Pearsonian Correlation Coefficients Associated with Least Squares Theory

Abstract

It is well known that the zero-order correlation between the predicted value of a variable and the observed value of the variable is the multiple correlation. It is also well known that the zero-order correlation between the residuals for two different variables, when the prediction is from a common set of variables, is the partial correlation. These considerations naturally lead to a systematic investigation of all the zero-order correlations involving the various variables associated with least squares theory. Such an investigation is the purpose of this paper. As a result of this study it appears that other zero-order correlations include the multiple alienation coefficient, the part correlation coefficient, and certain other coefficients which, as far as I am aware, have not been previously defined. The paper first examines the case of a single predicted variable and then continues with the case in which two or more variables are predicted simultaneously. The paper includes (1) a theoretical development of the different coefficients and the relations between them, (2) the expression of the formulas in determinantal form, (3) a matrix presentation of the material, and (4) an outline of the calculational techniques--with illustrations. It should be made clear at the start that this paper deals with populations (finite or infinite) and not with samples from those populations. The sampling distribution of each of the new correlation coefficients defined in this paper might well become the subject of a later investigation, but first we need to know what these correlation coefficients are.