Finding Pure Sub-Models for Improved Differentiation of Bi-Factor and Second-Order Models.

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