Coefficients of Factor Score Determinacy for Mean Plausible Values of Bayesian Factor Analysis

Abstract

In the context of Bayesian factor analysis, it is possible to compute plausible values, which might be used as covariates or predictors or to provide individual scores for the Bayesian latent variables. Previous simulation studies ascertained the validity of mean plausible values by the mean squared difference of the mean plausible values and the generating factor scores. However, the mean correlation of sets of single plausible values of different factors were shown to be an adequate estimator of the correlation between factors. Using sets of single plausible values to compute a mean prediction in secondary analysis implies that their determinacy should be known. Therefore, a plausible value-based determinacy coefficient allowing for estimation of the determinacy of single plausible values was proposed and evaluated by means of two simulation studies. The first simulation study demonstrated that the plausible value-based determinacy coefficient is an adequate estimate of the correlation of single plausible values with the population factor. It is also shown that the plausible value-based determinacy coefficient of mean plausible values approaches the conventional, model parameter-based determinacy coefficient with increasing number of imputations. The second simulation study revealed that the plausible value-based determinacy coefficient and the model parameter-based determinacy coefficient yield similar results even for misspecified models in small samples. It also revealed that for small sample sizes and a small salient loading size, the coefficients of determinacy overestimate the validity, so that it is recommended to report the determinacy coefficients together with a bias-correction to estimate the validity of plausible values in empirical settings.

The Trade-Off Between Factor Score Determinacy and the Preservation of Inter-Factor Correlations

“…Moreover, in the sample simulation study, determinacy coefficients based solely on model parameters were slightly larger than determinacy coefficients that were based on model parameters, observed scores, and the original factors used for generation of the observed scores. This result has already been reported (Beauducel & Hilger, 2022) and indicates that a correction proposed by Budescu (1982) may be used to reduce positive bias of determinacy coefficients that are only based on model parameters. This is relevant because in the context of empirical studies, only the model parameters are available for the computation of factor score determinacy.…”

“…Moreover, there are different possibilities to compute correlation-preserving factor score predictors. The correlation-preserving factor score predictor proposed by McDonald (1981) are computed from the model parameters whereas the correlation-preserving factor predictor proposed by Beauducel and Hilger (2022) is based on a transformation of a given factor score predictor. The latter can be useful in the context of Bayesian plausible values, when a software package only provides factor score predictors that are not correlation preserving (e.g., Mplus, Asparouhov & Muthén, 2010).…”

“…The latter can be useful in the context of Bayesian plausible values, when a software package only provides factor score predictors that are not correlation preserving (e.g., Mplus, Asparouhov & Muthén, 2010). The correlation-preserving factor predictor proposed by Beauducel and Hilger (2022) might also be of interest when only the factor score predictors and the model parameters are available, whereas the scores of the observed variables are unavailable. However, the correlation-preserving factor score predictor proposed by McDonald (1981), ten Berge, Krijnen, Wansbeek, and Shapiro (1999), and Beauducel and Hilger (2022b) has not been directly compared, so that, possible differences between their factor score determinacies are unknown.…”