Matrix Derivation of Multiple Correlation
IN MANY textbooks on statistics for students in the behavioral sciences, a rather full discussion can be found of the type of problem situation and as sumptions involved in multiple correlation. How ever, their authors usually then present without derivation the "normal equations" and finally dis cuss computational techniques for solving these equations to obtain constants for a regression equa tion (3:406; 4:178; 5:237). Thus, for students of be havioral science a formal derivation of the normal equations expressed in terms of m atr ix algebra will serve to fill this "logical gap". Assume that N persons have been measured on m predictor variables and one criterion variable. We can define the following vectors and m atr ices: j^isanNxi column vector of observed standard scores on the criterion, XisanNxm matrix of observed standard scores on the predictors, J3 is an m x 1 column vector of regression coefficients to be determined, and e is an N x 1 column vector of errors of estimate.