A Perspective on the Mathematical and Psychometric Aspects of Factor Indeterminacy

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.

Considering Horn's Parallel Analysis from a Random Matrix Theory Point of View

Horn's parallel analysis is a widely used method for assessing the number of principal components and common factors. We discuss the theoretical foundations of parallel analysis for principal components based on a covariance matrix by making use of arguments from random matrix theory. In particular, we show that (i) for the first component, parallel analysis is an inferential method equivalent to the Tracy-Widom test, (ii) its use to test high-order eigenvalues is equivalent to the use of the joint distribution of the eigenvalues, and thus should be discouraged, and (iii) a formal test for higher-order components can be obtained based on a Tracy-Widom approximation. We illustrate the performance of the two testing procedures using simulated data generated under both a principal component model and a common factors model. For the principal component model, the Tracy-Widom test performs consistently in all conditions, while parallel analysis shows unpredictable behavior for higher-order components. For the common factor model, including major and minor factors, both procedures are heuristic approaches, with variable performance. We conclude that the Tracy-Widom procedure is preferred over parallel analysis for statistically testing the number of principal components based on a covariance matrix.

Some Results on Mean Square Error for Factor Score Prediction

For the confirmatory factor model a series of inequalities is given with respect to the mean square error (MSE) of three main factor score predictors. The eigenvalues of these MSE matrices are a monotonic function of the eigenvalues of the matrix Γ _p = Φ ^1/2 Λ _p 'Ψ _p^-1 Λ _p Φ ^1/2. This matrix increases with the number of observable variables p. A necessary and sufficient condition for mean square convergence of predictors is divergence of the smallest eigenvalue of Γ _p or, equivalently, divergence of signal-to-noise (Schneeweiss & Mathes, 1995). The same condition is necessary and sufficient for convergence to zero of the positive definite MSE differences of factor predictors, convergence to zero of the distance between factor predictors, and convergence to the unit value of the relative efficiencies of predictors. Various illustrations and examples of the convergence are given as well as explicit recommendations on the problem of choosing between the three main factor score predictors.