Assessing moderator effects, main effects, and simple effects without collinearity problems in moderated regression models☆
Introduction
Most researchers have examined the moderation effect with the interaction effect between the focal predictor and the moderator in moderated regression analysis (e.g., Aiken and West, 1991, Cronbach, 1987, Irwin and McClelland, 2001, Sharma et al., 1981). Some researchers have considered two possible effects of a moderator on the form and strength of the relationship between a focal predictor and a dependent variable (e.g., Allison et al., 1992, Prescott, 1986, Sharma et al., 1981, Smithson, 2012). The form of relationship indicates the effect function of a focal predictor on a dependent variable. In contrast, the strength of relationship indicates the predictability of a focal predictor on a dependent variable. The strength of relationship is linked to the variance of an error term in a moderated regression model. However, it is convention to define the moderator based on the form of relationship in the literature. Thus, the moderated regression analysis defines a moderator as a third variable that alters the form of relationship between a focal predictor and a dependent variable. Consequently, it measures the moderation effect with the regression coefficient capturing the interaction effect. According to this conventional definition of a moderator, we examine moderation effects in this paper.
If the interaction effect is detected by the moderated regression analysis, one may subsequently assess the simple effect of the focal predictor at a specific value of the moderator in simple slopes analysis (e.g., Irwin and McClelland, 2001, Jaccard and Turrisi, 2003, Krishna, 2016, Liu et al., 2017, MacCallum et al., 2002, Spiller et al., 2013). The simple slopes analysis is referred to as a “pick-a-point” approach (Rogosa, 1980, Rogosa, 1981). It is also referred to as the “conditional effect” of the focal predictor in the conditional process analysis (e.g., Darlington and Hayes, 2017, Hayes, 2013, Preacher et al., 2007). The main effect of a predictor can be regarded as a specific conditional effect of the predictor, because it is the conditional effect of the predictor when the other predictors are zero.
However, there may exist considerable correlations among predictors in a moderated regression model and these correlations lead to considerable correlations among the individual effects of predictors. Thus, it may be difficult to assess the interaction effect and the simple effect with the moderated regression model and the simple slopes analysis, because the interaction effect is assessed independently from the other effects of predictors and the simple effect is assessed independently from the main effect of the moderator.
Moderated regression analysis (MRA) has been used in various academic fields including psychology and management. However, there are still many issues researchers should know about MRA. For example, Gardner et al. (2017) point out that there are three primary types of moderation effects: i.e., strengthening, weakening, and reversing effects. Dawson (2014) provides practical implications on MRA that may be helpful for researchers when they examine moderation effects with MRA. Aguinis et al. (2017) raise some issues in MRA such as measurement error, artificial categorization of continuous variables, and collinearity among predictors. This paper focuses on the collinearity problem in MRA. Simple slopes analysis (SSA) assesses simple effects with the total effect of a focal predictor (the sum of the main effect of the focal predictor and the interaction effect of the focal predictor with a moderator) not reflecting the correlation between the total effect and the main effect of the moderator. Thus, SSA may also suffer from the collinearity problem.
Some scholars argue that mean-centering can reduce the collinearity problem (Aiken and West, 1991, Cronbach, 1987, Jaccard et al., 1990), but other scholars argue that mean-centering does not alleviate the collinearity problem (e.g., Echambadi & Hess, 2007). Regardless of the debate on mean-centering, the moderated regression model and the simple slopes analysis cannot be free from the collinearity problem, because the two traditional analyses (MRA and SSA) do not reflect the correlations among the interaction effect and the other effects (e.g., the main effects of the focal predictor and the moderator). We will discuss this point in more detail later.
The moderated regression analysis may capture spurious moderation effects, because the interaction effect may capture the variance of a dependent variable due to the nonlinear effects of predictors (e.g., Daryanto & Lukas, 2022). One may further examine whether the identified interaction effect is a spurious moderation effect with alternative regression models including nonlinear covariates (indicating squared terms of predictors). However, we do not focus on the effects on nonlinear covariates in this paper.
The aim of this paper is to suggest alternative analyses assessing the interaction effect and the simple effect without the collinearity problem. We provide a practical guideline for assessing the simple effect, the main effects, and the interaction effect with the two traditional analyses as well as the alternative analyses.
More specifically, we show that it is possible to interpret the simple effect in view of the effect of the focal predictor on the overall effect size of predictors in an interval of the focal predictor and the interaction effect in view of the effect of the moderator on the simple effect in an interval of the moderator. Then, we introduce two approaches based on the overall effect size. One is the ‘sequential point estimation approach’ deriving the statistics for the simple effect and the interaction effect from the point estimate for the overall effect size. The other is the ‘sequential CI estimation approach’ deriving the statistics for the various effects from the confidence-interval estimate for the overall effect size.
