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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

Information

Type
Original Paper
Information
Psychometrika , Volume 79 , Issue 4 , October 2014 , pp. 701 - 732
Copyright
Copyright © 2013 The Psychometric Society

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