Multiple Rectilinear Prediction and the Resolution into Components: II

Given a battery of n tests that has already been resolved into r orthogonal common factors and n unique factors, procedures are outlined for computing the following types of linear multiple regressions directly from the factor loadings: (i) the regression of any one test on the n −1 remaining tests; (ii) all the n different regressions of order n −1 for the n tests, computed simultaneously; (iii) the regression of any common factor on the n tests; (iv) the regressions of all the common factors on the n tests computed simultaneously; (v) the regression of any unique factor on the n tests; (vi) the regressions of all the unique factors on the n tests, computed simultaneously. Multiple and partial correlations are then determined by ordinary formulas from the regression coefficients. A worksheet with explicit instructions is provided, with a completely worked out example. Computing these regressions directly from the factor loadings is a labor-saving device, the efficiency of which increases as the number of tests increases. The amount of labor depends essentially on the number of common factors. This is in contrast to computations based on the original test intercorrelations, where the amount of labor increases more than proportionately as the number of tests increases. The procedures evaluate formulas developed in a previous paper (2). They are based essentially on a shortened way of computing the inverse of the test intercorrelation matrix by use of the factor loadings.