(untitled) (GYMNASIUM, 2017)
Multivariate regression analysis of criterion variables from Zc in the space of Mahalanobis variables from M can be defined as a solution to the problem Mb = Zc + Eôtrag(EtE) = minimum. As MtM = I, the solution which 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 product-moment coefficients of correlation between the regressors transformed to a Mahalanobis form and criterion variables. Of course, that is why the asymptotic variance of coefficients bjp from matrix b is simply sjp2 = (1 - bjp2)2n-1, and the tests of hypotheses H0jp: bjp* = 0 are easily fjp = bjp2((n - 2)(1 - bjp2)-1), because under H0jp: bjp* = 0, variables fjp have the Fisher-Snedecor F-distribution with 1 and n - 2 degrees of freedom. Regression functions are now defined by the operation Y = Mb with the covariance matrix G = YtY = btb = RcrRrr-1Rrc and the diagonal elements of the matrix r2 = (rp2) = diag G are usual coefficients of determination; and since ZctY = RcrRrr-1Rrc = G, elements rp of matrix r are normal multiple correlation coefficients, so the 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 the Fisher – Snedecor F-distribution with m and n - m - 1 degrees of freedom.