Bifactor and other hierarchical models [in factor analysis] have become central to representing and explaining observations in psychopathology, health, and other areas of clinical science, as well as in the behavioral sciences more broadly. This prominence comes after a relatively rapid period of rediscovery, however, and certain features remain poorly understood.
Here, hierarchical models are compared and contrasted with other models of superordinate structure, with a focus on implications for model comparisons and interpretation. Issues pertaining to the specification and estimation of bifactor and other hierarchical models are reviewed in exploratory as well as confirmatory modeling scenarios, as are emerging findings about model fit and selection.
Bifactor and other hierarchical models provide a powerful mechanism for parsing shared and unique components of variance, but care is required in specifying and making inferences about them.
[Keywords: hierarchical model, higher order, bifactor, model equivalence, model complexity]
…Bifactor models are now ubiquitous in the structural modeling of psychopathology. They have been central to general factor models of psychopathology (eg. et al 2014, et al 2015, et al 2012, et al 2015) and have become a prominent focus in modeling a range of phenomena as diverse as internalizing psychopathology (Naragon- et al 2016), externalizing psychopathology ( et al 2007), psychosis ( et al 2017), somatic-related psychopathology ( et al 2016), cognitive functioning (2015), and constructs central to prominent therapeutic paradigms ( et al 2015). They have also become central to modeling method effects, such as informant ( et al 2013), keying ( et al 2017, 1999), and other effects (2006), and they have been used to explicate fundamental elements of measurement theory ( et al 2017).
Although bifactor and other hierarchical models are now commonplace, this was not always so. Their current ubiquity follows a long period of relative neglect (2012), having been derived in the early 20th century (1938, 1937) before being somewhat overlooked for a number of decades and then being rediscovered more recently. Bifactor models were mistakenly dismissed as equivalent to and redundant with other superordinate structural models (eg. 1964, 1981, 1959, 2012, et al 1999); as differences between bifactor models and other types of superordinate structural models became more recognized (Yung et al 199925ya), interest in bifactor models reemerged.
…Summary Points:
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Bifactor and other hierarchical models represent superordinate structure in terms of orthogonal general and specific factors representing distinct, non-nested components of shared variance among indicators. This contrasts with higher-order models, which represent superordinate structure in terms of specific factors that are nested in general factors, and correlated-factors models, which represent superordinate structure in terms of correlations among subordinate factors.
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Higher-order models can be approached as a constrained form of hierarchical models, in which direct relationships between superordinate factors and observed variables in the hierarchical model are constrained to equal the products of superordinate-subordinate paths and subordinate-observed variable paths.
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Multiple exploratory factor analytic approaches to the delineation of hierarchical structure are available, including rank-deficient transformations, analytic rotations, and targeted rotations. Among other things, these transformations and rotations differ in the number of factors being rotated, the nature of those factors, and how superordinate factor structures are approximated.
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Misspecification or under-specification of confirmatory bifactor and hierarchical models can occur for multiple reasons. Problems with model identification may occur (1) with specific patterns of homogeneity in estimated or observed covariances, (2) if factors are allowed to correlate in inadmissible ways, or (3) if covariate paths imply inadmissible correlations. Signs of model misspecification may be evident in anomalous estimates, such as loading estimates near boundaries, or estimates that are suggestive of other types of models.
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Common model fit statistics can overstate the fit of bifactor models due to the tendency of bifactor and other hierarchical models to overfit to data in general, regardless of plausibility or population structure. Hierarchical models are similar to exploratory factor models in their expansiveness of fit, and, in general, they are more expansive in fit than other confirmatory models.
Future Issues:
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Research is needed to determine how to best account for the flexibility of hierarchical models when comparing models and evaluating model fit, given that the relative flexibility of hierarchical models can only partly be accounted for by the number of parameters. Approaches based on minimum description length and related paradigms, such as Bayesian inference with reference priors, are promising in this regard.
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More research is needed to clarify the properties of hierarchical structures when they are embedded in longitudinal models and models with covariates. As with challenges of multicollinearity in regression, parsing unique general and specific factor components of explanatory paths may be inferentially challenging in the presence of strongly related predictors, covariates, and outcomes.
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More can be learned about the specification and identification of hierarchical models and the relationships between hierarchical models and other types of models, such as exploratory factor models. Similarities in overfitting patterns between exploratory and hierarchical models, approaches to hierarchical structure through bifactor rotations, and patterns of anomalous estimates that are sometimes obtained with hierarchical models, point to important relationships between exploratory and hierarchical models. Further explication of model specification principles with hierarchical models would also help clarify the appropriate structures to consider when evaluating models.