IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v85y2020i4d10.1007_s11336-020-09737-y.html
   My bibliography  Save this article

Partial Identification of Latent Correlations with Binary Data

Author

Listed:
  • Steffen Grønneberg

    (BI Norwegian Business School)

  • Jonas Moss

    (University of Oslo)

  • Njål Foldnes

    (BI Norwegian Business School)

Abstract

The tetrachoric correlation is a popular measure of association for binary data and estimates the correlation of an underlying normal latent vector. However, when the underlying vector is not normal, the tetrachoric correlation will be different from the underlying correlation. Since assuming underlying normality is often done on pragmatic and not substantial grounds, the estimated tetrachoric correlation may therefore be quite different from the true underlying correlation that is modeled in structural equation modeling. This motivates studying the range of latent correlations that are compatible with given binary data, when the distribution of the latent vector is partly or completely unknown. We show that nothing can be said about the latent correlations unless we know more than what can be derived from the data. We identify an interval constituting all latent correlations compatible with observed data when the marginals of the latent variables are known. Also, we quantify how partial knowledge of the dependence structure of the latent variables affect the range of compatible latent correlations. Implications for tests of underlying normality are briefly discussed.

Suggested Citation

  • Steffen Grønneberg & Jonas Moss & Njål Foldnes, 2020. "Partial Identification of Latent Correlations with Binary Data," Psychometrika, Springer;The Psychometric Society, vol. 85(4), pages 1028-1051, December.
    • Handle: RePEc:spr:psycho:v:85:y:2020:i:4:d:10.1007_s11336-020-09737-y
    DOI: 10.1007/s11336-020-09737-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11336-020-09737-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/s11336-020-09737-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. Albert Maydeu-Olivares, 2006. "Limited information estimation and testing of discretized multivariate normal structural models," Psychometrika, Springer;The Psychometric Society, vol. 71(1), pages 57-77, March.
    2. Yan, Jun, 2007. "Enjoy the Joy of Copulas: With a Package copula," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 21(i04).
    3. Tamer, Elie, 2010. "Partial Identification in Econometrics," Scholarly Articles 34728615, Harvard University Department of Economics.
    4. Ulf Olsson, 1979. "Maximum likelihood estimation of the polychoric correlation coefficient," Psychometrika, Springer;The Psychometric Society, vol. 44(4), pages 443-460, December.
    5. Peter Tankov, 2010. "Improved Frechet bounds and model-free pricing of multi-asset options," Papers 1004.4153, arXiv.org, revised Mar 2011.
    6. Rosseel, Yves, 2012. "lavaan: An R Package for Structural Equation Modeling," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i02).
    7. Steffen Grønneberg & Njål Foldnes, 2017. "Covariance Model Simulation Using Regular Vines," Psychometrika, Springer;The Psychometric Society, vol. 82(4), pages 1035-1051, December.
    8. Stanislav Kolenikov & Gustavo Angeles, 2009. "Socioeconomic Status Measurement With Discrete Proxy Variables: Is Principal Component Analysis A Reliable Answer?," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 55(1), pages 128-165, March.
    9. Bengt Muthén & Charles Hofacker, 1988. "Testing the assumptions underlying tetrachoric correlations," Psychometrika, Springer;The Psychometric Society, vol. 53(4), pages 563-577, December.
    10. Claeskens,Gerda & Hjort,Nils Lid, 2008. "Model Selection and Model Averaging," Cambridge Books, Cambridge University Press, number 9780521852258.
    11. Carlos Almeida & Michel Mouchart, 2014. "Testing normality of latent variables in the polychoric correlation," Statistica, Department of Statistics, University of Bologna, vol. 74(1), pages 3-22.
    12. Almeida Rodriguez, Carlos & Mouchart, Michel, 2014. "Testing Normality of latent variables in the polychoric correlation," LIDAM Reprints ISBA 2014046, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. Yoshio Takane & Jan Leeuw, 1987. "On the relationship between item response theory and factor analysis of discretized variables," Psychometrika, Springer;The Psychometric Society, vol. 52(3), pages 393-408, September.
    14. Elie Tamer, 2010. "Partial Identification in Econometrics," Annual Review of Economics, Annual Reviews, vol. 2(1), pages 167-195, September.
    15. Njål Foldnes & Steffen Grønneberg, 2019. "On Identification and Non-normal Simulation in Ordinal Covariance and Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 84(4), pages 1000-1017, December.
    16. Anders Christoffersson, 1975. "Factor analysis of dichotomized variables," Psychometrika, Springer;The Psychometric Society, vol. 40(1), pages 5-32, March.
    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. Jonas Moss & Steffen Grønneberg, 2023. "Partial Identification of Latent Correlations with Ordinal Data," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 241-252, March.

