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

Maxwell Mansolf, Steven P. Reise*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

83 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)120-129
Number of pages10
StatePublished - Mar 1 2017
Externally publishedYes

ASJC Scopus subject areas

  • Experimental and Cognitive Psychology
  • Developmental and Educational Psychology
  • Arts and Humanities (miscellaneous)


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