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

Milos Popovic, Hana Valkova, Veroljub Stankovic, Dragan Popovic, Evagelia Boli, Vladimir Savic, Jasna Popovic


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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