Abstract
Bi-factor analysis is a form of confirmatory factor analysis originally introduced by Holzinger and Swineford (Psychometrika 47:41–54, 1937). The bi-factor model has a general factor, a number of group factors, and an explicit bi-factor structure. Jennrich and Bentler (Psychometrika 76:537–549, 2011) introduced an exploratory form of bi-factor analysis that does not require one to provide an explicit bi-factor structure a priori. They use exploratory factor analysis and a bifactor rotation criterion designed to produce a rotated loading matrix that has an approximate bi-factor structure. Among other things this can be used as an aid in finding an explicit bi-factor structure for use in a confirmatory bi-factor analysis. They considered only orthogonal rotation. The purpose of this paper is to consider oblique rotation and to compare it to orthogonal rotation. Because there are many more oblique rotations of an initial loading matrix than orthogonal rotations, one expects the oblique results to approximate a bi-factor structure better than orthogonal rotations and this is indeed the case. A surprising result arises when oblique bi-factor rotation methods are applied to ideal data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We would like to thank Dr. David Huepe for facilitating the use of these data.
We would like to thank Mark Haviland for making these data available for analysis.
References
Browne, M.W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111–150.
Cai, L., Yang, J.S., & Hansen, M. (2011). Generalized full-information item bifactor analysis. Psychological Methods, 16, 221–248.
Chen, F.F., West, S.G., & Sousa, K.H. (2006). A comparison of bifactor and second-order models of the quality of life. Multivariate Behavioral Research, 41, 189–225.
CONACE (2007). Descargar Septimo estudio nacional de drogas en poblacion escolar de Chile 2007. http://www.conace.cl/portal/index.php?option=com_content&view=article&id=280&Itemid=370.
Golay, P., & Lecerf, T. (2011). On higher order structure and confirmatory factor analysis of the French Wechsler Adult Intelligence Scale (WAIS-III). Psychological Assessment, 23, 143–152.
Haviland, M.G., Warren, W.L., & Riggs, M.L. (2000). An observer scale to measure alexithymia. Psychosomatics, 41, 385–392.
Holzinger, K.J., & Swineford, S. (1937). The bi-factor method. Psychometrika, 47, 41–54.
Jennrich, R.I. (1973). Standard errors for obliquely rotated factor loadings. Psychometrika, 38, 593–604.
Jennrich, R.I. (2001). A simple general procedure for orthogonal rotation. Psychometrika, 66, 289–306.
Jennrich, R.I. (2002). A simple general method for oblique rotation. Psychometrika, 67, 7–20.
Jennrich, R.I., & Bentler, P.M. (2011). Exploratory bi-factor analysis. Psychometrika, 76, 537–549.
Reise, S.P., Moore, T.M., & Haviland, M.G. (2010). Bifactor models and rotations: exploring the extent to which multidimensional data yield univocal scores. Journal of Personality Assessment, 92, 544–559.
Schmid, J., & Leiman, J.M. (1957). The development of hierarchical factor solutions. Psychometrika, 22, 53–61.
Yates, A. (1987). Multivariate exploratory data analysis: a perspective on exploratory factor analysis. Albany: State University of New York Press.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by grants 5K05DA000017-33 and 5P01DA001070-37 from the National Institute on Drug Abuse to P.M. Bentler and grant 4R44CA137841-03 from the National Cancer Institute to P. Mair. Bentler acknowledges a financial interest in EQS and its distributor, Multivariate Software.
Appendix
Appendix
Computational formulas for the value and gradient of the bi-quartimin criterion are given in Jennrich and Bentler (2011).
Here we show how to use the Matlab code in the URL reference above to perform oblique bi-factor rotation of an initial loading matrix A. The procedure is similar when using the R, S, SAS PROC IML, and SPSS matrix code, also given in the URL.
The first step is to write a program to compute the value and gradient of the bi-factor criterion desired. For the bi-quartimin criterion this is
Because orthogonal and oblique bi-quartimin rotation use the same rotation criterion, this code is identical to that given in Jennrich and Bentler (2011).
The next step is to download the Matlab code for the general purpose oblique rotation program GPFoblq.
The final step is to compute an oblique bi-quartimin rotation Λ of A using
where T is a nonsingular matrix with columns of unit length used to start the rotation algorithm. It must have the same number of columns as A. A common choice for T is an identity matrix. An oblique random start can be generated using
where k is the number of columns of A and the function \(\mathtt{randn(k,k)}\) generates a k×k matrix of independent standard normal values. To deal with local minima problems the authors have found that it is a good idea to use the best of several random starts. On the problems discussed, 10 seemed to be sufficient. As usual, however, one should review the criterion values produced to see if the number of random starts seems sufficient. Bi-quartimin rotation seems to require more random starts than quartimin rotation, roughly the number required by geomin rotation. For the two real data examples the execution time on a Macintosh G5 was about 3 seconds for each random start.
Jennrich and Bentler (2011) show how to perform orthogonal bi-factor rotation. The only difference is downloading GPForth rather than GPFoblq, changing the rotation command to
and using
to generate random orthogonal starts.
Rights and permissions
About this article
Cite this article
Jennrich, R.I., Bentler, P.M. Exploratory Bi-factor Analysis: The Oblique Case. Psychometrika 77, 442–454 (2012). https://doi.org/10.1007/s11336-012-9269-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11336-012-9269-1