MULTIVARIATE REGRESSION ANALYSIS IN MAHALANOBIS OF ATHLETES AND INIFIED PARTNERS

Abstract

Multivariate regression analysis of criterion variables from Z_c in the space of Mahalanobis variables from M can be defined as a solution to the problem Mb = Z_c + Eôtrag(E^tE) = minimum. As M^tM = I, the solution which is easily obtained by differentiating the function trag (E^tE) is: b = M^tZ_c = R_rr^-1/2R_{rc ,} 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 b_jp from matrix b is simply s_jp² = (1 - b_jp²)²n^-1, and the tests of hypotheses  H_0jp: b_jp^* = 0  are easily f_jp = b_jp²((n - 2)(1 - b_jp²)^-1), because under  H_0jp: b_jp^* = 0, variables f_jp 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 = Y^tY = b^tb = R_crR_rr^-1R_rc  and the diagonal elements of the matrix r² = (r_p²) = diag G are usual coefficients of determination; and since  Z_c^tY = R_crR_rr^-1R_rc = G, elements r_p of matrix r are normal multiple correlation coefficients, so the tests of hypotheses  H_0p: r_p^* = 0 are defined by the functions f_p = (r_p²(1 - r_p²)^-1)((n - m - 1)m^-1)  because under H_0p: r_p^* = 0, functions f_p  have the Fisher – Snedecor F-distribution with  m  and  n - m - 1 degrees of freedom.