A Theoretical Comparison of the Predictive Power of the Multiple Regression and Equal Weighting Procedures

The multiple regression procedure (MRP) is compared with the equal weighting procedure (EWP) which forces the estimated regression weight to be the same for each of the k predictors (independent variables) after orienting each predictor to be in the same direction as the criterion (dependent variable) and standardizing each predictor to have zero mean and unit variance. An expression for the critical sample size n+< at which the expected mean squared error of prediction (EMSEP) is the same for the MRP and the EWP is derived. If the estimation sample size n>n*, then the MXP has smaller EMSEP than the EWP and the converse holds for nn* is interpreted in terms of testing the null hypothesis that the true regression weights are the same for all k predictors. It is shown that n* increases (i.e., the EWP becomes a more suitable procedure) as the population coefficient of determination and/or the coefficient of variation of the true beta weights decreases. The EWP also becomes more suitable as the number of predictors and/or the average predictor intercorrelation increases. Expressions are derived for the percent increase in EMSEP by using the inappropriate procedure. It is shown that the MRP has no more than twice the EPISEP of the EWP, and in most cases, the EWP has no more than twice the EMSEP of the LIRP. The expression for n* also holds approximately when EMSEP is replaced by predictive validity (correlation) as the measure of predictive power. The results of this paper (summarized in the last section), although derived under the assumption of nonrandom predictors, are shown to be good approximations to the case of random predictors.