Extending the Debate Between Spearman and Wilson 1929: When do Single Variables Optimally Reproduce the Common Part of the Observed Covariances?

Abstract

The covariances of observed variables reproduced from conventional factor score predictors are generally not the same as the covariances reproduced from the common factors. We sought to find a factor score predictor that optimally reproduces the common part of the observed covariances. It was found algebraically that—under some conditions—the single observed variable with highest loading on a factor reproduces the non-diagonal elements of the observed covariance matrix more exactly than the conventional factor score predictors. This finding is linked to Spearman's and Wilson's 1929 debate on the use of single variables as factor score predictors. A population-based and a sample-based simulation study confirmed the algebraic result that taking a single variable can outperform conventional factor score predictors in reproducing the non-diagonal covariances when the nonzero loading size and the number of nonzero loadings per factor are small. The results indicated that a weighted aggregation of variables does not necessarily lead to an improvement of the score over the variable with the highest loading.

KEYWORDS:

• factor analysis
• factor scores
• factor score predictors.