Exploratory Bifactor Analysis: The Schmid-Leiman Orthogonalization and Jennrich-Bentler Analytic Rotations

Abstract

Analytic bifactor rotations (Jennrich & Bentler, 2011, 2012) have been recently developed and made generally available, but are not well understood. The Jennrich-Bentler analytic bifactor rotations (bi-quartimin and bi-geomin) are an alternative to, and arguably an improvement upon, the less technically sophisticated Schmid-Leiman orthogonalization (Schmid & Leiman, 1957). We review the technical details that underlie the Schmid-Leiman and Jennrich-Bentler bifactor rotations, using simulated data structures to illustrate important features and limitations. For the Schmid-Leiman, we review the problem of inaccurate parameter estimates caused by the linear dependencies, sometimes called “proportionality constraints,” that are required to expand a p correlated factors solution into a (p+1) (bi)factor space. We also review the complexities involved when the data depart from perfect cluster structure (e.g., item cross-loading on group factors). For the Jennrich-Bentler rotations, we describe problems in parameter estimation caused by departures from perfect cluster structure. In addition, we illustrate the related problems of: (a) solutions that are not invariant under different starting values (i.e., local minima problems); and, (b) group factors collapsing onto the general factor. Recommendations are made for substantive researchers including examining all local minima and applying multiple exploratory techniques in an effort to identify an accurate model.

Psychometric Properties of the Externalizing Spectrum Inventory: Replication and Extension across Clinical and Non-Clinical Samples

“…When more than one dimension was found in the exploratory factor analysis, we conducted an exploratory bifactor analysis, using the Schmid-Leiman orthogonalization (Schmid & Leiman, 1957). An explanation of the Schmid-Leiman procedure is provided in Mansolf and Reise (2016). For this exploratory analysis the Root Mean Square of Residuals (RMSR) and Kelley's criterion (Kelley, 1935) were used as fit indices.…”

Quantifying the Strength of General Factors in Psychopathology: A Comparison of CFA with Maximum Likelihood Estimation, BSEM, and ESEM/EFA Bifactor Approaches

“…The bi-geomin rotation is recommended in cases where cross-loadings are likely to be present (e.g. Mansolf & Reise, 2016). Scaling and identification were achieved by fixing the latent factor variances to 1.…”

“…First, given the poor performance of the ESEM/EFA models we increased the random starts for the rotation algorithm, from the software default of 30 to 1000. Past research has suggested that bi-factor rotations in ESEM/EFA are prone to local minima and that within these solutions, general factor variance is liable to be overstated (Mansolf & Reise, 2016). We used a sample size of n=200.…”

“…In an ESEM or EFA model, all loadings are freely estimated; therefore, in principle, the covariance due to cross-loadings is appropriately modelled and does not inflate other parameters, especially those used to gauge the strength of a p-factor. However, in practice, Mansolf & Reise, (2016) noted that EFA models with bifactor rotations are liable to overattribute variance to the general factor because group factors can 'collapse' onto the general factor.…”

Appropriate Use of Bifactor Analysis in Psychopathology Research: Appreciating Benefits and Limitations

“…The bifactor model's flexibility can also enable it to show superior global fit than alternatives, even when the other models were themselves used to simulate data (33,(38)(39)(40)(41)(42)(43)(44)(45)(46). For example, skewed item distributions and unmodeled cross-loadings or correlated residuals can all lead fit statistics to favor the bifactor model over a correlated-factors model (with no general factor), even if the correlatedfactors model more accurately describes the true structure (46,47).…”