Moderated Mediation Analysis Using Bayesian Methods

Conventionally, moderated mediation analysis is conducted through adding relevant interaction terms into a mediation model of interest. In this study, we illustrate how to conduct moderated mediation analysis by directly modeling the relation between the indirect effect components including a and b and the moderators, to permit easier specification and interpretation of moderated mediation. With this idea, we introduce a general moderated mediation model that can be used to model many different moderated mediation scenarios including the scenarios described in Preacher, Rucker, and Hayes (2007). Then we discuss how to estimate and test the conditional indirect effects and to test whether a mediation effect is moderated using Bayesian approaches. How to implement the estimation in both BUGS and Mplus is also discussed. Performance of Bayesian methods is evaluated and compared to that of frequentist methods including maximum likelihood (ML) with 1st-order and 2nd-order delta method standard errors and mL with bootstrap (percentile or bias-corrected confidence intervals) via a simulation study. The results show that Bayesian methods with diffuse (vague) priors implemented in both BUGS and Mplus yielded unbiased estimates, higher power than the ML methods with delta method standard errors, and the ML method with bootstrap percentile confidence intervals, and comparable power to the ML method with bootstrap bias-corrected confidence intervals. We also illustrate the application of these methods with the real data example used in Preacher et al. (2007). Advantages and limitations of applying Bayesian methods to moderated mediation analysis are also discussed.

[1]  A. Brix.  Bayesian Data Analysis, 2nd edn , 2005 .

[2]  Kristopher J Preacher,et al.  Advantages of Monte Carlo Confidence Intervals for Indirect Effects , 2012 .

[3]  A. Hayes.  Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach , 2013 .

[4]  William J. Browne,et al.  Bayesian and likelihood-based methods in multilevel modeling 1 A comparison of Bayesian and likelihood-based methods for fitting multilevel models , 2006 .

[5]  Kristopher J Preacher,et al.  A general multilevel SEM framework for assessing multilevel mediation. , 2010, Psychological methods.

[6]  John Geweke,et al.  Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments , 1991 .

[7]  B. Muthén.  Bayesian Analysis In Mplus : A Brief Introduction , 2010 .

[8]  P. Shrout,et al.  Mediation in experimental and nonexperimental studies: new procedures and recommendations. , 2002, Psychological methods.

[9]  David P Mackinnon,et al.  Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods , 2004, Multivariate behavioral research.

[10]  Kristopher J Preacher.  Multilevel SEM Strategies for Evaluating Mediation in Three-Level Data , 2011, Multivariate behavioral research.

[11]  R. Dodhia.  A Review of Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd ed.) , 2005 .

[12]  J. Edwards,et al.  Methods for integrating moderation and mediation: a general analytical framework using moderated path analysis. , 2007, Psychological methods.

[13]  Heike Bruch,et al.  Affective mechanisms linking dysfunctional behavior to performance in work teams: a moderated mediation study. , 2008, The Journal of applied psychology.

[14]  J. Ibrahim,et al.  Power prior distributions for regression models , 2000 .

[15]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Lijuan Wang,et al.  Methods for Mediation Analysis with Missing Data , 2012, Psychometrika.

[17]  Ying Yuan,et al.  Bayesian mediation analysis. , 2009, Psychological methods.

[18]  John J. McArdle,et al.  A Simulation Study Comparison of Bayesian Estimation With Conventional Methods for Estimating Unknown Change Points , 2008 .

[19]  Andrew Thomas,et al.  WinBUGS - A Bayesian modelling framework: Concepts, structure, and extensibility , 2000, Stat. Comput..

[20]  C. Judd,et al.  When moderation is mediated and mediation is moderated. , 2005, Journal of personality and social psychology.

[21]  Kristopher J Preacher,et al.  SPSS and SAS procedures for estimating indirect effects in simple mediation models , 2004, Behavior research methods, instruments, & computers : a journal of the Psychonomic Society, Inc.

[22]  Kristopher J Preacher,et al.  Addressing Moderated Mediation Hypotheses: Theory, Methods, and Prescriptions , 2007, Multivariate behavioral research.

[23]  Zhiyong Zhang,et al.  Estimating and Testing Mediation Effects with Censored Data , 2011 .

[24]  Bradley P. Carlin,et al.  BAYES AND EMPIRICAL BAYES METHODS FOR DATA ANALYSIS , 1996, Stat. Comput..

[25]  John R. Nesselroade,et al.  Bayesian analysis of longitudinal data using growth curve models , 2007 .

[26]  D. A. Kenny,et al.  The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations. , 1986, Journal of personality and social psychology.

[27]  M. Sobel.  Asymptotic Confidence Intervals for Indirect Effects in Structural Equation Models , 1982 .

[28]  David J. Lunn,et al.  The BUGS Book: A Practical Introduction to Bayesian Analysis , 2013 .

[29]  J LunnDavid,et al.  WinBUGS A Bayesian modelling framework , 2000 .

[30]  Kristopher J Preacher,et al.  Alternative Methods for Assessing Mediation in Multilevel Data: The Advantages of Multilevel SEM , 2011 .

[31]  S. West,et al.  A comparison of methods to test mediation and other intervening variable effects. , 2002, Psychological methods.

[32]  Jacob Cohen,et al.  Applied multiple regression/correlation analysis for the behavioral sciences , 1979 .

[33]  C. Bayes.  Bayesian inference for the skewness parameter of the scalar skew-normal distribution , 2007 .

[34]  Zhiyong Zhang,et al.  Investigating Ceiling Effects in Longitudinal Data Analysis , 2008, Multivariate behavioral research.

[35]  Kristopher J Preacher,et al.  Conditional process modeling: Using structural equation modeling to examine contingent causal processes. , 2013 .

[36]  Victoria Savalei,et al.  Assessing Mediational Models: Testing and Interval Estimation for Indirect Effects , 2010, Multivariate behavioral research.

[37]  Lijuan Wang,et al.  A SAS Interface for Bayesian Analysis With WinBUGS , 2008 .

[38]  A. Rukhin.  Bayes and Empirical Bayes Methods for Data Analysis , 1997 .

[39]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[40]  D. Mackinnon.  Introduction to Statistical Mediation Analysis , 2008 .

[41]  David R. Jones,et al.  How vague is vague? A simulation study of the impact of the use of vague prior distributions in MCMC using WinBUGS , 2005, Statistics in medicine.

[42]  Ranjini Natarajan,et al.  Gibbs Sampling with Diffuse Proper Priors: A Valid Approach to Data-Driven Inference? , 1998 .

[43]  Sarah Depaoli,et al.  Mixture class recovery in GMM under varying degrees of class separation: frequentist versus Bayesian estimation. , 2013, Psychological methods.