# Introduction to BayesGmed

# Motivation

This vignette introduces the R package BayesGmed, a package designed for a Bayesian causal mediation analysis in Stan, a probabilistic programming language for Bayesian inference. BayesGmed uses a parametric approach for the specification of the outcome and mediator model, and uses the g-formula approach for estimation. In addition to the estimation of causal mediation effects, the package also allows researchers to conduct sensitivity analysis.

# Getting started

You can install from Github via devtools

    devtools::install_gitlab(belayb/BayesGmed)

# Example

BayesGmed comes with an example data, example_data, which contains a binary outcome \(Y\), a single binary mediator \(M\), a binary exposure \(A\), and two numeric confounding variables \(Z = (Z_1, Z_2)\). The first 6 rows of the data is visualized below.

    library(BayesGmed)

    head(example_data)
    #>   Z1 Z2 A M Y
    #> 1  0  1 1 1 0
    #> 2  1  0 0 1 0
    #> 3  0  0 0 0 0
    #> 4  1  1 1 0 0
    #> 5  0  1 1 0 0
    #> 6  1  0 1 1 0

The aim in this example data is to estimate the direct and indirect effect of \(A\) on \(Y\) adjusting for \(Z\). To do this, we may proced as follow.

\[logit( P(Y_{i} = 1|A_{i},M_{i},\mathbf{Z}_{i}) ) = \alpha_{0} + \mathbf{\alpha}_{Z}^{'}\mathbf{Z}_{i} + \alpha_{A}A_{i} + \alpha_{M}M_{i},\]

\[logit(P(M_{i} = 1| A_{i},\mathbf{Z}_{i} ) ) = \beta_{0} + \mathbf{\beta}_{Z}^{'}\mathbf{Z}_{i} + \beta_{A}A_{i}.\]

To complete the model specification, we assume the following priors for the model parameters.

\[ \begin{aligned} \beta &\sim MVN(location_m, scale_m)\\ \alpha &\sim MVN(location_y, scale_y) \end{aligned} \]

We then need to specify the parameters of the prior distrbutions or assume a hyper-prior. BayesGmed currently only allow fixed values and these values can be passed as part of the model call and if not the following defult values will be used

    P <- 3 # number of covariates plus the intercept term
    priors <- list(scale_m = 2.5*diag(P+1),
                   scale_y = 2.5*diag(P+2),
                   location_m = rep(0, P+1),
                   location_y = rep(0, P+2))

The model can then be fitted as follow

    fit1 <- bayesgmed(outcome = "Y", mediator =  "M", treat = "A", covariates = c("Z1", "Z2"), dist.y = "binary",
    dist.m = "binary", link.y = "logit", link.m = "logit", data = example_data)
    #>
    #> SAMPLING FOR MODEL 'BY_BM_single' NOW (CHAIN 1).
    #> Chain 1:
    #> Chain 1: Gradient evaluation took 0.00014 seconds
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    bayesgmed_summary(fit1)
    #>     Parameter  Mean    SE Median   2.5% 97.5% n_eff Rhat
    #> 1 NDE_control 0.263 0.069  0.263  0.127 0.397  5292    1
    #> 2 NDE_treated 0.286 0.069  0.287  0.150 0.420  5012    1
    #> 3 NIE_control 0.042 0.036  0.040 -0.027 0.117  3925    1
    #> 4 NIE_treated 0.065 0.048  0.063 -0.027 0.163  4434    1
    #> 5        ANDE 0.274 0.064  0.273  0.148 0.398  5482    1
    #> 6        ANIE 0.053 0.034  0.052 -0.010 0.123  4353    1
    #> 7          TE 0.327 0.065  0.327  0.200 0.453  4959    1