Bayesian Mediation for Prevention Data

Introduction:

Bayesian methods have become increasingly popular in the social sciences. In 2009 Yuan and MacKinnon outlined how to perform a Bayesian mediation analysis for the single mediator model by assigning prior distributions to the regression coefficients and variances of mediator and dependent variable. Enders, Fairchild, and MacKinnon (2013) developed a method for obtaining the posterior distribution of the variance-covariance matrix of the independent variable, the mediator, and the dependent variable, and subsequently computing the mediated effect using the posterior estimates of these variances and covariances. With non-informative prior distributions these two methods produce similar estimates to those obtained using the standard frequentist procedures, however, when the prior information is incorporated into the analysis the two Bayesian methods narrower interval estimates for the mediated effect.

Substantive Example:

When there is no prior information about the effect being studied, the researcher can use Bayesian methods with a diffuse prior. Data from the PHLAME (Promoting Healthy Lifestyles: Alternative Models’ Effects) Firefighter Study were used to compute the confidence limits of the mediated effect of stress on trouble sleeping through the mediator of depression. According to the normal theory confidence limits, the mediated effect lies between 0.16587 and 0.27183. By assigning diffuse prior distributions to regression coefficients and the variances of the mediator and dependent variable it was found that the mediated effect lies between 0.16402 and 0.26897 with 95% probability. Assigning a diffuse prior distribution to the covariance matrix of the variables indicates that the mediated effect lies between 0.16737 and 0.27245 with 95% probability.

With prior information available, the researcher can simply use the estimates from the prior study for the regression coefficients and variances of M and Y in the first approach, and the variance and covariance estimates in the second approach.

Conclusions:

Researchers can benefit from using Bayesian methods regardless of whether prior information is available. With the SAS syntax from this project, Bayesian mediation is easy to implement both when researchers have a regression equation and/or descriptive statistics from prior studies.