ISSN 0439-755X
CN 11-1911/B

心理学报 ›› 2021, Vol. 53 ›› Issue (3): 322-336.doi: 10.3724/SP.J.1041.2021.00322

• 研究报告 • 上一篇    


刘红云1,2, 袁克海3,4(), 甘凯宇1   

  1. 1北京师范大学心理学部, 北京 100875
    2北京师范大学心理学部应用实验心理北京市重点实验室, 北京 100875
    3南京邮电大学理学院, 南京 210023
    4美国圣母大学心理系, 印第安纳州46556, 美国
  • 收稿日期:2020-06-25 发布日期:2021-01-27 出版日期:2021-03-25
  • 通讯作者: 袁克海
  • 基金资助:
    * 国家自然科学基金项目资助(31971029);国家自然科学基金项目资助(32071091)

Two-level mediated moderation models with single level data
and new measures of effect sizes

LIU Hongyun1,2, YUAN Ke-Hai3,4(), GAN Kaiyu1   

  1. 1Faculty of Psychology, Beijing Normal University, Beijing 100875, China
    2Beijing Key Laboratory of Applied Experimental Psychology, Faculty of Psychology, Beijing Normal University, Beijing 100875, China
    3School of Science of Nanjing University of Posts and Telecommunications, Nanjing, 210023, China
    4Department of Psychology, University of Notre Dame, IN 46556, USA
  • Received:2020-06-25 Online:2021-01-27 Published:2021-03-25
  • Contact: YUAN Ke-Hai


传统的有中介的调节(mediated moderation, meMO)模型关于误差方差齐性的假设经常被违背, 应用研究中也缺乏测量meMO效应大小的指标。对于单层数据, 本文借助于两层建模的思想, 提出了一种可用于处理方差非齐性的两层有中介的调节(2meMO)模型; 给出了用于测量meMO分析中总调节效应、直接调节效应和有中介调节效应大小的效应量。通过Monte Carlo模拟研究, 比较了meMO和2meMO模型在参数和效应量估计上的表现。并通过实际案例解释了2meMO模型的应用以及效应量的计算和解释。

关键词: 有中介的调节, 方差齐性, 效应量, 贝叶斯估计


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 $di{{f}_{moME}}$ 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 $di{{f}_{meMO}}$. 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 $\phi _{meMO}^{(f)}$. 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