Further aspects of the theory of multiple regression

This paper may be regarded as a sequel to a previous papers(1) in these Proceedings . The vector and matrix notation of that paper used for a statistical sample is systematized somewhat further, so that while a sample S refers as before to the matrix of nm values (a sample of m observations in one variate only being a row vector), we write for the linear regression formula between the dependent and independent variates into which a sample is supposed partitioned (in place of equation (12) of (1)). More generally, a third submatrix S 0 is partitioned off, and its effect eliminated (corresponding to equation (13) of (1)), but without loss of generality we assume that S 2 in equation (1) above can always stand for S 2.0 if necessary.