Robust Nonparametric Methods for Regression to the Mean Model

Regression to the mean is a statistical phenomenon that often confounds treatment effects in experiments. Consider an experiment involving a treatment, in which a response is measured (baseline) on a subject then a treatment is applied and a second measurement is taken. Then under many bivariate models for the pair of responses (including the bivariate normal), the predicted response of the second measurement will regress to the mean. In experiments where the second response is only taken for a select sample, say above a cutoff value, then this regression to the mean effect may mistakenly be thought of as a treatment effect.\nIn this investigation, we consider a model of the treatment effect which also takes into account this regression to the mean effect. In particular, we consider the multiplicative model of Naranjo and McKean (2001). Naranjo and McKean developed a bootstrap test for treatment effect based on least squares methods for bivariate normal distributions. We developed robust procedures to assess treatment effects for this multiplicative model. Our procedures are based on rank-based methods, for general score functions. Our preliminary Monte Carlo investigations show that our procedures are robust. We extend this robust and traditional development to models other than the bivariate normal, including the multivariate t distributions. We investigate the finite sample properties of these methods and compare their empirical behavior over a variety of models and situations.