Structural Causal Models

Even if you don't have to, it can be useful to define a Structural Causal Model (SCM) for your problem. A SCM is a directed acyclic graph that describes the causal relationships between the random variables under study.

Incremental Construction

All models are wrong? Well maybe not the following:

scm = SCM()

SCM
---

This model does not say anything about the random variables and is thus not really useful. Let's assume that we are interested in an outcome $Y$ and that this outcome is determined by 8 other random variables. We can add this assumption to the model

add_equation!(scm, :Y => [:T₁, :T₂, :W₁₁, :W₁₂, :W₂₁, :W₂₂, :W, :C])

Let's now assume that we have a more complete knowledge of the problem and we also know how T₁ and T₂ depend on the rest of the variables in the system.

add_equations!(scm, :T₁ => [:W₁₁, :W₁₂, :W], :T₂ => [:W₂₁, :W₂₂, :W])

One Step Construction

Instead of constructing the SCM incrementally, one can provide all the specified equations at once:

scm = SCM([
    :Y  => [:T₁, :T₂, :W₁₁, :W₁₂, :W₂₁, :W₂₂, :W, :C],
    :T₁ => [:W₁₁, :W₁₂, :W],
    :T₂ => [:W₂₁, :W₂₂, :W]
])

SCM
---
T₁ = f₂(W₁₂, W, W₁₁)
T₂ = f₃(W₂₂, W₂₁, W)
Y = f₁(C, W₁₂, W₂₂, W₂₁, W₁₁, W, T₁, T₂)

Classic Structural Causal Models

There are many cases where we are interested in estimating the causal effect of a some treatment variables on a some outcome variables. If all treatment variables share the same set of confounders, we can quickly define the associated SCM with the StaticSCM interface:

scm = StaticSCM(
    outcomes=[:Y₁, :Y₂],
    treatments=[:T₁, :T₂],
    confounders=[:W₁, :W₂];
)

SCM
---
Y₂ = f₆(W₁, T₁, W₂, T₂)
T₁ = f₂(W₁, W₂)
T₂ = f₃(W₁, W₂)
Y₁ = f₁(W₁, T₁, W₂, T₂)

where outcome_extra_covariates is a set of extra variables that are causal of the outcomes but are not of direct interest in the study.