Abstract
The commentaries by Burke and Johnston (this issue), Eid (this issue), Junghänel et al. (this issue), and Willoughby (this issue) on Burns et al. (this issue) provide useful context for comparing three latent variable modeling approaches to understanding psychopathology—the correlated first-order syndrome-specific factors model, the bifactor S – 1 model, and the symmetrical bifactor model. The correlated first-order syndrome-specific factors model has proven useful in constructing explanatory models of psychopathology. The bifactor S – 1 model is also useful for examining the latent structure of psychopathology, especially in contexts with clear theoretical predictions. Joint use of correlated first-order syndrome-specific model and bifactor S – 1 model provides leverage for explaining psychopathology, and both models can also guide individual clinical assessment. In this reply, we further clarify reasons why the symmetrical bifactor model should not be used to study the latent structure of psychopathology and also discuss a restricted bifactor S – 1 model that is equivalent to the first-order syndrome-specific factors model.
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07 September 2020
The original version of this article unfortunately contained a mistake. The supplementary material was not captured to our commentary article.
Notes
This can be seen by writing out the measurement and structural equations in the model shown in Figure 1C. The measurement equation for mean-centered variables is Yif = λifFf + εif (i = indicator, f = facet). The structural (latent regression) equation for the non-reference facets is Ff = βfF1 + Sf, where f = 1 indicates the reference facet. Inserting the structural equation for Ff into the measurement equation for the non-reference facets yields Yif = λif (βfF1 + Sf) + εif = λifβfF1 + λifSf + εif. This is the measurement equation for the restricted bifactor S – 1 model shown in Figure 1D.
Note that the fact that factors have the same loadings does not always mean that the factors have the same meaning. This can be seen by comparing the factors Ff and Sf in Figure 1A and D. Notice that F2 and F3 in Figure 1A have the same loadings as do S2 and S3 in Figure 1D. Nonetheless, the factors Ff and Sf have different meanings. The Ff factors are true score variables, whereas the Sf factors are true score regression residual variables. This is because the indicators measuring the Sf factors also load on the reference factor F1 in Figure 1D.
The application of the symmetrical bifactor model to structurally different symptom facets can yield an admissible solution if the correlations among the symptom indicators are homogenous, a pattern that is common with randomly selected and thereby inter-changeable raters. Such an admissible solution, however, still does not yield a psychometrically sound definition of the general factor (see Eid et al. 2017; Heinrich et al. 2018).
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Acknowledgments
This research was supported by two grants from the Ministry of Economy and Competitiveness of Spanish Government under award numbers PSI2014-52605-R and PSI2017-82550-R (AEI/FEDER, UE), and a predoctoral fellowship co-financed by MINECO (Spanish Government) and the European Social Fund (BES-2015-075142). Stephen Becker (K23MH108603) and Theodore Beauchaine (UL1TR002733, UH2DE025980) are supported by grants from the National Institutes of Health. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health (NIH) or the Spanish Government. We thank Cristina Trias for assistance with the study.
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Burns, G.L., Geiser, C., Servera, M. et al. Promises and Pitfalls of Latent Variable Approaches to Understanding Psychopathology: Reply to Burke and Johnston, Eid, Junghänel and Colleagues, and Willoughby. J Abnorm Child Psychol 48, 917–922 (2020). https://doi.org/10.1007/s10802-020-00656-1
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DOI: https://doi.org/10.1007/s10802-020-00656-1