Bayesian Third-Variable Effect Analysis

The Bayesian method is important in statistics analysis since prior knowledge can be used and the explanation of statistics inference from the Bayesian method is straightforward. In the third-variable effect analysis, the Bayesian method can provide further advantages since it takes third-variables as random variables directly and the goodness-of-fit for the final model is given with third-variable effects embedded in the hierarchical data structure. In this chapter, we provide the third-variable analysis method in the Bayesian setting. Three Bayesian methods are provided: 1) third-variable effects as a function of estimated coefficients, 2) product of partial differences for X-M effect and M-Y effect separately and 3) the resampling method. We provide R codes using WinBUGS models and explain how to extend the method to different types of exposure, third-variable and outcome variables. In the provided R codes, vague informative priors are used in the analysis, which can be adjusted easily. The three Bayesian methods are compared using simulations.