while relaxing potentially restrictive assumptions commonly used in past research. An expression for the total effect is derived in general and decomposed into RTM and intervention effects. Derivation for a p parameter exponential family is separately considered. Examples of some selected bivariate distributions are given for illustrative purposes. Statistical properties of the MLE of the total, RTM and intervention effects are established theoretically where possible. The proposed and existing methods are compared using data on cholesterol levels by estimating the total, RTM and intervention effects.