---
title: "Mediation Analysis using Bayesian Regression Models"
output: 
  rmarkdown::html_vignette:
    toc: true
    fig_width: 10.08
    fig_height: 6
tags: [r, bayesian, posterior, mediation]
vignette: >
  \usepackage[utf8]{inputenc}
  %\VignetteIndexEntry{Mediation Analysis using Bayesian Regression Models}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
bibliography: bibliography.bib
csl: apa.csl
---

```{r, SETTINGS-knitr, echo = FALSE, warning = FALSE, message = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE, 
  comment = "#>", 
  dev = "png", 
  fig.width = 7, 
  fig.height = 5, 
  message = FALSE, 
  warning = FALSE
)
options(width = 800)
if (!requireNamespace("mediation", quietly = TRUE) ||
    !requireNamespace("httr", quietly = TRUE) ||
    !requireNamespace("lavaan", quietly = TRUE) ||
    !requireNamespace("brms", quietly = TRUE) ||
    !requireNamespace("rstanarm", quietly = TRUE) ||
    !requireNamespace("insight", quietly = TRUE)) {
  knitr::opts_chunk$set(eval = FALSE)
}
```

This vignettes demonstrates the `mediation()`-function. Before we start, we fit
some models, including a mediation-object from the _mediation_-package and a
structural equation modelling approach with the _lavaan_-package, both of which
we use for comparison with _brms_ and _rstanarm_.

## Mediation Analysis in brms and rstanarm

```{r}
library(bayestestR)
library(mediation)
library(brms)
library(rstanarm)

# load sample data
data(jobs)

set.seed(123)
# linear models, for mediation analysis
b1 <- lm(job_seek ~ treat + econ_hard + sex + age, data = jobs)
b2 <- lm(depress2 ~ treat + job_seek + econ_hard + sex + age, data = jobs)

# mediation analysis, for comparison with brms
m1 <- mediate(b1, b2, sims = 1000, treat = "treat", mediator = "job_seek")
```
```{r eval=FALSE}
# Fit Bayesian mediation model in brms
f1 <- bf(job_seek ~ treat + econ_hard + sex + age)
f2 <- bf(depress2 ~ treat + job_seek + econ_hard + sex + age)
m2 <- brm(f1 + f2 + set_rescor(FALSE), data = jobs, cores = 4)

```
```{r echo=FALSE}
m2 <- insight::download_model("brms_mv_6")
```
```{r eval=FALSE}
# Fit Bayesian mediation model in rstanarm
m3 <- stan_mvmer(
  list(job_seek ~ treat + econ_hard + sex + age + (1 | occp),
       depress2 ~ treat + job_seek + econ_hard + sex + age + (1 | occp)),
  data = jobs,
  cores = 4,
  refresh = 0
)
```
```{r echo=FALSE}
m3 <- insight::download_model("stanmvreg_2")
```

`mediation()` is a summary function, especially for mediation analysis, i.e. for
multivariate response models with casual mediation effects.

In the models `m2` and `m3`, `treat` is the treatment effect and `job_seek` is
the mediator effect. For the *brms* model (`m2`), `f1` describes the mediator
model and `f2` describes the outcome model. This is similar for the *rstanarm*
model.

`mediation()` returns a data frame with information on the _direct effect_
(median value of posterior samples from treatment of the outcome model),
_mediator effect_ (median value of posterior samples from mediator of the
outcome model), _indirect effect_ (median value of the multiplication of the
posterior samples from mediator of the outcome model and the posterior samples
from treatment of the mediation model) and the _total effect_ (median value of
sums of posterior samples used for the direct and indirect effect). The
_proportion mediated_ is the indirect effect divided by the total effect.

The simplest call just needs the model-object.

```{r, message=TRUE}
# for brms
mediation(m2)

# for rstanarm
mediation(m3)
```

Typically, `mediation()` finds the treatment and mediator variables
automatically. If this does not work, use the `treatment` and `mediator`
arguments to specify the related variable names. For all values, the 89%
credible intervals are calculated by default. Use `ci` to calculate a different
interval.

## Comparison to the mediation package

Here is a comparison with the _mediation_ package. Note that the
`summary()`-output of the _mediation_ package shows the indirect effect first,
followed by the direct effect.

```{r}
summary(m1)

mediation(m2, ci = .95)

mediation(m3, ci = .95)
```

If you want to calculate mean instead of median values from the posterior
samples, use the `centrality`-argument. Furthermore, there is a
`print()`-method, which allows to print more digits.

```{r, message=TRUE}
m <- mediation(m2, centrality = "mean", ci = .95)
print(m, digits = 4)
```

As you can see, the results are similar to what the _mediation_ package produces
for non-Bayesian models.

## Comparison to SEM from the lavaan package

Finally, we also compare the results to a SEM model, using *lavaan*. This
example should demonstrate how to "translate" the same model in different
packages or modeling approached.

```{r}
library(lavaan)
data(jobs)
set.seed(1234)

model <- ' # direct effects
             depress2 ~ c1*treat + c2*econ_hard + c3*sex + c4*age + b*job_seek
             
           # mediation
             job_seek ~ a1*treat + a2*econ_hard + a3*sex + a4*age
             
           # indirect effects (a*b)
             indirect_treat := a1*b
             indirect_econ_hard := a2*b
             indirect_sex := a3*b
             indirect_age := a4*b
             
           # total effects
             total_treat := c1 + (a1*b)
             total_econ_hard := c2 + (a2*b)
             total_sex := c3 + (a3*b)
             total_age := c4 + (a4*b)
         '
m4 <- sem(model, data = jobs)
summary(m4)

# just to have the numbers right at hand and you don't need to scroll up
mediation(m2, ci = .95)
```

The summary output from *lavaan* is longer, but we can find the related numbers
quite easily:

- the _direct effect_ of treatment is `treat (c1)`, which is `-0.040`
- the _indirect effect_ of treatment is `indirect_treat`, which is `-0.016`
- the _mediator effect_ of job_seek is `job_seek (b)`, which is `-0.240`
- the _total effect_ is `total_treat`, which is `-0.056`