Method of Correlated Vectors
The method of correlated vectors is one way of testing whether the g factor extracted from a battery of diverse tests is related to some variable, X, which is external to the battery of tests. If the degree to which each of the various tests is loaded on g significantly predicts the relative magnitudes of the various tests' correlations with the external variable X, it is concluded that variable X is related to g (independently of whether or not it is related to other factors or test specificity). The significance level is determined from the rank-order correlation between the elements in the column vector of the various tests' g loadings and the elements in the column vector of the tests' correlations with variable X.
As the size of a test's factor loading (in this case its g loading) and the size of the test's correlation with an external variable (X) are both affected by the test's reliability, and as various tests may differ in reliability, it is necessary to rule out the possibility that the correlation between the vector of the tests' g loadings and the vector of the tests' correlations with X is not attributable to the tests' differing reliability coefficients. This is accomplished by correcting the g loadings and the correlations for attenuation, or by obtaining the correlation of the column vector of the tests' reliability coefficients with both the vector of g loadings and the vector of correlations, and using the three correlations between the three vectors to calculate the partial correlation between the g vector and the X vector (with the vector of reliability coefficients partialed out). If the partial correlation is large enough to be statistically significant, the tests' varying reliability coefficients are not responsible for the correlation between the g and X vectors. (Note: The degrees of freedom for testing the significance of the correlation [or the partial correlation] is based, not on the number of subjects in the study, but on the number of tests [i.e., number of elements in the vector of g loadings]).
An actual example of the use of correlated vectors is shown in connection with the information in Table B.1, from a study ( Schafer, 1985) on the relation