Correlations between Similar Sets of Measurements

Abstract

The correlation between, for instance, father-and-son pairs with respect to one variable (e.g., height) is easily calculated. When we consider p variables simultaneously, let X1, * * *, X, be a father's measurements, and let Y1, * * *, Yp, be his son's measurements of the same p variables. The problem is to find the father-son correlation with respect to the p variables as a whole. If U = a1X1 + * * * + apXp and V = a1Yi + * * * + aY, are 'linear characteristics', the problem then reduces to finding a set of coefficients (a, , * * *, a,) that maximizes the correlation between U and V. The present problem is thus a restricted case of the general problem of finding the maximum correlation between two sets of variables linearly combined as U = a1Xi + * * + apX, and V = b1Yi + * * * + bqYq. In the restricted case, let 1xx be the p X p covariance matrix of X and ryy be the p X p covariance matrix of Y; these two matrices are symmetric and positive definite, and we assume I;Xx = 1yy = S. Next, let sxy be the p X p matrix of covariance of each Xi with each Yj and construct the symmetric matrix T = 2(Ixy + V y)4 Then the correlation between U and V for a particular set of coefficients is p(a) = a'Ta/a'Sa, which is to be maximized with respect to a = (a, , * * *, ap)'. The solution yields a set of canonical correlations pi > P2 > ... > pp and corresponding pairs of canonical variates (a'X, a'Y), (a'X, a1Y), * , (aX, a1Y). It is shown that the desired coefficients are eigenvectors corresponding to the eigenvalues X1 > X2 > * > Xp of the matrix S-'T. Maximum likelihood estimates of the elements of the matrices S and T are given when s y = 11 y . These turn out to be simply the familiar estimates of the elements of an arbitrary covariance matrix, averaged whenever necessary to obtain the hypothesized symmetry. A numerical example with p = 2 variables has been worked out.