Revisiting the Bi-Factor Model: Can Mixture Modeling Help Assess Its Applicability?

Abstract

This article revisits from the perspective of finite mixture modeling the increasingly popular bi-factor model applied in contemporary behavioral and social research. It is pointed out that in a population with substantial unobserved heterogeneity resulting from a mixture of latent classes, and where the unidimensional model holds along with models that markedly differ from the bi-factor model, the latter may turn out to be spuriously plausible. To raise caution about this possibility, an example of a 3-class setting is provided, where correspondingly (a) the single (global) factor model, (b) a model with a global factor and a single local factor, and (c) a model with a global factor and two local factors hold, while the bi-factor model with a global factor and three local factors is also plausible for the analyzed data overall. Examination of population heterogeneity prior to testing the bi-factor model is therefore recommendable in empirical research, in order to avoid spurious findings of its plausibility when ignoring substantial unobserved heterogeneity in studied populations.

Keywords:

• bi-factor model
• global factor
• latent class
• local factor
• mixture
• unobserved heterogeneity

ACKNOWLEDGMENTS

This research was conducted in part while Tenko Raykov was visiting GESIS – Leibniz Institute for the Social Sciences, Mannheim, Germany, and its support is greatly appreciated. We thank S. Reise for valuable discussions on the bi-factor model. We are grateful to an anonymous referee for valuable comments and criticism on an earlier version of the note, which have contributed significantly to its improvement.

Notes

¹ The bi-factor model is potentially seriously mis-specified in any of the three classes owing to the following direct observation. This model can be treated as differing from the models used to generate the class-specific data by at least one added factor possessing the same variance as the factor(s) in the class-specific models, as seen from their definitional Equations 2–4 (and surrounding discussion; see also Appendix A). (We also note in passing that the use of equal values for nearly all loadings in the data generation models is not essential for the intended message of this article and was primarily decided for convenience reasons while consistent with the goal of the note to show one example where plausibility of the bi-factor model is spurious; see also Conclusion section.)

² The bootstrap LRT for the 4-class solution was associated with a numerical warning indicating that its p-value, reported in Table 1, “may not be trustworthy due to local maxima”. Since this test’s p-value for the 5-class solution was nonsignificant (see Table 1), and given the results reported in the current section of the main text, this numerical issue result was interpreted as consistent with the selection of the 3-class model, as discussed in the main text. Such a numerical warning is also consistent with an attempt to “over-extract” classes that are not supported by the analyzed data, which were based on a 3-class model here.

³ Considering the seven dependable p-values in Table 1 as resulting from multiple testing carried out on the same data set (see also Footnote 2), adjustment for this multiplicity can be made using the Benjamini–Hochberg procedure (e.g., Raykov & Marcoulides, 2017). The latter determines as significant only the four tests associated with p-values of 0.

⁴ In the exploratory factor analyses reported next, we do not interpret software provided p-values due to the involved misclassification error, as mentioned, and do not pursue confirmatory factor analysis within the latent classes (see also Raykov & Pohl, 2013a, 2013b; with respect to “dominance” of the global factor, as well as Equations 2 and 4 and surrounding discussion on the data generation process). Instead, we focus only on the structure-related, qualitative findings consistent strictly speaking with unidimensionality rather than the bi-factor model with regard to the observed measures; these findings are of relevance for the main conclusion stated at the end of the current paragraph in the main text. None of the discussion in the last paragraph implies or is to be taken to imply a suggestion that a single factor model holds in each of the classes.