Fit indices are insensitive to multiple minor violations of perfect simple structure in confirmatory factor analysis

Classic confirmatory factor analysis (CFA) models are theoretically superior to exploratory factor analysis (EFA) models because they specify that each indicator only measures one factor. In contrast, in EFA, all loadings are permitted to be nonzero. In this article, we show that when fit to EFA structures and other models with many cross-loadings, classic CFA models often produce excellent fit. A key requirement for breaking this pattern is to have highly variable ratios of main loadings to corresponding cross-loadings in the true data-generating structure-and strongest misfit results when cross-loadings are of mixed sign. We show mathematically that EFA structures that are rotatable to a CFA representation are those where the main loadings and the cross-loadings are proportional for each group of indicators. With the help of a ShinyApp, we show that unless these proportionality constraints are violated severely in the true data structure, CFA models will fit well to most true models containing many cross-loadings by commonly accepted fit index cutoffs. We also show that fit indices are nonmonotone functions of the number of positive cross-loadings, and the relationship becomes monotone only when cross-loadings are of mixed sign. Overall, our findings indicate that good fit of a CFA model rules out that the true model is an EFA model with highly variable ratios of main and cross-loadings, but does not rule out most other plausible EFA structures. We discuss the implications of these findings. (PsycInfo Database Record (c) 2025 APA, all rights reserved).