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Unrestricted factor analysis: A powerful alternative to confirmatory factor analysis

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Abstract

The gold standard for modeling multiple indicator measurement data is confirmatory factor analysis (CFA), which has many statistical advantages over traditional exploratory factor analysis (EFA). In most CFA applications, items are assumed to be pure indicators of the construct they intend to measure. However, despite our best efforts, this is often not the case. Cross-loadings incorrectly set to zero can only be expressed through the correlations between the factors, leading to biased factor correlations and to biased structural (regression) parameter estimates. This article introduces a third approach, which has emerged in the psychometric literature, viz., unrestricted factor analysis (UFA). UFA borrows strengths from both traditional EFA and CFA. In simulation studies, we show that ignoring cross-loadings even as low as .2 can substantially bias factor correlations when CFA is used and that even the commonly used guideline RMSEA ≤ .05 may be too lenient to guard against non-negligible bias in factor correlations in CFA. Next, we present two empirical applications using Schwartz’s value theory, and electronic service quality. In the first case, UFA leads to much better model fit and more plausible regression estimates. In the second case, the difference is less dramatic but nevertheless, UFA provides richer results. We provide recommendations on when to use UFA vs. CFA.

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Notes

  1. Recall that in regression/SEM analysis, β = R−1r, where β is the vector of standardized regression coefficients, R is the correlation matrix of the independent variables and r is the vector of correlations between each independent variable and the dependent variable. Bias in R leads to a different estimate of its inverse (R−1), and hence to bias in the regression coefficients used in theory testing. Standard errors of the regression coefficients are also biased because they involve the square multiple correlation of each predictor i with the other predictors, \({R}_{i}^{2}\):

    SEβi = \(\sqrt{(1-{R}_{Y}^{2})/(N-k-1)}\) * \(\sqrt{1/(1-{R}_{i}^{2})}\) where \({R}_{Y}^{2}\) is the explained variance in the dependent variable, N is the sample size, and k is the number of predictors (Cohen et al., 2003, p. 86).

  2. These identification constraints are automatically implemented by Mplus. The researcher can review them with the command TECH1 on the OUTPUT line.

  3. For ease of exposition, we describe the estimation of UFA models as a two-step process as it is the approach used in traditional EFA. In fact, the rotated solution can be estimated directly (Browne 2001), and whether a one-step or two-step estimation approach is used, depends on computational/programming convenience.

  4. Because we use MLR, we use a modified chi-square difference test (Satorra and Bentler 2001). A convenient program to do the calculations is available on https://www.thestatisticalmind.com/calculators/SBChiSquareDifferenceTest.htm.

  5. By June 2022, the article had garnered nearly 6,000 Google Scholar citations.

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Acknowledgements

We thank the editor and the reviewers for their constructive comments, AiMark for providing the Schwartz Value Survey data and Arvind Malhotra for providing the E-S-QUAL data. The participation of Alberto Maydeu-Olivares was supported, in part, by the Research Center for Child Well-Being (NIGMS P20GM130420), grant PID2020-119755GB-I00 funded by AEI 10.13039/501100011033, and AGAUR grant 1237SGR2017. The cross-national data for the Schwartz Value Survey as well as Mplus and R input files for both empirical applications can be found on: https://www.dropbox.com/sh/tg6573rsx27o21c/AAApQpxdvtZwWoKgosPVUVu-a?dl=0.

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  1. University of North Carolina at Chapel Hill, Campus Box 3490, McColl Building, Chapel Hill, NC, 27599-3490, USA

    Jan-Benedict E. M. Steenkamp

  2. University of South Carolina, Barnwell College, 1512 Pendleton St., Columbia, SC, 29208, USA

    Alberto Maydeu-Olivares

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Steenkamp, JB.E.M., Maydeu-Olivares, A. Unrestricted factor analysis: A powerful alternative to confirmatory factor analysis. J. of the Acad. Mark. Sci. 51, 86–113 (2023). https://doi.org/10.1007/s11747-022-00888-1

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