The t-statistics for the simple effect and the interaction effect derived by the sequential point estimation approach are identical to those used in the two traditional analyses. In contrast, the t-statistics for the simple effect and the interaction effect derived by the sequential CI estimation approach are different from those used in the two traditional analyses. More specifically, one can consider alternative analyses (corresponding to the two traditional analyses) based on the sequential CI estimation approach. The alternative analyses provide the t-statistics for the simple effect, the interaction effect, and the main effects while reflecting the correlations among the individual effects of predictors. The correlations are reflected in the standard errors for the simple effect, the interaction effect, and the main effects. Furthermore, the alternative analyses provide the t-statistics with which one can assess the various effects reflecting all the possible correlations among the individual effects of predictors including the intercept in a moderated regression model.
Section snippets
Moderation effects and simple effects
Let us consider a typical moderated regression model written as:where
: a value of predictor (i.e., ),
: a value of moderator (i.e., ),
: a value of dependent variable (i.e., ),
: an error term [i.e., ].
The traditional MRA assesses the moderation effect based on the point estimate for the interaction effect. More specifically, it uses the t-statistic calculated with the ratio of the point estimate to its standard error. The t-statistic is
Basic notations and formulae
We start with basic notations and formulae for the simple effect and the interaction effect. The basic notations can be summarized as follows.
- (a)
and : the lower and upper bounds of the focal predictor (),
- (b)
and : the lower and upper bounds of the moderator (),
- (c)
: the overall effect size of predictors at ()
where is given as a specific value of (),
- (d)
: the confidence interval of ,
- (e)
,
The sequential point estimation approach
The sequential point estimation approach derives the point estimate for (presented in Eq. (8)) and its standard error and then assesses with the t-statistic calculated with the estimate and its standard error. We summarize the statistic in Theorem 1. Theorem 1 shows that the t-statistic in the sequential point estimation approach is identical to that in the traditional SSA. Theorem 1 The estimate for is represented as:
The sequential CI estimation approach
The sequential CI estimation approach derives the estimate for (presented in Eq. (8)) and its standard error based on the CI estimate for where the CI estimate is derived from the CI estimates for the OESs at the two bounds of the focal predictor. Then, it assesses the simple effect with the t-statistic calculated with the estimate and the standard error. We summarize the t-statistic for derived by the sequential CI estimation approach in Theorem
Comparing the traditional analyses and the alternative analyses
In this section, we compare the t-statistics between the traditional MRA and the traditional SSA (based on the point estimation approach) and the corresponding alternative analyses (based on the sequential CI estimation approach), and derive conditions under which they lead to the identical t-statistics for the various effects. The alternative analyses assess the various effects by considering the correlations between the inner and the outer components corresponding to the various effects,
Empirical illustration
The statistics for the various effects in the two alternative analyses are different from those used in the two traditional analyses in two ways. First, unlike the traditional analyses assessing the various effects in view of marginal effects, the alternative analyses assess the various effects in view of average effects in particular intervals of the focal predictor and the moderator. Second, unlike the traditional analyses, the alternative analyses reflect the interval effects of the focal
Theoretical contributions
Previous research has examined the interaction effect between a focal predictor and a moderator and the main effects of the variables with moderated regression analysis and subsequently has examined simple effects of the focal predictor with simple slopes analysis. However, the two traditional analyses may suffer from correlated individual effects of predictors because the product term of the focal predictor and the moderator is likely to be correlated with the focal predictor and the moderator.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Sang-June Park is Professor of Marketing at Jeonbuk National University and holds a Ph.D. in marketing from the KAIST. He has published in Marketing Science, Technological Forecasting and Social Change, Journal of Mathematical Psychology, Journal of Economic Psychology, Corporate Social Responsibility and Environmental Management, and other journals. His research interests include marketing models and advanced research methods.
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Sang-June Park is Professor of Marketing at Jeonbuk National University and holds a Ph.D. in marketing from the KAIST. He has published in Marketing Science, Technological Forecasting and Social Change, Journal of Mathematical Psychology, Journal of Economic Psychology, Corporate Social Responsibility and Environmental Management, and other journals. His research interests include marketing models and advanced research methods.
Youjae Yi is Professor of Marketing in the College of Business Administration at Seoul National University. After receiving his PhD from Stanford University, he had been on the faculty of University of Michigan, Ann Arbor (1987-93). His work has appeared in the Journal of Marketing Research, Journal of Consumer Research, Journal of Applied Psychology, Journal of the Academy of Marketing Science, Journal of Consumer Psychology, Journal of Business Research, and other journals. He has served as editor of the Korean Journal of Consumer Studies, co-editor of Service Industries Journal, and associate editor of Journal of Consumer Psychology.
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This research is supported by the Institute of Management Research at Seoul National University.