# Through the front door

    if(!requireNamespace("fabricatr", quietly = TRUE)) {
      install.packages("fabricatr")
    }

    library(CausalQueries)
    library(fabricatr)
    library(knitr)

Here is an example of a model in which `X` causes `M` and `M` causes `Y`. There is, in addition, unobservable confounding between `X` and `Y`. This is an example of a model in which you might use information on `M` to figure out whether `X` caused `Y` making use of the “front door criterion.”

The DAG is defined using `dagitty` syntax like this:

    model <- make_model("X -> M -> Y <-> X")

We might set priors thus:

    model <- set_priors(model, distribution = "jeffreys")
    #> No specific parameters to alter values for specified. Altering all parameters.

You can plot the dag thus.

    plot(model)

Updating is done like this:

    # Lets imagine highly correlated data; here an effect of .9 at each step
    data <- fabricate(N = 5000,
                      X = rep(0:1, N/2),
                      M = rbinom(N, 1, .05 + .9*X),
                      Y = rbinom(N, 1, .05 + .9*M))

    # Updating
    model <- model |> update_model(data, refresh = 0)

Finally you can calculate an estimand of interest like this:

    query_model(
        model = model,
        using = c("priors", "posteriors"),
        query = "Y[X=1] - Y[X=0]",
        ) |>
      kable(digits = 2)

query | given | using | case_level | mean | sd | cred.low | cred.high
--- | --- | --- | --- | --- | --- | --- | ---
Y[X=1] - Y[X=0] | - | priors | FALSE | 0.00 | 0.15 | -0.34 | 0.35
Y[X=1] - Y[X=0] | - | posteriors | FALSE | 0.81 | 0.01 | 0.78 | 0.83

This uses the posterior distribution and the model to assess the average treatment effect estimand.

Let’s compare now with the case where you do not have data on `M`:

    model |>
      update_model(data |> dplyr::select(X, Y), refresh = 0) |>
      query_model(
        using = c("priors", "posteriors"),
        query = "Y[X=1] - Y[X=0]") |>
      kable(digits = 2)

query | given | using | case_level | mean | sd | cred.low | cred.high
--- | --- | --- | --- | --- | --- | --- | ---
Y[X=1] - Y[X=0] | - | priors | FALSE | 0.00 | 0.15 | -0.35 | 0.33
Y[X=1] - Y[X=0] | - | posteriors | FALSE | 0.09 | 0.15 | -0.03 | 0.55

Here we update much less and are (relatively) much less certain in our beliefs precisely because we are aware of the confounded related between `X` and `Y`, without having the data on `M` we could use to address it.

# Try it

Say `X`, `M`, and `Y` were perfectly correlated. Would the average treatment effect be identified?