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Dealing with Reflection Invariance in Bayesian Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Elena A. Erosheva  and
S. McKay Curtis
Elena A. Erosheva*
Affiliation:
University of Washington
S. McKay Curtis
Affiliation:
University of Washington
*
Correspondence should be made to Elena A. Erosheva, Department of Statistics, University of Washington, Box 354320, Seattle, WA 98195, USA. Email: erosheva@u.washington.edu
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Abstract

This paper considers the reflection unidentifiability problem in confirmatory factor analysis (CFA) and the associated implications for Bayesian estimation. We note a direct analogy between the multimodality in CFA models that is due to all possible column sign changes in the matrix of loadings and the multimodality in finite mixture models that is due to all possible relabelings of the mixture components. Drawing on this analogy, we derive and present a simple approach for dealing with reflection in variance in Bayesian factor analysis. We recommend fitting Bayesian factor analysis models without rotational constraints on the loadings—allowing Markov chain Monte Carlo algorithms to explore the full posterior distribution—and then using a relabeling algorithm to pick a factor solution that corresponds to one mode. We demonstrate our approach on the case of a bifactor model; however, the relabeling algorithm is straightforward to generalize for handling multimodalities due to sign invariance in the likelihood in other factor analysis models.

Keywords

identifiability constraintslabel-switchingMarkov chain Monte Carlorelabelingrotationrotational invariance
Type
Original Paper
Information
Psychometrika , Volume 82 , Issue 2 , June 2017 , pp. 295 - 307
Copyright
Copyright © 2017 The Psychometric Society

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