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Exploratory Bi-factor Analysis: The Oblique Case

Published online by Cambridge University Press:  01 January 2025

Robert I. Jennrich  and
Peter M. Bentler
Robert I. Jennrich*
Affiliation:
University of California, Los Angeles
Peter M. Bentler
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Robert I. Jennrich, University of California, Los Angeles, 3400 Purdue Ave., Los Angeles, CA, USA. E-mail: rij@stat.ucla.edu
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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.

Keywords

bi-factor rotationgeneral factorgroup factorgradient projection algorithmsoblique rotationorthogonal rotation
Type
Original Paper
Information
Psychometrika , Volume 77 , Issue 3 , July 2012 , pp. 442 - 454
Copyright
Copyright © 2012 The Psychometric Society

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Footnotes

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.

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