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

ABSTRACT

Analytic bifactor rotations have been recently developed and made generally available, but they 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. 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.

KEYWORDS:

• Exploratory bifactor
• local minima
• Schmid-Leiman

Article information

Conflict of Interest Disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical Principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was supported by Grant 1317428 from the DMS - CDS&E-MSS and Grant AR052177 from the National Institutes of Health.

Role of the Funders/Sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors' institution, the National Institutes of Health, or the DMS - CDS&E-MSS is not intended and should not be inferred. The authors thank Peter M. Bentler for his comments on an earlier draft of the manuscript.

Notes

1 While oblique versions of the JB rotations exist (Jennrich & Bentler, 2012), oblique bifactor models run contrary to the conceptualization of the group factors as representing residual variance, which is by definition orthogonal to the general factor. For this and other very technical reasons, we ignore the oblique case in this tutorial.

2 The initial estimation of communalities (e.g., squared multiple correlations) and the choice of extraction method (e.g., maximum likelihood), and the number of factors extracted are of course important in terms of the accuracy of the ultimate bifactor solution. However, these topics are not important for understanding the relative limitations of SL or JB rotations and thus will not be discussed further.

3 Yung et al. (1999) present a more complete treatment of the Schmid-Leiman orthogonalization, in which there are arbitrarily many levels of factors.

4 The true factor loading was .9973; the results in the table were rounded to 2 decimal places.

5 GPArotation returns solutions with seemingly random orderings and signs (positive or negative) of group factors. Because the factors are orthogonal, any reordering of group factors or change in sign of any factor is mathematically equivalent to any other. Unique solutions were thus identified as follows. First, each column of a solution was multiplied by the sign (i.e., 1 or -1) of the mean of the loadings in that column to produce only positive factors. Next, all permutations of the columns of the solution were compared to the population loading matrix, and the permutation with the lowest mean squared error (MSE) was kept. These solutions were then compared to identify unique solutions, which we examined afterward to guarantee that no duplicate solutions were erroneously identified.

6 Jennrich and Bentler (2011) proved that JB rotation criterion are always minimized when the structure is bifactor, but only if there are no cross-loadings.

7 Here, we use dimensionality to describe the number of factors with meaningful variance; if the constraints do not hold, the dimensionality of the reduced correlation matrix is, strictly speaking, p + 1, but if the constraints almost hold, the dimensionality is still “essentially” p.

8 In popular psychometric packages, such as the R psych library (Revelle, 2015), the Schmid-Leiman routine only allows two possible rotations—oblimin and Promax. Researchers wishing to explore alternative rotations from the CF family should use the GPArotation library and then transform the correlated factors solution into an SL via multiplication of Λ by L, as done here. This, of course, will not automatically yield indices such as omega, omega hierarchical, factor determinacy, and so on, as the psych package does by default.