# Exploratory Bi-factor Analysis: The Oblique Case

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## 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.

## Key words

bi-factor rotation general factor group factor gradient projection algorithms oblique rotation orthogonal rotation## References

- Browne, M.W. (2001). An overview of analytic rotation in exploratory factor analysis.
*Multivariate Behavioral Research*,*36*, 111–150. CrossRefGoogle Scholar - Cai, L., Yang, J.S., & Hansen, M. (2011). Generalized full-information item bifactor analysis.
*Psychological Methods*,*16*, 221–248. PubMedCrossRefGoogle Scholar - 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. CrossRefGoogle Scholar - 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. PubMedCrossRefGoogle Scholar - Haviland, M.G., Warren, W.L., & Riggs, M.L. (2000). An observer scale to measure alexithymia.
*Psychosomatics*,*41*, 385–392. PubMedCrossRefGoogle Scholar - Holzinger, K.J., & Swineford, S. (1937). The bi-factor method.
*Psychometrika*,*47*, 41–54. CrossRefGoogle Scholar - Jennrich, R.I. (1973). Standard errors for obliquely rotated factor loadings.
*Psychometrika*,*38*, 593–604. CrossRefGoogle Scholar - Jennrich, R.I. (2001). A simple general procedure for orthogonal rotation.
*Psychometrika*,*66*, 289–306. CrossRefGoogle Scholar - Jennrich, R.I. (2002). A simple general method for oblique rotation.
*Psychometrika*,*67*, 7–20. CrossRefGoogle Scholar - Jennrich, R.I., & Bentler, P.M. (2011). Exploratory bi-factor analysis.
*Psychometrika*,*76*, 537–549. PubMedCrossRefGoogle Scholar - 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. PubMedCrossRefGoogle Scholar - Schmid, J., & Leiman, J.M. (1957). The development of hierarchical factor solutions.
*Psychometrika*,*22*, 53–61. CrossRefGoogle Scholar - Yates, A. (1987).
*Multivariate exploratory data analysis: a perspective on exploratory factor analysis*. Albany: State University of New York Press. Google Scholar