Moments of the Nonnegative Adjusted Estimator of Squared Multiple Correlation

Abstract

I present the moments of the nonnegative adjusted estimator of the squared multiple correlation ρ 2 , the coefficient of determination for random-predictor regression. This estimator, first proposed by Ezekiel, replaces with zero the negative estimates from the well-known adjusted estimator proposed by Fisher that, in turn, corrects the positive bias of the sample R 2. Although Fisher’s version is presented in texts, Ezekiel’s version is used in practice. Each moment comprises a binomial sum of a negative binomial series of incomplete beta functions. Numerical computations, for which an R function is provided, are required to examine the moments. Ezekiel’s estimator is positively biased for smaller ρ 2 and negatively biased for larger. It dominates Fisher’s via MSE. It does not dominate R 2, but the MSE of Ezekiel’s estimator can be substantially smaller but at most negligibly larger. Possible applications to powers of ρ 2 and to other adjusted estimators are briefly discussed.