IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v82y2017i2d10.1007_s11336-017-9564-y.html
   My bibliography  Save this article

Dealing with Reflection Invariance in Bayesian Factor Analysis

Author

Listed:
  • Elena A. Erosheva

    (University of Washington)

  • S. McKay Curtis

    (University of Washington)

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.

Suggested Citation

  • Elena A. Erosheva & S. McKay Curtis, 2017. "Dealing with Reflection Invariance in Bayesian Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 82(2), pages 295-307, June.
    • Handle: RePEc:spr:psycho:v:82:y:2017:i:2:d:10.1007_s11336-017-9564-y
    DOI: 10.1007/s11336-017-9564-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11336-017-9564-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11336-017-9564-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Terrance Savitsky & Daniel McCaffrey, 2014. "Bayesian Hierarchical Multivariate Formulation with Factor Analysis for Nested Ordinal Data," Psychometrika, Springer;The Psychometric Society, vol. 79(2), pages 275-302, April.
    2. Robert Gibbons & Donald Hedeker, 1992. "Full-information item bi-factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 57(3), pages 423-436, September.
    3. James Martin & Roderick McDonald, 1975. "Bayesian estimation in unrestricted factor analysis: A treatment for heywood cases," Psychometrika, Springer;The Psychometric Society, vol. 40(4), pages 505-517, December.
    4. Geweke, John & Zhou, Guofu, 1996. "Measuring the Pricing Error of the Arbitrage Pricing Theory," The Review of Financial Studies, Society for Financial Studies, vol. 9(2), pages 557-587.
    5. Bafumi, Joseph & Gelman, Andrew & Park, David K. & Kaplan, Noah, 2005. "Practical Issues in Implementing and Understanding Bayesian Ideal Point Estimation," Political Analysis, Cambridge University Press, vol. 13(2), pages 171-187, April.
    6. Karl Holzinger & Frances Swineford, 1937. "The Bi-factor method," Psychometrika, Springer;The Psychometric Society, vol. 2(1), pages 41-54, March.
    7. Richard Scheines & Herbert Hoijtink & Anne Boomsma, 1999. "Bayesian estimation and testing of structural equation models," Psychometrika, Springer;The Psychometric Society, vol. 64(1), pages 37-52, March.
    8. Sik-Yum Lee, 1981. "A bayesian approach to confirmatory factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 46(2), pages 153-160, June.
    9. Matthew Stephens, 2000. "Dealing with label switching in mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(4), pages 795-809.
    10. Jackman, Simon, 2001. "Multidimensional Analysis of Roll Call Data via Bayesian Simulation: Identification, Estimation, Inference, and Model Checking," Political Analysis, Cambridge University Press, vol. 9(3), pages 227-241, January.
    11. Douglas Clarkson, 1979. "Estimating the standard errors of rotated factor loadings by jackknifing," Psychometrika, Springer;The Psychometric Society, vol. 44(3), pages 297-314, September.
    12. Rosseel, Yves, 2012. "lavaan: An R Package for Structural Equation Modeling," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i02).
    13. Quinn, Kevin M., 2004. "Bayesian Factor Analysis for Mixed Ordinal and Continuous Responses," Political Analysis, Cambridge University Press, vol. 12(4), pages 338-353.
    14. Robert Jennrich, 1978. "Rotational equivalence of factor loading matrices with specified values," Psychometrika, Springer;The Psychometric Society, vol. 43(3), pages 421-426, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Adrian Quintero & Emmanuel Lesaffre & Geert Verbeke, 2024. "Bayesian Exploratory Factor Analysis via Gibbs Sampling," Journal of Educational and Behavioral Statistics, , vol. 49(1), pages 121-142, February.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Aßmann, Christian & Boysen-Hogrefe, Jens & Pape, Markus, 2014. "Bayesian analysis of dynamic factor models: An ex-post approach towards the rotation problem," Kiel Working Papers 1902, Kiel Institute for the World Economy (IfW Kiel).
    2. David Kaplan & Chansoon Lee, 2018. "Optimizing Prediction Using Bayesian Model Averaging: Examples Using Large-Scale Educational Assessments," Evaluation Review, , vol. 42(4), pages 423-457, August.
