Estimation of the Proportion of the Variance Explained by Regression, When the Number of Parameters in the Model May Depend on the Sample Size

Abstract

The multiple correlation coefficient is widely utilized in the physical and biological sciences. Its statistical properties are well known and discussed in [1] and [5]. Its principle practical drawback, however, is that it is sensitive to the number of parameters utilized in the regression model. In this note, it is shown that if the number of parameters in the regression model is a function of the total sample size, then the usual estimator of the multiple correlation coefficient, R2, is not an asymptotically consistent estimator of the population multiple correlation coefficient, R2. An alternative estimator, p2, is proposed and it is shown to be consistent under all conditions. Some of the properties of the proposed estimator are discussed and an example of its application is presented.