Regression To The Mean for a New Bivariate Binomial Distribution

Abstract

When measurements are made twice on the same subject / person, RTM is noticed when relatively high or low observations are likely to be followed by less extreme observations that are closer to the true mean. When subjects are selected based on some cut-off points, the observed mean difference of the pre-post variable is called the total effect. The total effect is equal to the sum of the RTM effect and treatment effect and should be accounted for RTM. Bi-variate binomial-binomial distribution models the data when the two groups, say i and j, have two possible outcomes in a fixed number of trials in which the number of success follows a binomial distribution. It is the result of the convolution of two independent binomial marginals. The study considers the Bi-variate binomial-binomial distribution. Formula for the total , treatment and RTM effect are derived. Using R’s optimize function, the log-likelihood function was maximized to provide the maximum likelihood estimators. The results of the simulation study showed that the maximum likelihood (ML) estimators of RTM are unbiased and consistent. The proposed method is applied to the real data of the number of patients dealt with in one of two boxes of critical care Emergency service of San Agustin Hospital, in Linares (Spain).The parameters of real data are estimated through MLE and substitute in the derived RTM formula. This shows that the Total Effect = Treatment Effect + RTM Effect