When a common battery of predictor variables is used to estimate criterion measures of examinees for a number of separate activities, a system of regression equations may be established. It may be possible to establish a smaller number of composite predictor variables without serious loss in efficiency of the estimates of criterion performances. These composite predictor variables may have simple relations to both the original predictor variables and the estimation of the criterion measures. A method for the investigation of such systems of regression equations has been devised. The procedures depend on the simple proposition that predictor variables may be subjected to nonsingular linear transformations without altering the estimates of criterion performances. This proposition may be outlined in matrix algebra, involving scores on the predictor variables, estimates of the criterion performances of the individuals, regression weights, and a square nonsingular transformation matrix of order equal to the number of predictors. Steps involved in the procedure are: (1) establishment of orthogonal (uncorrelated) reference predictor composites; (2) establishment of a reduced space containing significant principal predictor composites; and (3) rotation of predictor composites in the reduced space. Data from the manual for the Differential Aptitude Test are presented as an illustration.