Abstract: Mediation and moderation analyses are commonly used methods for studying the relationship between an independent variable (X) and a dependent variable (Y) in conducting empirical research. To better understand the relationships among variables, there is an increasing demand for a more general theoretical framework that combines moderation and mediation analyses. Recently, statistical analysis of mediated moderation (meMO) effects has become a powerful tool for scientists to investigate complex processes. However, the traditional meMO model is formulated based on the homoscedasticity assumption, which is most likely to be violated when moderation effects exist. In addition, routinely reporting effect sizes has been recommended as the primary solution to the issue of overemphasis on significance testing. Appropriate effect sizes (ES) for measuring meMO effects are very important in reporting and interpreting inferential results. However, there does not exist an effective measure that allows us to answer the question regarding the extent to which a variable Z moderates the effect of X on Y via the mediator variable (M) in the meMO model.
The article is organized as follows. First, the two-level moderated regression model proposed by Yuan, Cheng, & Maxwell (2014) was extended to a two-level mediated moderation (2meMO) model with single-level data, the statistical path diagram was structured according to the conceptual model and the equations. Second, several effect sizes were developed for the 2meMO effect by decomposing the total variance of the moderation effect. Third, to estimate the parameters of the 2meMO model and the ES measures of the meMO effects, we developed a Bayesian estimation method to estimate the parameters of the 2meMO model. Fourth, a Monte Carlo simulation study was conducted to evaluate the performance of the 2meMO model and the proposed ES measures against those with the meMO model. Finally, we illustrate the application of the new model and measures with a real data example.
The simulation results indicate that the size of bias and MSE for parameter estimates are small under both meMO and 2meMO models whether the homoscedasticity assumption hold or not. The results of the coverage rate of the 95% CI for dif_moME following 2meMO is comparable to those following meMO when the variance of moderation error is zero, which is the assumption the meMO model is based. However, when the moderation-error variance is nonzero, 2meMO yields more accurate estimates for dif_moMO. than meMO does, the advantages of 2meMO over meMO become more obvious as the moderation-error variance increases. The results of Type I error rate indicate that 2meMO controls Type I error rather well, and the rates are close to 0.05 or below 0.05 under all the conditions. However, the Type I error rates of meMO tend to be higher than 0.05 when the moderation-error variance is nonzero. The power rates following the meMO and 2meMO models are comparable for the medium or large sample size, or when there is a large difference in meMO effects. While the value of power following 2meMO is slightly lower than that following meMO at small sample se, this result is mostly due to the inflated Type I error rate of meMO, and larger sample sizes and the smaller moderation-error variances correspond to more accurate estimates of $\phi_{me MO}^{(f)} $. The results also indicate that, when the homoscedasticity assumption of the meMO model is satisfied, the effect size estimates following the two models are about the same. However, when the moderation-error variance is not zero, the results following 2meMO are more accurate than those following meMO.
In summary, the article developed a 2meMO model with single-level data and proposed several measures to evaluate the size of the meMO effect explained by moderator variables in total, directly, or indirectly. The performance of the 2meMO model is compared against that of the traditional meMO model via Monte Carlo simulations. Results indicate that, when the assumption of homoscedasticity holds, 2meMO yields comparable results with those under meMO. When the homoscedasticity assumption is violated, estimates under 2meMO are more accurate than those under meMO. More importantly, the measures of the size of the meMO effect proposed in this article can be used as a supplement to the test of meMO effects and will meet the needs for reporting ES in practice. Consequently, the 2meMO model is recommended for the analysis of mediated moderation, and the effect sizes (ESs) for the interpretation of the effect according to the questions of interest are better reported.

Key words: mediated moderation, heteroscedasticity, effect size, Bayesian estimation