Abstract:

In recent years, multiple regression has been widely used in social sciences to analyze the moderating effect. However, this practice was found to have at least four weaknesses. First, the concept of moderation is artificially treated as interaction. Second, the role of the predictor X is confounded with that of the moderator Z. That is, the roles of the predictor and the moderator are statistically indistinguishable, or the moderation effect of Z equals that of X. Third, the assumption of homoscedasticity of error variances across different values of X and Z is often violated by data in social and behavioral sciences. Violating this assumption often results in an inflated Type Ⅱ error rate and a low power. Fourth, △R² does not directly measure the effect of moderation as conceptually defined. That is, a measure that reflects the impact of Z on the relationship between X and Y (i.e., X→Y) should be used to quantify the moderating effect.
Compared to multiple regression, the two-level regression model has many advantages in the analysis of moderating effect. First, the two-level regression model does not require the homoscedasticity assumption in moderation analysis. Second, the two-level regression model allows the regression coefficients of a dependent variable Y on predictor X (i.e., X→Y) are further regressed on moderator variables Z. Therefore, the two-level regression model permits estimating the percentage of variance of each regression coefficient that is due to moderator variables (i.e., the moderation effect size). The two-level regression model directly shows us to what extent a moderator explains the variance of the regression coefficient between the dependent variable and the predictor.
At the present study, we propose a procedure to analyze the moderating effect based on the two-level regression model. The first step is to determine the moderating effect using the two-level regression model. If the variance of the error εi1 of the level 2 slope equation is statistically significant, the result of the two-level regression model and the corresponding effect size should be reported. Otherwise, go to the second step. In the second step, multiple regression was used to work out the moderating effect. If the Bayesian information criterion of the two-level regression model is smaller than that of the multiple regression, the result of the two-level regression model and the corresponding effect size should be reported. Otherwise, the result of the multiple regression and the corresponding effect size △R² should be reported.
We exemplify how to conduct the proposed procedure by using Mplus. It is noteworthy that, with this software, a two-level regression model could be built via a “trick” for 2-level model with single level data. The Mplus syntax is offered to facilitate the implementation of two-level regression model in analyzing moderated mediation effects. The program can be managed easily by empirical researchers.
Directions for future study on two-level regression model are discussed at the end of the paper. First, if data contains outliers or heavy tails, robust methods of two-Level regression model should be adapted. Second, if the measurement error of the variables needs to be taken into account, two-level moderated latent variable model should be adapted. Third, two-level regression model could be used to analyze the mediated moderation model and moderated mediation model.

Key words: moderating effect, two-level regression model, multiple regression, effect size