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title: 'Mediation Analysis'
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bibliography: refs.bib
---

Mediation analysis [@yuan2009bayesian] allows researchers to investigate the mechanism by which an independent variable ($X$) influences a dependent variable ($Y$). 
Rather than just asking "Does X affect Y?", mediation asks "Does X affect Y through an intermediate variable M?"

Common examples include:

 - **Psychology:** Does a therapy ($X$) reduce anxiety ($M$), which in turn improves sleep quality ($Y$)?
- **Medicine:** Does a new drug ($X$) lower blood pressure ($M$), thereby decreasing the risk of heart attack ($Y$)?

In this vignette, we demonstrate how to estimate a simple mediation model using `{INLAvaan}`.
We will fit a standard three-variable mediation model:

```{mermaid}
%%| fig-align: center
graph LR
    X((X)) -->|a| M((M))
    M -->|b| Y((Y))
    X -->|c| Y
```

- $a$: The effect of $X$ on $M$.
- $b$: The effect of $M$ on $Y$.
- $c$: The direct effect of $X$ on $Y$.
- $a \times b$: The indirect effect (the mediation effect).

In a mediation model, the *Total Effect* represents the overall impact of $X$ on $Y$, ignoring the specific pathway. 
It answers the question: 
"If I change $X$, how much does $Y$ change in _total_, regardless of whether it goes through $M$ or not?".

## Data Simulation 

To verify that `{INLAvaan}` recovers the correct parameters, we simulate data where the "truth" is known. 
The logic is as follows: Generate...

1. $X$ normally; 
2. $M$ dependent on $X$ with a coefficient of 0.5; and
3. $Y$ dependent only on $M$ with a coefficient of 0.7.

Critically, we do not add $X$ to the generation of $Y$. 
This means the true direct effect ($c$) is 0, and the relationship is fully mediated.
We expect our model to estimate $a \approx 0.5$, $b \approx 0.7$, and the indirect effect $ab \approx 0.35$. The direct effect $c$ should be close to zero.

```{r}
set.seed(11)
n <- 100  # sample size

# 1. Predictor
X <- rnorm(n)

# 2. Mediator (Path a = 0.5)
M <- 0.5 * X + rnorm(n)

# 3. Outcome (Path b = 0.7, Path c = 0)
Y <- 0.7 * M + rnorm(n) 

dat <- data.frame(X = X, Y = Y, M = M)
```

## Model Specification and Fit

The standard `lavaan` syntax for a mediation model is straightforward (note the use of the `:=` operator to define the indirect effect as a new parameter.):

```{r}
mod <- "
  # Direct effect (path c)
  Y ~ c*X

  # Mediator paths (path a and b)
  M ~ a*X
  Y ~ b*M

  # Define Indirect effect (a*b)
  ab := a*b

  # Define Total effect
  total := c + (a*b)
"
```

The model is fit using `asem()`.
The `meanstructure = TRUE` argument is supplied to estimate intercepts for the variables.

```{r}
library(INLAvaan)
fit <- asem(mod, dat, meanstructure = TRUE)
```

The user may wish to specify different prior distributions for the parameters.
See the relevant section in the [Get started](https://inlavaan.haziqj.ml/articles/INLAvaan.html#setting-priors) vignetted for further details.

## Results 

The summary output provides the posterior mean, standard deviation, and 95% credible intervals 
for all paths.

```{r}
summary(fit)
```

Looking at the Regressions and Defined Parameters sections of the output:

```{r}
#| include: false
summ <- INLAvaan:::get_inlavaan_internal(fit)$summary
fmt <- function(x) sprintf("%.3f", x)

a     <- fmt(summ["a", "Mean"])
b     <- fmt(summ["b", "Mean"])
c     <- fmt(summ["c", "Mean"])
c_lo  <- fmt(summ["c", "2.5%"])
c_hi  <- fmt(summ["c", "97.5%"])
ab    <- fmt(summ["ab", "Mean"])
ab_lo <- fmt(summ["ab", "2.5%"])
ab_hi <- fmt(summ["ab", "97.5%"])
tot   <- fmt(summ["total", "Mean"])
```


- Both intercepts are non-significant, since we simulated data with true means of zero.
- Path $a$ (`M ~ X`) estimated at `r a` (true value 0.5).
- Path $b$ (`Y ~ M`) estimated at `r b` (true value 0.7).
- Path $c$ (`Y ~ X`) estimated at `r c`. The 95% Credible Interval [`r c_lo`, `r c_hi`] includes zero, correctly identifying that there is no direct effect.
- Indirect Effect $ab$ estimated at `r ab` (true value 0.35). The interval [`r ab_lo`, `r ab_hi`] does not cross zero, indicating significant mediation.
- Total Effect estimated at `r tot`.
    * This is the sum of the direct and indirect effects ($c + ab$). 
    * It tells us that a 1-unit increase in $X$ leads to a total increase of roughly `r tot` in $Y$.
    * **Note:** In this simulation, even though the *direct* effect is non-significant (close to zero), the *total* effect is significant because the mechanism via $M$ is strong. This illustrates a "full mediation" scenario: $X$ affects $Y$, but *only* because of $M$.

## References