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.

Keywords:

• Factor indeterminacy
• common factor analysis
• generalized inverses
• psychometrics

Article information

Conflict of interest disclosures: The author signed a form for disclosure of potential conflicts of interest. The author did not report any financial or other conflicts of interest in relation to the work described.

Ethical principles: The author affirms having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was not supported.

Role of the funders/sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The ideas and opinions expressed herein are those of the author alone, and endorsement by the author’s institution is not intended and should not be inferred. This study is based on a Presidential Address presented to the Society of Multivariate Experimental Psychology. Richmond, VA, October 2016. While a graduate Student at Purdue University in the late 1960’s, one of my graduate advisors, Peter Schönemann, told the graduate students majoring in psychometrics (both of us), that, as Ph.D. psychometricians, it was a role-expectation that we provide a unique derivation of coefficient alpha and make a contribution to the literature on factor indeterminacy. I have fulfilled the coefficient alpha expectation, and the present paper hopefully satisfies Dr. Schönemann second one. The Author thanks James Steiger and Niels Waller for the many hours that they graciously spent with me in reviewing and discussing the nature of factor indeterminacy. This effort owes so very much to their wise counsel; however, they disagree with some of the positions taken here.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 The results in Equations (11)–(16) can also be derived using vectors of rescaled, random numbers to serve as the arbitrary vectors, z_k and z_j. Such an approach was suggested by McDonald (1974, p.219).

2 The author thanks A. Beauducel for pointing out that the Anderson-Rubin (1956) predictor of the common factors, ${\widehat{\mathbf{\theta }}}_T,$ where ${\widehat{\mathbf{\theta }}}_T={({\mathbf{\Lambda }}^T{\mathbf{\Sigma }}^{-1}\mathbf{\Lambda })}^{-1/2}\widehat{\mathbf{\theta }},$ has the desirable property that Corr(${\widehat{\mathbf{\theta }}}_T$) = I. However, ${\widehat{\mathbf{\theta }}}_T$ will not satisfy the CFM equations–$\mathbf{y}=\mathbf{\Lambda }\mathbf{\theta }+\mathbf{\Psi }\mathbf{\nu };$ it is strictly an estimate or predictor of $\mathbf{\theta }$ [See Beauducel (2015)].

3 Since the variables in a factor analysis may not be behavioral in nature, a more general term for a collection of correlated, non-behavioral measures might be labelled as a “Coherent Variable Domain” (CHD). Such a domain would be identical to an IBD in that it entails variables measuring the same common factors, that when held constant, cause the observed-variable correlations to vanish.