    3. Terrance Savitsky & Daniel McCaffrey, 2014. "Bayesian Hierarchical Multivariate Formulation with Factor Analysis for Nested Ordinal Data," Psychometrika, Springer;The Psychometric Society, vol. 79(2), pages 275-302, April.
    4. Aßmann, Christian & Boysen-Hogrefe, Jens & Pape, Markus, 2016. "Bayesian analysis of static and dynamic factor models: An ex-post approach towards the rotation problem," Journal of Econometrics, Elsevier, vol. 192(1), pages 190-206.
    5. Pellegrina, Lucia Dalla & Garoupa, Nuno & Gómez-Pomar, Fernando, 2017. "Estimating judicial ideal points in the Spanish Supreme Court: The case of administrative review," International Review of Law and Economics, Elsevier, vol. 52(C), pages 16-28.
    6. Nana Kim & Daniel M. Bolt & James Wollack, 2022. "Noncompensatory MIRT For Passage-Based Tests," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 992-1009, September.
    7. Li Cai, 2015. "Lord–Wingersky Algorithm Version 2.0 for Hierarchical Item Factor Models with Applications in Test Scoring, Scale Alignment, and Model Fit Testing," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 535-559, June.
    8. Piotr Tarka, 2018. "An overview of structural equation modeling: its beginnings, historical development, usefulness and controversies in the social sciences," Quality & Quantity: International Journal of Methodology, Springer, vol. 52(1), pages 313-354, January.
    9. Aßmann, Christian & Boysen-Hogrefe, Jens & Pape, Markus, 2012. "The directional identification problem in Bayesian factor analysis: An ex-post approach," Kiel Working Papers 1799, Kiel Institute for the World Economy (IfW Kiel).
    10. Diana Mindrila, 2023. "Bayesian Latent Class Analysis: Sample Size, Model Size, and Classification Precision," Mathematics, MDPI, vol. 11(12), pages 1-18, June.
    11. Bjørn Høyland, 2010. "Procedural and party effects in European Parliament roll-call votes," European Union Politics, , vol. 11(4), pages 597-613, December.
    12. Rockstuhl, Thomas & Van Dyne, Linn, 2018. "A bi-factor theory of the four-factor model of cultural intelligence: Meta-analysis and theoretical extensions," Organizational Behavior and Human Decision Processes, Elsevier, vol. 148(C), pages 124-144.
    13. Victoria T. Tanaka & George Engelhard & Matthew P. Rabbitt, 2020. "Using a Bifactor Model to Measure Food Insecurity in Households with Children," Journal of Family and Economic Issues, Springer, vol. 41(3), pages 492-504, September.
    14. Jiang, Xiaomo & Mahadevan, Sankaran, 2009. "Bayesian structural equation modeling method for hierarchical model validation," Reliability Engineering and System Safety, Elsevier, vol. 94(4), pages 796-809.
    15. Chun Wang & Steven W. Nydick, 2020. "On Longitudinal Item Response Theory Models: A Didactic," Journal of Educational and Behavioral Statistics, , vol. 45(3), pages 339-368, June.
    16. Michael Edwards, 2010. "A Markov Chain Monte Carlo Approach to Confirmatory Item Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 75(3), pages 474-497, September.
    17. Kan, Kees-Jan & van der Maas, Han L.J. & Levine, Stephen Z., 2019. "Extending psychometric network analysis: Empirical evidence against g in favor of mutualism?," Intelligence, Elsevier, vol. 73(C), pages 52-62.
    18. Hix, Simon & Hoyland, Bjorn & Vivyan, Nick, 2007. "From doves to hawks: a spatial analysis of voting in the Monetary Policy Committee of the Bank of England, 1997-2007," LSE Research Online Documents on Economics 25199, London School of Economics and Political Science, LSE Library.
    19. Bertomeu Juan González & Pellegrina Lucia Dalla & Garoupa Nuno, 2017. "Estimating Judicial Ideal Points in Latin America: The Case of Argentina," Review of Law & Economics, De Gruyter, vol. 13(1), pages 1-35, March.
    20. Richard F. Potthoff, 2018. "Estimating Ideal Points from Roll-Call Data: Explore Principal Components Analysis, Especially for More Than One Dimension?," Social Sciences, MDPI, vol. 7(1), pages 1-27, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:psycho:v:82:y:2017:i:2:d:10.1007_s11336-017-9564-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.