Regression to the mean: Estimation and adjustment under the bivariate normal distribution

Abstract

Regression to the mean (RTM) is a statistical phenomenon that happens when subjects having relatively high or low observations upon remeasurement are found closer to the population mean. RTM can erroneously influence the conclusion in a pre-post study design. Expressions are available for quantifying RTM when the distribution of pre-post variables is bivariate normal; however, these methods assume the pre and post observations are identically distributed and strictly positively correlated. This study generalizes previous results to include non-stationary, normally distributed random variables that are potentially negatively correlated, and also provides the decomposition of the conditional mean difference into RTM and unbiased intervention effects. In addition, the maximum likelihood estimators are derived and the unbiasedness, consistency and normality of these estimators are established. A simulation study is conducted to asses the accuracy of estimating the RTM and intervention effects using existing and the proposed methods. Data on the blood lead level in both intervention and placebo groups are used for decomposing the total change in blood lead level on pre-post occasions into RTM and intervention effects.