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Moderation Analysis Using a Two-Level Regression Model

Published online by Cambridge University Press:  01 January 2025

Ke-Hai Yuan ,
Ying Cheng  and
Scott Maxwell
Ke-Hai Yuan*
Affiliation:
University of Notre Dame
Ying Cheng
Affiliation:
University of Notre Dame
Scott Maxwell
Affiliation:
University of Notre Dame
*
Requests for reprints should be sent to Ke-Hai Yuan, Department of Psychology, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail: kyuan@nd.edu
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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.

Keywords

biasconsistencyefficiencyiteratively reweighted least squaressandwich-type covariance matrixR-square
Type
Original Paper
Information
Psychometrika , Volume 79 , Issue 4 , October 2014 , pp. 701 - 732
Copyright
Copyright © 2013 The Psychometric Society

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References

Aguinis, H. (2004). Regression analysis for categorical moderators. New York: Guilford.Google Scholar
Aguinis, H., Petersen, S.A., & Pierce, C.A. (1999). Appraisal of the homogeneity of error variance assumption and alternatives for multiple regression for estimating moderated effects of categorical variables. Organizational Research Methods, 2, 315–339.CrossRefGoogle Scholar
Aiken, L.S., & West, S.G. (1991). Multiple regression: testing and interpreting interactions. Thousand Oaks: Sage.Google Scholar
Baron, R.M., & Kenny, D.A. (1986). The moderator-mediator variable distinction in social psychological research: concept, strategic and statistical considerations. Journal of Personality and Social Psychology, 51, 1173–1182.CrossRefGoogle Scholar
Bast, J., & Reitsma, P. (1998). Analyzing the development of individual differences in terms of Matthew effects in reading: results from a Dutch longitudinal study. Developmental Psychology, 34, 1373–1399.CrossRefGoogle ScholarPubMed
Carroll, R.J., & Ruppert, D. (1988). Transformation and weighting in regression. New York: Chapman & Hall/CRC.CrossRefGoogle Scholar
Casella, G., & Berger, R.L. (2002). Statistical inference (2nd ed.). Pacific Grove: Duxbury Press.Google Scholar
Chaplin, W.F. (2007). Moderator and mediator models in personality research: a basic introduction. In Robins, R.W., Fraley, R.C., & Krueger, R.F. (Eds.), Handbook of research methods in personality psychology (pp. 602–632). New York: Guilford.Google Scholar
Cohen, J. (1978). Partialed products are interactions; partialed powers are curve components. Psychological Bulletin, 85, 858–866.CrossRefGoogle Scholar
Cribari-Neto, F. (2004). Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics & Data Analysis, 45, 215–233.CrossRefGoogle Scholar
Darlington, R.B. (1990). Regression and linear models. New York: McGraw-Hill.Google Scholar
Davidson, R., & MacKinnon, J.G. (1993). Estimation and inference in econometrics. Oxford: Oxford University Press.Google Scholar
Davison, M.L., Kwak, N., Seo, Y.S., & Choi, J. (2002). Using hierarchical linear models to examine moderator effects: person-by-organization interactions. Organizational Research Methods, 5, 231–254.CrossRefGoogle Scholar
Dent, W.T., & Hildreth, C. (1977). Maximum likelihood estimation in random coefficient models. Journal of the American Statistical Association, 72, 69–72.CrossRefGoogle Scholar
DeShon, R.P., & Alexander, R.A. (1996). Alternative procedures for testing regression slope homogeneity when group error variances are unequal. Psychological Methods, 1, 261–277.CrossRefGoogle Scholar
Dretzke, B.J., Levin, J.R., & Serlin, R.C. (1982). Testing for regression homogeneity under variance heterogeneity. Psychological Bulletin, 91, 376–383.CrossRefGoogle Scholar
Efron, B., & Tibshirani, R.J. (1993). An introduction to the bootstrap. New York: Chapman & Hall.CrossRefGoogle Scholar
Fisicaro, S.A., & Tisak, J. (1994). A theoretical note on the stochastics of moderated multiple regression. Educational and Psychological Measurement, 54, 32–41.CrossRefGoogle Scholar
Froehlich, B.R. (1973). Some estimators for a random coefficient regression model. Journal of the American Statistical Association, 68, 329–335.Google Scholar
Hayes, A.F., & Cai, L. (2007). Using heteroscedasticity-consistent standard error estimators in OLS regression: an introduction and software implementation. Behavior Research Methods, 39, 709–722.CrossRefGoogle Scholar
