Bifactor Models and Rotations: Exploring the Extent to Which Multidimensional Data Yield Univocal Scale Scores

Abstract

The application of psychological measures often results in item response data that arguably are consistent with both unidimensional (a single common factor) and multidimensional latent structures (typically caused by parcels of items that tap similar content domains). As such, structural ambiguity leads to seemingly endless “confirmatory” factor analytic studies in which the research question is whether scale scores can be interpreted as reflecting variation on a single trait. An alternative to the more commonly observed unidimensional, correlated traits, or second-order representations of a measure's latent structure is a bifactor model. Bifactor structures, however, are not well understood in the personality assessment community and thus rarely are applied. To address this, herein we (a) describe issues that arise in conceptualizing and modeling multidimensionality, (b) describe exploratory (including Schmid–Leiman [Schmid & Leiman, 1957] and target bifactor rotations) and confirmatory bifactor modeling, (c) differentiate between bifactor and second-order models, and (d) suggest contexts where bifactor analysis is particularly valuable (e.g., for evaluating the plausibility of subscales, determining the extent to which scores reflect a single variable even when the data are multidimensional, and evaluating the feasibility of applying a unidimensional item response theory (IRT) measurement model). We emphasize that the determination of dimensionality is a related but distinct question from either determining the extent to which scores reflect a single individual difference variable or determining the effect of multidimensionality on IRT item parameter estimates. Indeed, we suggest that in many contexts, multidimensional data can yield interpretable scale scores and be appropriately fitted to unidimensional IRT models.

Acknowledgments

This work was supported by the Consortium for Neuropsychiatric Phenomics (National Institute of Health [NIH] Roadmap for Medical Research grants UL1-DE019580 (principal investigator [PI]: Robert Bilder) and RL1DA024853 (PI: Edythe London). Additional research support was obtained through the NIH Roadmap for Medical Research grant (AR052177; PI: David Cella) and from a National Cancer Institute [NCI] grant 4R44CA137841-03 (PI: Patrick Mair) for IRT software development for health outcomes and behavioral cancer research. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NCI or the NIH.

Notes

¹This does not surprise us given the nature of the alexithymia construct and how it is captured by the OAS. Alexithymia refers to deficits in the processing of emotionally charged information. The construct emerged from the clinical literature and has never, to our knowledge, emerged in any empirically based major taxonomies of personality or psychopathology. In short, its behavioral penetrance probably is low, and thus, we do not expect indicators (which are very distal from the trait) to be highly correlated. Second, this is an observer-report measure that attempts to indirectly capture the construct by collecting ratings of its observable manifestations in a variety of domains; for example, interpersonal matters and relationships, insight and self-understanding, health worries, humor, and rigidity. We recognize, actually expect, that individual differences in alexithymia is just one possible common source of individual differences on these variables. For this reason as well, we did not expect high factor intercorrelations.

²Technically this is standardized coefficient omega hierarchical, and the previously reported alpha is standardized alpha (i.e., based on polychoric correlations). In this study, we worked exclusively with a polychoric correlation matrix to conduct the factor analyses, and so our estimated factor loadings are standardized. The appropriate raw score aggregate for interpretation of coefficient omega hierarchical in this case is the sum of standardized items.

³In theory, we could calculate omega based on the Schmid–Leiman results or the target bifactor rotation. In fact, the R psych package omega command cited earlier makes this easy. In this data, omega hierarchical drops to around .65 in the exploratory analysis. On the other hand, as described previously, one cannot fully trust the exploratory results, especially the Schmid–Leiman parameters. For this reason, we argue that omega is most wisely calculated only after a confirmatory model has been established.