Search

Analytic bifactor rotations (Jennrich & Bentler, 2011, 2012) have been recently developed and made generally available, but are not well understood. The Jennrich-Bentler analytic bifactor rotations (bi-quartimin and bi-geomin) are an alternative to, and arguably an improvement upon, the less technically sophisticated Schmid-Leiman orthogonalization (Schmid & Leiman, 1957). We review the technical details that underlie the Schmid-Leiman and Jennrich-Bentler bifactor rotations, using simulated data structures to illustrate important features and limitations. For the Schmid-Leiman, we review the problem of inaccurate parameter estimates caused by the linear dependencies, sometimes called “proportionality constraints,” that are required to expand a p correlated factors solution into a (p+1) (bi)factor space. We also review the complexities involved when the data depart from perfect cluster structure (e.g., item cross-loading on group factors). For the Jennrich-Bentler rotations, we describe problems in parameter estimation caused by departures from perfect cluster structure. In addition, we illustrate the related problems of: (a) solutions that are not invariant under different starting values (i.e., local minima problems); and, (b) group factors collapsing onto the general factor. Recommendations are made for substantive researchers including examining all local minima and applying multiple exploratory techniques in an effort to identify an accurate model.

Although the structure of the Rosenberg Self-Esteem Scale (RSES; Rosenberg, 1965) has been exhaustively evaluated, questions regarding dimensionality and direction of wording effects continue to be debated. To shed new light on these issues, we ask: (1) for what percentage of individuals is a unidimensional model adequate, (2) what additional percentage of individuals can be modeled with multidimensional specifications, and (3) what percentage of individuals respond so inconsistently that they cannot be well modeled? To estimate these percentages, we applied iteratively reweighted least squares (IRLS; Yuan & Bentler, 2000) to examine the structure of the RSES in a large, publicly available dataset. A distance measure, ds, reflecting a distance between a response pattern and an estimated model, was used for case weighting. We found that a bifactor model provided the best overall model fit, with one general factor and two wording-related group factors. But, based on dr values, a distance measure based on individual residuals, we concluded that approximately 86% of cases were adequately modeled through a unidimensional structure, and only an additional 3% required a bifactor model. Roughly 11% of cases were judged as “unmodelable” due to their significant residuals in all models considered. Finally, analysis of ds revealed that some, but not all, of the superior fit of the bifactor model is owed to that model’s ability to better accommodate implausible and possibly invalid response patterns, and not necessarily because it better accounts for the effects of direction of wording.

Bifactor latent structures were introduced over 70 years ago, but only recently has bifactor modeling been rediscovered as an effective approach to modeling construct-relevant multidimensionality in a set of ordered categorical item responses. I begin by describing the Schmid-Leiman bifactor procedure (Schmid & Leiman, 1957), and highlight its relations with correlated-factors and second-order exploratory factor models. After describing limitations of the Schmid-Leiman, two newer methods of exploratory bifactor modeling are considered, namely, analytic bifactor (Jennrich & Bentler, 2011) and target bifactor rotations (Reise, Moore, & Maydeu-Olivares, 2011). In section two, I discuss limited and full-information estimation approaches to confirmatory bifactor models that have emerged from the item response theory and factor analysis traditions, respectively. Comparison of the confirmatory bifactor model to alternative nested confirmatory models and establishing parameter invariance for the general factor also are discussed. In the final section, important applications of bifactor models are reviewed. These applications demonstrate that bifactor modeling potentially provides a solid foundation for conceptualizing psychological constructs, constructing measures, and evaluating a measure's psychometric properties. However, some applications of the bifactor model may be limited due to its restrictive assumptions.

Group-level parametric and non-parametric item response theory models were applied to the Consumer Assessment of Healthcare Providers and Systems (CAHPS(®)) 2.0 core items in a sample of 35,572 Medicaid recipients nested within 131 health plans. Results indicated that CAHPS responses are dominated by within health plan variation, and only weakly influenced by between health plan variation. Thus, although the CAHPS 2.0 survey has acceptable psychometric properties when analyzed at the individual level, large sample sizes are needed to reliably differentiate among health plans. These results illustrate why it is important to study evaluations of health care, such as CAHPS, at multiple levels of analyses.

The mixed-measurement IRT model and the traditional factor analytic model are two fundamentally different ways of representing the structure underlying an item response matrix. To contrast these approaches, the parameters of full-information item factor models of varying dimensionality and mixed-measurement models of varying numbers of latent classes were estimated in a sample of 1,000 responses to a 14-item measure of Positive Interpersonal Engagement. Findings indicated that either a two latent factor or a two latent class mixed-model provided the most appropriate representation of the data. We also found that the factor models were, in general, more parsimonious than the mixed-measurement models. Nevertheless, we argue that deciding between a dimensional and a mixed- measurement representation of item response heterogeneity should not rest solely on statistical criteria. In the discussion, we suggest some research contexts in which the mixed- measurement model may be conceptually more appropriate than the traditional factor model.

We explored between gender invariance on the NEO PI-R Neuroticism scale. Sorbom's (1974;1982) mean and covariance structural analysis was used to evaluate weak and strong forms of factorial invariance for the six facet scales when treated as indicators of a common neuroticism factor. Item response theory procedures were used to evaluate differential item functioning (DIF) within each of the six facets. The structural modeling results indicated that the Neuroticism scale is weakly invariant (i.e., factor loadings were equivalent for men and women), but the hypothesis of strong invariance was not supported. The within-facet DIF analysis was difficult to interpret. Although several items displayed significant DIF, it was difficult to associate DIF with specific aspects of item content. Also, even when substantial item-level DIF was identified on the Anxiety facet, its effects when considered at the facet level appeared inconsequential. This latter finding illustrates that item-level DIF does not necessarily ...

I describe how multilevel logistic regression can be used to assess the consistency of an individual's response pattern with an item response theory measurement model. Specifically, by treating item responses as being nested within individuals, multilevel logistic regression is used to estimate a person-response curve that models how an individual's item endorsement rate decreases as a function of item difficulty. The slope of an individual's person-response curve is used as an indicator of the degree of response consistency or person-fit. I argue that the proposed multilevel modeling approach to person-fit assessment has several potential advantages over traditional techniques. The most important advantage being that the multilevel modeling approach allows explanatory variables to be entered into the model so that the causes of response inconsistency or differential test functioning can be investigated.

Bifactor latent structures were introduced over 70 years ago, but only recently has bifactor modeling been rediscovered as an effective approach to modeling construct-relevant multidimensionality in a set of ordered categorical item responses. I begin by describing the Schmid-Leiman bifactor procedure (Schmid & Leiman, 1957), and highlight its relations with correlated-factors and second-order exploratory factor models. After describing limitations of the Schmid-Leiman, two newer methods of exploratory bifactor modeling are considered, namely, analytic bifactor (Jennrich & Bentler, 2011) and target bifactor rotations (Reise, Moore, & Maydeu-Olivares, 2011). In section two, I discuss limited and full-information estimation approaches to confirmatory bifactor models that have emerged from the item response theory and factor analysis traditions, respectively. Comparison of the confirmatory bifactor model to alternative nested confirmatory models and establishing parameter invariance for the general factor also are discussed. In the final section, important applications of bifactor models are reviewed. These applications demonstrate that bifactor modeling potentially provides a solid foundation for conceptualizing psychological constructs, constructing measures, and evaluating a measure’s psychometric properties. However, some applications of the bifactor model may be limited due to its restrictive assumptions.

(2001). Introduction to the Special Issue on Multilevel Models. Multivariate Behavioral Research: Vol. 36, No. 2, pp. 153-153.