Causal inference

What is Causal inference

• is prediction of intervention: What if I do this?
• is imputation of missing observations: What if I had done something else?

Terminology

Counterfactuals

• A counterfactual is a hypothetical scenario used to reason about what would have happened if a different action or condition had been present.
• It’s fundamental in defining causal effects.
• It is different from potential outcomes:
  • potential outcomes framework defines causal effects by comparing what would happen under different treatments or conditions
  • counterfactuals specifically refer to the unobserved outcome that would have occurred under a different scenario than what actually happened

Confounder

• A confounder is a variable that influences both the treatment (or exposure) and the outcome, potentially leading to a spurious association if not properly controlled.

Causal Effects

• Average Treatment Effect (ATE)
  • = the average difference in outcomes between if everyone was treated with A=1 and if everyone was treated with A=0:
  • formula:
  $$E(Y^1-Y^0)$$
  • note that it is different from conditioning (which compares subpopulation)
  $$E(Y^1-Y^0)\ne E(Y|A=1)-E(Y|A=0)$$
• Causal relative risk
  • formula: $$
    E(Y^1 / Y^0)
  $$$$
  • = the average treatment effect conditional on a set of covariates or specific subgroups
  • formula: $$
    E(Y^1 - Y^0 | V=v)
  $$$$
• the problem: we only observe one treatment and one outcome for each person

Causal assumptions

• = the foundational beliefs or conditions necessary to establish valid causal inferences from data
• key causal assumptions
  • stable unit treatment value assumption (SUTVA)
    • no interference (spillover/contagion): the treatment assigned to one individual does not affect the potential outcomes of another individual
    • no hidden variations: there is only one version of each treatment level
  • consistency
    • assumes that the potential outcomes for an individual under a given treatment are exactly what would be observed if that treatment were actually applied
  • ignorability ('no unmeasured confounders')
    • given pre-treatment covariates X, treatment assignment is independent from the potential outcomes
  • positivity
    • assumes that for every combination of confounders, there is a non-zero probability of receiving each treatment level
    • this means that there must be sufficient variation in treatment assignment across all levels of the confounding variables

Confounds in causal inference

The Four Elemental Confounds

Causal graphs