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Breaking Our Silence on Factor Score Indeterminacy

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  • Niels G. Waller

    (University of Minnesota)

Abstract

Although many textbooks on multivariate statistics discuss the common factor analysis model, few of these books mention the problem of factor score indeterminacy (FSI). Thus, many students and contemporary researchers are unaware of an important fact. Namely, for any common factor model with known (or estimated) model parameters, infinite sets of factor scores can be constructed to fit the model. Because all sets are mathematically exchangeable, factor scores are indeterminate. Our professional silence on this topic is difficult to explain given that FSI was first noted almost 100 years ago by E. B. Wilson, the 24th president (1929) of the American Statistical Association. To help disseminate Wilson’s insights, we demonstrate the underlying mathematics of FSI using the language of finite-dimensional vector spaces and well-known ideas of regression theory. We then illustrate the numerical implications of FSI by describing new and easily implemented methods for transforming factor scores into alternative sets of factor scores. An online supplement (and the fungible R library) includes R functions for illustrating FSI.

Suggested Citation

  • Niels G. Waller, 2023. "Breaking Our Silence on Factor Score Indeterminacy," Journal of Educational and Behavioral Statistics, , vol. 48(2), pages 244-261, April.
    • Handle: RePEc:sae:jedbes:v:48:y:2023:i:2:p:244-261
    DOI: 10.3102/10769986221128810
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    References listed on IDEAS

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    1. James Steiger, 1979. "Factor indeterminacy in the 1930's and the 1970's some interesting parallels," Psychometrika, Springer;The Psychometric Society, vol. 44(2), pages 157-167, June.
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    3. Walter Lederman, 1938. "The orthogonal transformations of a factorial matrix into itself," Psychometrika, Springer;The Psychometric Society, vol. 3(3), pages 181-187, September.
    4. Stanley Mulaik & Roderick McDonald, 1978. "The effect of additional variables on factor indeterminacy in models with a single common factor," Psychometrika, Springer;The Psychometric Society, vol. 43(2), pages 177-192, June.
    5. James Steiger, 1979. "The relationship between external variables and common factors," Psychometrika, Springer;The Psychometric Society, vol. 44(1), pages 93-97, March.
    6. Roderick McDonald, 1974. "The measurement of factor indeterminacy," Psychometrika, Springer;The Psychometric Society, vol. 39(2), pages 203-222, June.
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