Moderation analysis using a two-level regression model. (English) Zbl 1303.62110

Summary: 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.


62P15 Applications of statistics to psychology
62J05 Linear regression; mixed models


SASmixed; bootstrap; alr3; SAS; R
Full Text: DOI


[1] Aguinis, H. (2004). Regression analysis for categorical moderators. New York: Guilford.
[2] Aguinis, H.; Petersen, S.A.; Pierce, C.A., 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, (1999)
[3] Aiken, L.S., & West, S.G. (1991). Multiple regression: testing and interpreting interactions. Thousand Oaks: Sage.
[4] Baron, R.M.; Kenny, D.A., The moderator-mediator variable distinction in social psychological research: concept, strategic and statistical considerations, Journal of Personality and Social Psychology, 51, 1173-1182, (1986)
[5] Bast, J.; Reitsma, P., Analyzing the development of individual differences in terms of matthew effects in Reading: results from a Dutch longitudinal study, Developmental Psychology, 34, 1373-1399, (1998)
[6] Carroll, R.J., & Ruppert, D. (1988). Transformation and weighting in regression. New York: Chapman & Hall/CRC. · Zbl 0666.62062
[7] Casella, G., & Berger, R.L. (2002). Statistical inference (2nd ed.). Pacific Grove: Duxbury Press.
[8] Chaplin, W.F.; Robins, R.W. (ed.); Fraley, R.C. (ed.); Krueger, R.F. (ed.), Moderator and mediator models in personality research: a basic introduction, 602-632, (2007), New York
[9] Cohen, J., Partialed products are interactions; partialed powers are curve components, Psychological Bulletin, 85, 858-866, (1978)
[10] Cribari-Neto, F., Asymptotic inference under heteroskedasticity of unknown form, Computational Statistics & Data Analysis, 45, 215-233, (2004) · Zbl 1429.62184
[11] Darlington, R.B. (1990). Regression and linear models. New York: McGraw-Hill. · Zbl 0715.62270
[12] Davidson, R., & MacKinnon, J.G. (1993). Estimation and inference in econometrics. Oxford: Oxford University Press. · Zbl 1009.62596
[13] Davison, M.L.; Kwak, N.; Seo, Y.S.; Choi, J., Using hierarchical linear models to examine moderator effects: person-by-organization interactions, Organizational Research Methods, 5, 231-254, (2002)
[14] Dent, W.T.; Hildreth, C., Maximum likelihood estimation in random coefficient models, Journal of the American Statistical Association, 72, 69-72, (1977) · Zbl 0346.62049
[15] DeShon, R.P.; Alexander, R.A., Alternative procedures for testing regression slope homogeneity when group error variances are unequal, Psychological Methods, 1, 261-277, (1996)
[16] Dretzke, B.J.; Levin, J.R.; Serlin, R.C., Testing for regression homogeneity under variance heterogeneity, Psychological Bulletin, 91, 376-383, (1982)
[17] Efron, B., & Tibshirani, R.J. (1993). An introduction to the bootstrap. New York: Chapman & Hall. · Zbl 0835.62038
[18] Fisicaro, S.A.; Tisak, J., A theoretical note on the stochastics of moderated multiple regression, Educational and Psychological Measurement, 54, 32-41, (1994)
[19] Froehlich, B.R., Some estimators for a random coefficient regression model, Journal of the American Statistical Association, 68, 329-335, (1973)
[20] Hayes, A.F.; Cai, L., Using heteroscedasticity-consistent standard error estimators in OLS regression: an introduction and software implementation, Behavior Research Methods, 39, 709-722, (2007)
[21] Hinkley, D.V., Jackknifing in unbalanced situations, Technometrics, 19, 285-292, (1977) · Zbl 0367.62085
[22] Hildreth, C.; Houck, J., Some estimators for a linear model with random coefficients, Journal of the American Statistical Association, 63, 584-595, (1968) · Zbl 0162.49804
[23] Holmbeck, G.N., 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, (1997)
[24] Kenny, D.; Judd, C.M., Estimating the nonlinear and interactive effects of latent variables, Psychological Bulletin, 96, 201-210, (1984)
[25] Littell, R., Milliken, G., Stroup, W., Wolfinger, R., & Schabenberger, O. (2006). SAS for mixed models (2nd ed.). Cary: SAS Institute.
[26] Long, J.S.; Ervin, L.H., Using heteroscedasticity consistent standard errors in the linear regression model, American Statistician, 54, 217-224, (2000)
[27] MacKinnon, J.G.; White, H., Some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties, Journal of Econometrics, 29, 305-325, (1985)
[28] Marsh, H.W.; Wen, Z.; Hau, K.-T., Structural equation models of latent interactions: evaluation of alternative estimation strategies and indicator construction, Psychological Methods, 9, 275-300, (2004)
[29] Nelson, E.A.; Dannefer, D., Aged heterogeneity: fact or fiction? the fate of diversity in gerontological research, The Gerontologist, 32, 17-23, (1992)
[30] Newsom, J.T.; Prigerson, H.G.; Schulz, R.; Reynolds, C.F., 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, (2003)
[31] Ng, M.; Wilcox, R.R., Comparing the regression slopes of independent groups, British Journal of Mathematical & Statistical Psychology, 63, 319-340, (2010)
[32] Overton, R.C., Moderated multiple regression for interactions involving categorical variables: a statistical control for heterogeneous variance across two groups, Psychological Methods, 6, 218-233, (2001)
[33] Ping, R.A., Latent variable interaction and quadratic effect estimation: a two-step technique using structural equation analysis, Psychological Bulletin, 119, 166-175, (1996)
[34] Preacher, K.J.; Merkle, E.C., The problem of model selection uncertainty in structural equation modeling, Psychological Methods, 17, 1-14, (2012)
[35] Shieh, G., 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, (2009)
[36] Singh, B.; Nagar, A.L.; Choudhry, N.K.; Raj, B., On the estimation of structural change: a generalization of the random coefficients regression model, International Economic Review, 17, 340-361, (1976) · Zbl 0357.62040
[37] Tang, W.; Yu, Q.; Crits-Christoph, P.; Tu, X.M., A new analytic framework for moderation analysis-moving beyond analytic interactions, Journal of Data Science, 7, 313-329, (2009)
[38] Weisberg, S. (1980). Applied linear regression. New York: Wiley. · Zbl 0529.62054
[39] White, H., A heteroskedastic-consistent covariance matrix estimator and a direct test of heteroskedasticity, Econometrica, 48, 817-838, (1980) · Zbl 0459.62051
[40] Wooldridge, J.M. (2010). Econometric analysis of cross section and panel data (2nd ed.). Cambridge: MIT Press. · Zbl 1327.62009
[41] Wu, C.F.J., Jackknife, bootstrap and other resampling methods in regression analysis, The Annals of Statistics, 14, 1261-1295, (1986) · Zbl 0618.62072
[42] Yuan, K.-H.; Bentler, P.M., Improving parameter tests in covariance structure analysis, Computational Statistics & Data Analysis, 26, 177-198, (1997) · Zbl 1003.62523
[43] Yuan, K.-H.; Bentler, P.M., Finite normal mixture SEM analysis by Fitting multiple conventional SEM models, Sociological Methodology, 40, 191-245, (2010)
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