IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v79y2014i4p701-732.html
   My bibliography  Save this article

Moderation Analysis Using a Two-Level Regression Model

Author

Listed:
  • Ke-Hai Yuan
  • Ying Cheng
  • Scott Maxwell

Abstract

Moderation analysis is widely used in social and behavioral research. The most commonly used model for moderation analysis is moderated multiple regression (MMR) in which the explanatory variables of the regression model include product terms, and the model is typically estimated by least squares (LS). This paper argues for a two-level regression model in which the regression coefficients of a criterion variable on predictors are further regressed on moderator variables. An algorithm for estimating the parameters of the two-level model by normal-distribution-based maximum likelihood (NML) is developed. Formulas for the standard errors (SEs) of the parameter estimates are provided and studied. Results indicate that, when heteroscedasticity exists, NML with the two-level model gives more efficient and more accurate parameter estimates than the LS analysis of the MMR model. When error variances are homoscedastic, NML with the two-level model leads to essentially the same results as LS with the MMR model. Most importantly, the two-level regression model permits estimating the percentage of variance of each regression coefficient that is due to moderator variables. When applied to data from General Social Surveys 1991, NML with the two-level model identified a significant moderation effect of race on the regression of job prestige on years of education while LS with the MMR model did not. An R package is also developed and documented to facilitate the application of the two-level model. Copyright The Psychometric Society 2014

Suggested Citation

  • Ke-Hai Yuan & Ying Cheng & Scott Maxwell, 2014. "Moderation Analysis Using a Two-Level Regression Model," Psychometrika, Springer;The Psychometric Society, vol. 79(4), pages 701-732, October.
  • Handle: RePEc:spr:psycho:v:79:y:2014:i:4:p:701-732
    DOI: 10.1007/s11336-013-9357-x
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s11336-013-9357-x
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1007/s11336-013-9357-x?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. Cribari-Neto, Francisco, 2004. "Asymptotic inference under heteroskedasticity of unknown form," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 215-233, March.
    2. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-838, May.
    3. Davidson, Russell & MacKinnon, James G., 1993. "Estimation and Inference in Econometrics," OUP Catalogue, Oxford University Press, number 9780195060119.
    4. Jeffrey M Wooldridge, 2010. "Econometric Analysis of Cross Section and Panel Data," MIT Press Books, The MIT Press, edition 2, volume 1, number 0262232588, April.
    5. Singh, Balvir, et al, 1976. "On the Estimation of Structural Change: A Generalization of the Random Coefficients Regression Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 17(2), pages 340-361, June.
    6. MacKinnon, James G. & White, Halbert, 1985. "Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties," Journal of Econometrics, Elsevier, vol. 29(3), pages 305-325, September.
    7. Yuan, Ke-Hai & Bentler, Peter M., 1997. "Improving parameter tests in covariance structure analysis," Computational Statistics & Data Analysis, Elsevier, vol. 26(2), pages 177-198, December.
    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. Larissa Batrancea & Anca Nichita & Ioan Batrancea & Lucian Gaban, 2018. "The Strenght of the Relationship Between Shadow Economy and Corruption: Evidence from a Worldwide Country-Sample," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 138(3), pages 1119-1143, August.

    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. Sin, C.Y. (Chor-yiu) & Lee, Cheng-Few, 2021. "Using heteroscedasticity-non-consistent or heteroscedasticity-consistent variances in linear regression," Econometrics and Statistics, Elsevier, vol. 18(C), pages 117-142.
    2. Pötscher, Benedikt M. & Preinerstorfer, David, 2021. "Valid Heteroskedasticity Robust Testing," MPRA Paper 107420, University Library of Munich, Germany.
    3. Francisco Cribari-Neto & Maria da Gloria Lima, 2010. "Approximate inference in heteroskedastic regressions: A numerical evaluation," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(4), pages 591-615.
    4. Romano, Joseph P. & Wolf, Michael, 2017. "Resurrecting weighted least squares," Journal of Econometrics, Elsevier, vol. 197(1), pages 1-19.
    5. Hausman, Jerry & Palmer, Christopher, 2012. "Heteroskedasticity-robust inference in finite samples," Economics Letters, Elsevier, vol. 116(2), pages 232-235.
    6. Uchôa, Carlos F.A. & Cribari-Neto, Francisco & Menezes, Tatiane A., 2014. "Testing inference in heteroskedastic fixed effects models," European Journal of Operational Research, Elsevier, vol. 235(3), pages 660-670.
    7. Francisco Cribari-Neto & Wilton Silva, 2011. "A new heteroskedasticity-consistent covariance matrix estimator for the linear regression model," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 95(2), pages 129-146, June.
    8. Hartigan, Luke, 2018. "Alternative HAC covariance matrix estimators with improved finite sample properties," Computational Statistics & Data Analysis, Elsevier, vol. 119(C), pages 55-73.
    9. James G. MacKinnon, 2012. "Thirty Years Of Heteroskedasticity-robust Inference," Working Paper 1268, Economics Department, Queen's University.
    10. José Curto & José Pinto & Ana Morais & Isabel Lourenço, 2011. "The heteroskedasticity-consistent covariance estimator in accounting," Review of Quantitative Finance and Accounting, Springer, vol. 37(4), pages 427-449, November.
    11. Francisco Cribari-Neto & Maria Lima, 2010. "Sequences of bias-adjusted covariance matrix estimators under heteroskedasticity of unknown form," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(6), pages 1053-1082, December.
    12. Davidson, Russell & Flachaire, Emmanuel, 2008. "The wild bootstrap, tamed at last," Journal of Econometrics, Elsevier, vol. 146(1), pages 162-169, September.
    13. Psaradakis, Zacharias & Sola, Martin, 1996. "On the power of tests for superexogeneity and structural invariance," Journal of Econometrics, Elsevier, vol. 72(1-2), pages 151-175.
    14. Katarzyna Jabłońska, 2018. "Dealing With Heteroskedasticity Within The Modeling Of The Quality Of Life Of Older People," Statistics in Transition New Series, Polish Statistical Association, vol. 19(3), pages 423-452, September.
    15. Richard H. Spady & Sami Stouli, 2018. "Simultaneous Mean-Variance Regression," Bristol Economics Discussion Papers 18/697, School of Economics, University of Bristol, UK.
    16. Maurice J. G. Bun & Teresa D. Harrison, 2019. "OLS and IV estimation of regression models including endogenous interaction terms," Econometric Reviews, Taylor & Francis Journals, vol. 38(7), pages 814-827, August.
    17. Steven Saeger, 1997. "Globalization and deindustrialization: Myth and reality in the OECD," Review of World Economics (Weltwirtschaftliches Archiv), Springer;Institut für Weltwirtschaft (Kiel Institute for the World Economy), vol. 133(4), pages 579-608, December.
    18. Cheng, Tsung-Chi, 2012. "On simultaneously identifying outliers and heteroscedasticity without specific form," Computational Statistics & Data Analysis, Elsevier, vol. 56(7), pages 2258-2272.
    19. Russell Davidson & Victoria Zinde‐Walsh, 2017. "Advances in specification testing," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 50(5), pages 1595-1631, December.
    20. Emmanuel Flachaire, 2002. "Bootstrapping heteroskedasticity consistent covariance matrix estimator," Computational Statistics, Springer, vol. 17(4), pages 501-506, December.

    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:79:y:2014:i:4:p:701-732. 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.