Multiple Rectilinear Prediction and the Resolution into Components

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Abstract
It is assumed that a battery of n tests has been resolved into components in a common factor space of r dimensions and a unique factor space of at most n dimensions, where r is much less than n. Simplified formulas for ordinary multiple and partial correlation of tests are derived directly in terms of the components. The best (in the sense of least squares) linear regression equations for predicting factor scores from test scores are derived also in terms of the components. Spearman's “single factor” prediction formulas emerge as special cases. The last part of the paper shows how the communality is an upper bound for multiple correlation. A necessary and sufficient condition is established for the square of the multiple correlation coefficient of test j on the remaining n−1 tests to approach the communality of test j as a limit asn increases indefinitely while r remains constant. Limits are established for partial correlation and regression coefficients and for the prediction of factor scores.