```
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:

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

We might set priors thus:

```
<- set_priors(model, distribution = "jeffreys")
model #> 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
<- fabricate(N = 5000,
data X = rep(0:1, N/2),
M = rbinom(N, 1, .05 + .9*X),
Y = rbinom(N, 1, .05 + .9*M))
# Updating
<- model |> update_model(data, refresh = 0) model
```

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.

Say `X`

, `M`

, and `Y`

were perfectly
correlated. Would the average treatment effect be identified?