Hinkley, D.V. (1977). Jackknifing in unbalanced situations. Technometrics, 19, 285–292.CrossRefGoogle Scholar
Hildreth, C., & Houck, J. (1968). Some estimators for a linear model with random coefficients. Journal of the American Statistical Association, 63, 584–595.CrossRefGoogle Scholar
Holmbeck, G.N. (1997). Toward terminological, conceptual and statistical clarity in the study of mediators and moderators: examples from the child-clinical and pediatric psychology literatures. Journal of Consulting and Clinical Psychology, 65, 599–610.CrossRefGoogle Scholar
Kenny, D., & Judd, C.M. (1984). Estimating the nonlinear and interactive effects of latent variables. Psychological Bulletin, 96, 201–210.CrossRefGoogle Scholar
Littell, R., Milliken, G., Stroup, W., Wolfinger, R., & Schabenberger, O. (2006). SAS for mixed models (2nd ed.). Cary: SAS Institute.Google Scholar
Long, J.S., & Ervin, L.H. (2000). Using heteroscedasticity consistent standard errors in the linear regression model. American Statistician, 54, 217–224.CrossRefGoogle Scholar
MacKinnon, J.G., & White, H. (1985). Some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics, 29, 305–325.CrossRefGoogle Scholar
Marsh, H.W., Wen, Z., & Hau, K.-T. (2004). Structural equation models of latent interactions: evaluation of alternative estimation strategies and indicator construction. Psychological Methods, 9, 275–300.CrossRefGoogle ScholarPubMed
Nelson, E.A., & Dannefer, D. (1992). Aged heterogeneity: fact or fiction? The fate of diversity in gerontological research. The Gerontologist, 32, 17–23.CrossRefGoogle ScholarPubMed
Newsom, J.T., Prigerson, H.G., Schulz, R., & Reynolds, C.F. (2003). Investigating moderator hypotheses in aging research: statistical methodological, and conceptual difficulties with comparing separate regressions. The International Journal of Aging & Human Development, 57, 119–150.CrossRefGoogle ScholarPubMed
Ng, M., & Wilcox, R.R. (2010). Comparing the regression slopes of independent groups. British Journal of Mathematical & Statistical Psychology, 63, 319–340.CrossRefGoogle ScholarPubMed
Overton, R.C. (2001). Moderated multiple regression for interactions involving categorical variables: a statistical control for heterogeneous variance across two groups. Psychological Methods, 6, 218–233.CrossRefGoogle ScholarPubMed
Ping, R.A. (1996). Latent variable interaction and quadratic effect estimation: a two-step technique using structural equation analysis. Psychological Bulletin, 119, 166–175.CrossRefGoogle Scholar
Preacher, K.J., & Merkle, E.C. (2012). The problem of model selection uncertainty in structural equation modeling. Psychological Methods, 17, 1–14.CrossRefGoogle ScholarPubMed
Shieh, G. (2009). Detection of interactions between a dichotomous moderator and a continuous predictor in moderated multiple regression with heterogeneous error variance. Behavior Research Methods, 41, 61–74.CrossRefGoogle Scholar
Singh, B., Nagar, A.L., Choudhry, N.K., & Raj, B. (1976). On the estimation of structural change: a generalization of the random coefficients regression model. International Economic Review, 17, 340–361.CrossRefGoogle Scholar
Tang, W., Yu, Q., Crits-Christoph, P., & Tu, X.M. (2009). A new analytic framework for moderation analysis–moving beyond analytic interactions. Journal of Data Science, 7, 313–329.CrossRefGoogle ScholarPubMed
Weisberg, S. (1980). Applied linear regression. New York: Wiley.Google Scholar
White, H. (1980). A heteroskedastic-consistent covariance matrix estimator and a direct test of heteroskedasticity. Econometrica, 48, 817–838.CrossRefGoogle Scholar
Wooldridge, J.M. (2010). Econometric analysis of cross section and panel data (2nd ed.). Cambridge: MIT Press.Google Scholar
Wu, C.F.J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. The Annals of Statistics, 14, 1261–1295.Google Scholar
Yuan, K.-H., & Bentler, P.M. (1997). Improving parameter tests in covariance structure analysis. Computational Statistics & Data Analysis, 26, 177–198.CrossRefGoogle Scholar
Yuan, K.-H., & Bentler, P.M. (2010). Finite normal mixture SEM analysis by fitting multiple conventional SEM models. Sociological Methodology, 40, 191–245.CrossRefGoogle ScholarPubMed