When and why the second-order and bifactor models are distinguishable

- Previous research has explored conditions where statistical comparison of a second-order and bifactor model is challenging.
- The models imply nested sets of tetrad constraints, and comparing them depends on whether these constraints are violated.
- How tetrads relate to proportionality and why fit is biased in favor of the bifactor when data contain added complexities.

Previous studies (Gignac, 2016; Murray & Johnson, 2013) have explored conditions for which accurate statistical comparison of a second-order and bifactor model is challenging, if not impossible. However, a mathematical basis for this problem has not been offered. We show that a second-order model implies a unique set of tetrad constraints (Bollen & Ting, 1993, 1998, 2000) that the bifactor model does not, and that the two models are distinguishable to the degree that these unique tetrad constraints are violated. Simulated population matrices, and mathematical proofs, are used to demonstrate that: (a) when a second-order model with cross-loadings or correlated residuals is true, fitting a pure (misspecified) second-order model leads to violation of the tetrad constraints, which in turn leads to the chi-square and Bayesian Information Criterion favoring a (misspecified) bifactor model, and (b) a true bifactor model can be identified only when the tetrad constraints of the second-order model are violated, which is mainly a function of the proportionality of loadings. Three model-comparison approaches are offered for applied researchers.