Replacing Variables in Correlation Problems
Vincent I. West · Journal of the American Statistical Association · 1952
T HE importance of using matrix-inverting methods in solving normal equations has often been emphasized. Computation of the matrix inverse to the complete correlation matrix or to the matrix of correlations among the independent variables is usually the easiest way to evaluate the standard errors of the regression constants. For special problems the complete inverse may not be needed [10, page 456]. Because of the need to compute the standard errors, and the frequent desirability of working later with slightly altered sets of variables, however, it would seem that the inversion of the matrix of correlations (or covariances) should be the initial step in solving most problems involving normal equations [5]. After solving the normal equations of correlation problems one may wish to delete a variable, to replace a variable, or to add another variable. In some situations these operations may be performed most efficiently by making the desired deletions, substitutions, or additions and then re-solving the normal equations. In other situations the desired changes may be effected more efficiently by operations on the inverse matrix already computed, without directly re-solving the normal equations. One of the advantages of using matrix-inverting methods is the possibility of utilizing the inverse in deriving solutions for altered sets of variables. The purpose of this paper is to describe one method which may be useful in situations where the direct re-solving of the normal equations would be laborious. The solution will first be indicated by the problem of replacing the fourth variable in a five-variable problem. The problem will then be generalized to the replacing of the kth variable in an n-variable problem. In discussing the five-variable problem the matrix inverse to the correlation matrix will be called the P table so that this discussion may be more easily understood by computers who are accustomed to the Waugh Method [1].