Quanti cation and Estimation of Regression to the mean for Bivariate Lognormal Distribution

Abstract

As a natural phenomenon, regression to the mean (RTM) happens when extreme observations are picked at the initial measurement and get closer to the mean throughout the subsequent measurements. Regression to the mean is a potential problem in data analysis that could lead to incorrect conclusions. Identifying and accounting for the RTM e ect is an essential objective in any statistical study. RTM expressions for the Normal, Poisson and Binomial distributions are accessible in the literature. RTM expression is not available when the pre and post-variables are distributed according to the bivariate lognormal distribution. The RTM e ect becomes more severe when the correlation between the two variables becomes weaker. Based on the correlation function, our derivations showed that a bivariate lognormal distribution behaved like a bivariate normal distribution. In pre-post experiments, the RTM impact decreases linearly as the correlation between variables increases. In a lognormal distribution, the RTM for the left and right cut-o points decrease di erently with correlation. The proposed formulation for the RTM e ect under bivariate lognormal distribution is substantially more satisfying than the Edgeworth series and Saddlepoint approximation. We conducted a simulation analysis to compare our suggested RTM expression to previously published approaches for non-normal populations. The RTM e ect was assessed for 56 cyclosporin test pairs at various cut-o values. The study included blood samples from organ transplant patients. We get parameter estimates using maximum likelihood. It is unreasonable to assess cyclosporin's real e cacy without considering the RTM e ect. The RTM e ect becomes increasingly noticeable as the cut-o point approaches the tail of the distribution.