A Perspective on the Mathematical and Psychometric Aspects of Factor Indeterminacy

Abstract

It is generally known that the factors in the common factor model (CFM) cannot be solved for uniquely even if the factor model fits perfectly and the factor loadings are known. This indeterminacy of the factor variables (often referred to as ‘factor scores’) stems from the fact that the CFM is a set of underdetermined linear equations that has more unknowns than equations—a situation which produces an infinity of alternative solutions. In the first section of this inquiry, factor indeterminacy for all factors, common and unique, is examined in the context of The Theory of Generalized Inverses (TGI). Casting factor indeterminacy in the context of TGI provides some simpler results than older approaches. It is shown, for example, that essentially all statistical properties of the infinity of solutions are summarized in the elements of a single, idempotent matrix—symbolized here as H. In the final section of this article, a psychometric view of indeterminacy is presented in which it is argued that the two causes of indeterminacy are low reliabilities and low communalities of the observed variables. It is shown that the diagonal elements of the matrix H are reliability coefficients; therefore, the Spearman-Brown equation can be brought to bear on how much indeterminacy can be reduced by repeated sampling from an Infinite Behavior Domain (IBD). A brief discussion is presented on the effects of factor indeterminacy in IRT models.