    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. Njål Foldnes & Steffen Grønneberg, 2019. "On Identification and Non-normal Simulation in Ordinal Covariance and Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 84(4), pages 1000-1017, December.
    2. Jonas Moss & Steffen Grønneberg, 2023. "Partial Identification of Latent Correlations with Ordinal Data," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 241-252, March.
    3. Yang Yixin & Lü Xin & Ma Jian & Qiao Han, 2014. "A Robust Factor Analysis Model for Dichotomous Data," Journal of Systems Science and Information, De Gruyter, vol. 2(5), pages 437-450, October.
    4. Sayed H. Kadhem & Aristidis K. Nikoloulopoulos, 2023. "Factor Tree Copula Models for Item Response Data," Psychometrika, Springer;The Psychometric Society, vol. 88(3), pages 776-802, September.
    5. Dylan Molenaar, 2015. "Heteroscedastic Latent Trait Models for Dichotomous Data," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 625-644, September.
    6. Steffen Grønneberg & Njål Foldnes, 2019. "A Problem with Discretizing Vale–Maurelli in Simulation Studies," Psychometrika, Springer;The Psychometric Society, vol. 84(2), pages 554-561, June.
    7. Shaobo Jin & Fan Yang-Wallentin & Kenneth A. Bollen, 2021. "A unified model-implied instrumental variable approach for structural equation modeling with mixed variables," Psychometrika, Springer;The Psychometric Society, vol. 86(2), pages 564-594, 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. Maydeu-Olivares, Albert, 2002. "Limited information estimation and testing of Thurstonian models for preference data," Mathematical Social Sciences, Elsevier, vol. 43(3), pages 467-483, July.
    10. Sung Jae Jun & Sokbae Lee, 2020. "Causal Inference under Outcome-Based Sampling with Monotonicity Assumptions," Papers 2004.08318, arXiv.org, revised Oct 2023.
    11. Vikesh Amin & Jere R. Behrman & Jason M. Fletcher & Carlos A. Flores & Alfonso Flores-Lagunes & Hans-Peter Kohler, 2022. "Does Schooling Improve Cognitive Abilities at Older Ages: Causal Evidence from Nonparametric Bounds," PIER Working Paper Archive 22-016, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    12. Millimet, Daniel L. & Roy, Jayjit, 2015. "Multilateral environmental agreements and the WTO," Economics Letters, Elsevier, vol. 134(C), pages 20-23.
    13. Breunig, Christoph & Mammen, Enno & Simoni, Anna, 2018. "Nonparametric estimation in case of endogenous selection," Journal of Econometrics, Elsevier, vol. 202(2), pages 268-285.
    14. Christian Bontemps & Raquel Menezes Bezerra Sampaio, 2020. "Entry games for the airline industry," Post-Print hal-02137358, HAL.
    15. Donald S. Poskitt & Xueyan Zhao, 2023. "Bootstrap Hausdorff Confidence Regions for Average Treatment Effect Identified Sets," Monash Econometrics and Business Statistics Working Papers 9/23, Monash University, Department of Econometrics and Business Statistics.
    16. Albert Maydeu-Olivares, 2006. "Limited information estimation and testing of discretized multivariate normal structural models," Psychometrika, Springer;The Psychometric Society, vol. 71(1), pages 57-77, March.
    17. Laura Forastiere & Patrizia Lattarulo & Marco Mariani & Fabrizia Mealli & Laura Razzolini, 2021. "Exploring Encouragement, Treatment, and Spillover Effects Using Principal Stratification, With Application to a Field Experiment on Teens’ Museum Attendance," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(1), pages 244-258, January.
    18. Undral Byambadalai, 2022. "Identification and Inference for Welfare Gains without Unconfoundedness," Papers 2207.04314, arXiv.org.
    19. Ali Fakih & Paul Makdissi & Walid Marrouch & Rami V. Tabri & Myra Yazbeck, 2020. "Confidence in public institutions and the run up to the October 2019 uprising in Lebanon," Discussion Papers Series 629, School of Economics, University of Queensland, Australia.
    20. Laurens Cherchye & Bram De Rock & Arthur Lewbel & Frederic Vermeulen, 2015. "Sharing Rule Identification for General Collective Consumption Models," Econometrica, Econometric Society, vol. 83(5), pages 2001-2041, September.

    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:85:y:2020:i:4:d:10.1007_s11336-020-09737-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.