Moderated Mediation Analysis Using Bayesian Methods

Abstract

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.

Keywords:

• Bayesian methods
• moderated mediation
• model specification

Notes

¹ The posterior probability distribution is the conditional probability distribution of a random variable θ after relevant evidence is taken into account. Mathematically, we have where p(θ) is the prior probability, X is data, is the likelihood of the data, and is the posterior probability.

² Depending on the form and precision of the vague prior, the choice of vague priors could have impacts on the posterior distributions of parameters, especially on variance components in models with random effects (e.g., Depaoli, 2013; Lambert, Sutton, Burton, Abrams, & Jones, 2005; Natarajan & McCulloch, 1998).

³ The full conditional distribution of a parameter is in the same distribution family of the specified prior distribution of the parameter. The full conditional distribution, , is the distribution of the mth component of the parameter vector conditioning on all the remaining components and the data.

⁴ The Bayesian approach described in this paper is different from the two distribution approaches used by MacKinnon et al. (2004), also called the Monte Carlo method (Preacher & Selig, 2012). For the two distribution approaches, we first input estimates of a and b (e.g., from ML) and their standard errors as fixed values to simulate the empirical distribution for ab, in which a and b follow normal distributions in the simulations by assumption. The Bayesian strategy uses one or more MCMC approaches (e.g., Gibbs sampling) to draw samples for parameters a and b from their full conditional distributions, and thus we can obtain the empirical posterior distribution of a, b, and ab without inputting ML or other estimates of a and b first. From the posterior distributions, we can obtain point estimates for a, b, and ab and their standard deviations. In the Bayesian framework, a and b can follow any distributions.

⁵ There are two versions of BUGS, WinBUGS and OpenBUGS, which share common coding except for a few exceptions (e.g., Dirichlet priors). In this study, code can be run in both WinBUGS and OpenBUGs. WinBUGS is no longer supported, whereas OpenBUGS is being supported for development.

⁶ We did not include SCMATH8 for the equation to be consistent with the model specification in Preacher et al. (2007).