SAS ® for Bayesian Mediation Analysis

Recent statistical mediation analysis research focuses on using Bayesian methods for several reasons including the ability to incorporate prior information about model parameters, ease of constructing intervals for simple and complex models, and exact inference for small samples without restrictive assumptions of conventional mediation analysis (Yuan & MacKinnon, 2009).The focus of this paper is how the Bayesian framework offers a solution to mediation analysis with small samples; namely, using diffuse prior distributions when there is no prior literature on the phenomenon being studied, and incorporating prior information into the analysis when there is existing knowledge about the expected magnitude of mediation effects. The SAS MCMC procedure allows researchers to use one of two simple and effective methods to incorporate their prior knowledge into the statistical analysis and obtain the posterior probabilities for quantities of interest, such as the mediated effect. This paper presents four examples of using SAS PROC MCMC to analyze a single mediator model using: (1) diffuse prior information for each regression coefficient in the model, (2) informative prior distributions for each regression coefficient, (3) diffuse prior distribution for the covariance matrix of variables in the model, and (4) informative prior distribution for the covariance matrix. INTRODUCTION Statistical mediation analysis is common in business, social sciences, epidemiology, and other related fields, because it helps explain how and why two variables are related (see MacKinnon, 2008). A mediating variable (M) is a variable that transmits or carries the effect of an independent variable (X) to a dependent variable (Y). Mediation analysis can be used to investigate how product presentation affects liking of the product, which then affects purchase of the product. Mediation analysis can also be used to identify the mechanism by which a health intervention changes norms that then change health behavior. There are many motivations for mediation analysis including the investigation of the process through which X affects Y, explanation of when and why the relation of X and Y occurs in different settings and populations, identification of key ingredients in interventions, and to interpret why interventions achieve beneficial effects and why they do not (Judd & Kenny, 1981; MacKinnon, 1994). Research on mediation analysis methods is an active area of research. This paper focuses on four methods to apply Bayesian statistics to the single mediator model using PROC MCMC. SINGLE MEDIATOR MODEL The goal of many research projects is to identify and describe a relation between two variables, X and Y. Sometimes a third variable can improve the understanding of the relation between two variables. When a third variable is intermediate between X and Y in a causal chain, it is called a mediator (James & Brett, 1984; MacKinnon, 2008). More specifically, mediators are operationally defined as variables that transmit the influence that one variable (X) exerts on another (Y). The simplest mediation model is the single mediator model and consists of three variables: the independent variable (X) related to the mediator (M), which is related to the dependent variable (Y) (MacKinnon, 2008). The single mediator model is portrayed in Figure 1 and can be described using the following three regression equations:  Y  i1 cX e1 (1) 2 2 e aX i M    (2) 3 3 ' e X c bM i Y     (3) where c represents the total effect of X on Y, c’ represents the effect of X on Y adjusted for the effect of the mediator M, b measures the relation between the mediator M and the dependent variable Y adjusted for the independent variable X, and a measures the relation between X and M. The intercepts are i1, i2, and i3, and it is assumed that the three error terms, e1, e2, and e3 follow a normal distribution with a mean of zero and variance 2 1  ,  2 2 and 2 3  , respectively.