Extending the debate between Spearman and Wilson 1929: When do single variables optimally reproduce the common part of the observed covariances?

TL;DR

This paper investigates when using a single observed variable as a factor score predictor best reproduces the common part of observed covariances, challenging conventional methods especially with small loadings.

Contribution

It demonstrates that under certain conditions, a single variable with the highest loading can optimally reproduce covariances, extending Spearman and Wilson's 1929 debate.

Findings

Single variable with highest loading can perfectly reproduce non-diagonal covariances.

Single variable outperforms conventional factor scores with small loadings.

Simulation studies confirm the theoretical results.

Abstract

Because the covariances of observed variables reproduced from conventional factor score predictors are generally not the same as the covariances reproduced from the common factors, it is proposed to find a factor score predictor that optimally reproduces the common part of the observed covariances. It is shown that, under some conditions, the single observed variable with highest loading on a factor perfectly reproduces the non-diagonal observed covariances. This refers to Spearman's and Wilson's 1929 debate on the use of single variables as factor score predictors. The implications of this finding were investigated in a population based and in a sample based simulation study confirming that taking a single variable outperforms conventional factor score predictors in reproducing the observed covariances when the salient loading size and the number of salient loadings per factor are small. Implications of this finding for factor score predictors are discussed.