Estimating counterfactuals

Lucy D’Agostino McGowan

Potential outcomes

  • Prior to some “cause” occurring, the potential outcomes are all of the potential things that could occur depending on what you end up exposed to

Potential outcomes

  • Let’s assume an exposure has two levels:
    • \(X=1\) if you are exposed
    • \(X=0\) if you are not exposed

Potential outcomes

  • Under this simple scenario, there are two potential outcomes:
    • \(Y(1)\) the potential outcome if you are exposed
    • \(Y(0)\) the potential outcome if you are not exposed

Potential outcomes

  • Only one of these potential outcomes will actually be realized
  • It is important to remember here that these exposures are defined at a particular instance in time, so only one can happen to any individual
  • In the case of a binary exposure, this leaves one potential outcome as observable and one missing

Potential outcomes

  • Our causal effect of interest is often some difference in potential outcomes \(Y(1) - Y(0)\), averaged over a particular population

Counterfactuals

  • Early causal inference methods were often framed as missing data problems
  • We need to make certain assumptions about the missing counterfactuals, the value of the potential outcome corresponding to the exposure(s) that did not occur
  • We wish we could observe the conterfactual outcome that would have occurred in an alternate universe

Counterfactuals

  • To do this, we attempt to control for all factors that are related to an exposure and outcome such that we can construct (or estimate) such a counterfactual outcome.

Ice-T and Spike

Split Decision: Life Stories

Award-winning actor, rapper, and producer Ice-T unveils a compelling memoir of his early life robbing jewelry stores until he found fame and fortune—while a handful of bad choices sent his former crime partner down an incredibly different path.

Vicky, CC BY 2.0 https://creativecommons.org/licenses/by/2.0, via Wikimedia Commons

Ice-T and Spike

Ice-T and Spike Causal Map

flowchart LR
  A{Ice-T} --> |observed| B(Abandons criminal life)
  A -.-> |missing counterfactual| C(Does one more heist)
  C -.-> D[35 years in prison]
  B --> E[Fame & Fortune]
  
  classDef grey fill:#ddd
  class D,C grey

flowchart LR
  A{Spike} -.-> |missing counterfactual| B(Abandons criminal life)
  A --> |observed| C(Does one more heist)
  C --> D[35 years in prison]
  B -.-> E[Fame & Fortune]
  classDef grey fill:#ddd
  class E,B grey

Ice-T and Spike

  • What would need to be true for us to draw a causal conclusion?
  • Can we really conclude that Spike’s life would have turned out exactly like Ice-T’s if he had made the exact same choices as Ice-T?

In practice

  • We could conduct an experiment where we randomize many individuals to leave criminal life (or not) and see how this impacts their outcomes on average
  • This randomized trial seems to present some ethical issues, perhaps we need to look to observational studies to help answer this question
  • We must rely on statistical techniques to help construct these unobservable counterfactuals

Slides by Dr. Lucy D’Agostino McGowan