A rigorous and an approximate solution are found for the problem: Given a primary trait matrix for n tests and r1 traits, and a matrix for the same n tests and r2 reference axes, to discover the transformation which will transform the second matrix into the first, or primary trait matrix. Formulas for determining the limits of the effect of using the approximate solution are presented. The method is applied to a set of twenty hypothetical tests, defined by their loadings on four orthogonal primary traits. After factoring the inter-correlations of these variables by Thurstone's centroid method, approximating the diagonals, the original hypothetical matrix is reproduced with a root mean square discrepancy of .014 by assuming as known the primary trait loadings of only the first eight tests. The method is applied to the results of factoring two batteries of 14 tests, having 8 tests in common, to give the factor loadings of the two batteries on the same reference axes. The method provides a means of comparing directly and quantitatively the results of two different factor studies, provided they have tests in common, and of testing the stability of simple structure under changes in the battery. The relations of the method here developed to certain problems in multiple correlation are shown.

A method is indicated by which multiple factor analysis may be used in determining a number, r, and then in selecting r “predicting” variables out of n variables so that each of the remaining n-r variables may be predicted almost as well from the r variables as it could be predicted from all the n−1 variables.