The bifactor model, originally proposed by Holzinger and Swineford (1937), posits that each observed variable is influenced by a general factor common to all items and by one (and only one) group-specific factor shared with a subset of items. The general and group factors are specified as orthogonal, meaning each item's variance is partitioned into three independent sources: general factor variance, specific group factor variance, and unique (error) variance. This structure is particularly well-suited for constructs hypothesized to have both a broad overarching dimension and narrower subdimensions.
Model Structure
Cov(G, S_k) = 0 for all k
Cov(S_j, S_k) = 0 for all j ≠ k
Σ = λ_G λ′_G + Σ_k (λ_Sk λ′_Sk) + Θ
In the bifactor model, λ_iG is the loading of item i on the general factor G, and λ_iS is its loading on group factor S_k. The orthogonality constraints mean that the contribution of the general factor to item covariance is completely separable from the contributions of group factors. This decomposition enables direct computation of how much variance in total scores is attributable to the general factor versus specific group factors.
Model-Based Indices
The bifactor model yields several important psychometric indices. The explained common variance (ECV) is the proportion of common variance attributable to the general factor: ECV = Σ λ²_iG / (Σ λ²_iG + Σ Σ_k λ²_iSk). When ECV is high (e.g., > 0.70), the data are essentially unidimensional, and total scores primarily reflect the general factor. Omega hierarchical (ω_h) estimates the reliability of the total score as a measure of the general factor, while omega subscale (ω_hs) estimates the reliability of subscale scores after removing general factor variance.
ω_hs = (Σ λ_iSk)² / (1′Σ1)
where 1′Σ1 = total score variance from the model-implied covariance matrix
A common alternative to the bifactor model is the higher-order model, where first-order factors load on a second-order general factor. Mathematically, the higher-order model is a constrained version of the bifactor model: it imposes proportionality constraints on the general-factor loadings (each item's general loading equals its first-order loading times its factor's second-order loading). The bifactor model is thus more flexible, and likelihood-ratio tests can compare the two. In practice, the bifactor model often fits better because it relaxes these proportionality constraints, but interpretation requires care — the orthogonality assumption may not be theoretically defensible in all applications.
Applications
Bifactor models have become widely used in intelligence research, where a general factor (g) coexists with specific ability factors (verbal, spatial, processing speed). In clinical psychology, the bifactor model has been applied to psychopathology, with a general factor (p-factor) capturing shared variance across psychiatric symptoms and specific factors capturing disorder-specific variance. In personality measurement, bifactor models evaluate whether facet subscales contain useful specific variance beyond the broad domain factor.
The practical implications are significant. If omega hierarchical is high and omega subscale values are low, subscale scores add little beyond the total score, and reporting subscale scores may be misleading. Conversely, if group factors have substantial reliable specific variance, subscale scores convey unique information. Bifactor modeling thus provides a principled, model-based answer to the perennial question of whether to report total scores, subscale scores, or both — replacing ad hoc decisions with quantitative evidence about the dimensionality and scoring of psychological measures.