Regression to the mean for bivariate distributions

Abstract

Regression to the mean is said to have occurred when subjects having relatively high or low measurements are remeasured closer to the population mean. This phenomenon can influence the conclusion about the effectiveness of a treatment in a pre-post study design. The mean difference of the pre- and post-variables, conditioned on the initial variable being above or below a cut-point, is the sum of regression to the mean and treatment effects. Expressions for regression to the mean are available for the bivariate normal distribution under restrictive assumptions, and for the bivariate Poisson and binomial distributions, more generally. This article derives expressions for regression to the mean for any bivariate distribution while making fewer restrictive assumptions than previous methods. Maximum likelihood estimators are derived, and the unbiasedness, consistency, and asymptotic normality of these estimators are shown for exponential families, where possible. Data on the cholesterol levels in men aged 35-39 are used for decomposing the conditional mean difference in cholesterol level on pre-post occasions into regression to the mean and treatment effects. In another example, data on diastolic blood pressure for 341 patients are used to demonstrate the fraction of change due to regression to the mean and the treatment effects, respectively.