### Multivariate Regression Analysis of the Criterion Variables of Zc in the Space of Mahalanobis Variables of M.

#### Abstract

Multivariate regression analysis of the criterion variables of Zc in the space of Mahalanobis variables of M can be defined as a solution to the problem Mb = Zc + E|trag(EtE) = minimum. As MtM = I, the solution that is easily obtained by differentiating the function trag (EtE) is b = MtZc = Rrr-1/2Rrc and the matrix of partial regression coefficients is, in fact, a matrix of ordinary prod¬uct - moment coefficients of correlation between the regressors transformed into Ma¬halanobis form and criterion variables. Of course, therefore, the asymptotic variance of coefficients bjp of matrix b is simply sjp2 = (1 - bjp2)2n-1, and the tests of hypotheses H0jp: bjp* = 0 are clearly fjp = bjp2((n - 2)(1 - bjp2)-1), because under H0jp: bjp* = 0 variables fjp have Fisher-Snedecor F distribution with 1 and n2 degrees of freedom Regression functions are now defined by the operation Y = Mb with a matrix of covariances G = YtY = btb = RcrRrr-1Rrc,s the diagonal elements of the matrix r2 = (rp2) = diag G are normal coefficients of determination; and since ZctY = RcrRrr-1Rrc = G, then rp elements of r matrix are ordinary multiple correlation coefficients, and tests of hypotheses H0p: rp* = 0 are defined by the functions fp = (rp2(1 - rp2)-1)((n - m - 1) m-1), because under H0p: rp* = 0 functions fp have Fisher-Snedecor F distribution with m and nm1 degrees of freedom.

**Keywords: **/multivariate / regression / analysis / criterion /variables /

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