A Bayesian Approach to Factor Scores

• In Event: Composite/Factor Scores Analyses With Bias Correction: Good Alternatives to Full Structural Equation Modeling?

Abstract

One strategy for conducting structural equation modeling involves proceeding in stages. In the first stage, a factor analytic measurement model is employed to obtain estimates of the measurement model parameters (possibly factor loadings, intercepts, error variances, and factor variances, covariances, and means). In a second stage, these estimated parameters are used to obtain estimates of the latent variables (factor scores) for participants. In a third stage, these factor scores are used as the values of the latent variables in structural models, relating these variables to covariates and/or outcomes, which may themselves be observed variables or factor scores from analogous factor analytic measurement models. One potential drawback of these approaches is the (mis)characterization of uncertainty in the factor scores and subsequent analyses using them. Scholars have suggested using predictive densities for factor scores to better characterize the uncertainty (Skrondal & Kuha, 2012). Conceptually, this approach allows for uncertainty in the factor scores emerging from the second stage, given the measurement model parameters.

In the current work, this is framed as an instance of Bayesian reasoning, in ways analogous to empirical Bayesian approaches in which frequentist point estimates are used for values of the measurement model parameters. However, this approach does not account for the uncertainty in these point estimates of the measurement model parameters emerging from the first stage. A fully Bayesian approach models and incorporates this source of uncertainty throughout all subsequent analyses. The current work describes a fully Bayesian approach, emphasizing how it expands on the above-mentioned approaches.

Early research on this in structural equation modeling has shown that this approach can characterize the uncertainty properly, and yield better estimates of structural parameters (Levy, 2017). Furthermore, this approach can be implemented in freely available software, precludes problems of interpretational confounding (Burt, 1976), and supports analyses of model-data fit (Levy, 2017). These findings suggest this is a promising approach for estimating factor scores and conducting structural equation modeling analyses, all the while acknowledging and expressing uncertainty throughout the multiple stages of fitting the models, and subsequent analyses such as model-data fit.

Author

• Roy Levy, Arizona State University