Finding Pure Submodels for Improved Differentiation of Bifactor and Second-Order Models

Several studies have indicated that bifactor models fit a broad range of psychometric data better than alternative multidimensional models such as second-order models (e.g., Carnivez, 2016; Gignac, 2016; Rodriguez, Reise, & Haviland, 2016). Murray and Johnson (2013) and Gignac (2016) argued that this phenomenon is partially due to unmodeled complexities (e.g., unmodeled cross-factor loadings) that induce a bias in standard statistical measures that favors bifactor models over second-order models. We extend the Murray and Johnson simulation studies to show how the ability to distinguish second-order and bifactor models diminishes as the amount of unmodeled complexity increases. By using theorems about rank constraints on the covariance matrix to find submodels of measurement models that have less unmodeled complexity, we are able to reduce the statistical bias in favor of bifactor models; this allows researchers to reliably distinguish between bifactor and second-order models.