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

Ice-T and Spike

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:#fff
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:#fff
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

Consistency

• We assume that the causal question you claim you are answering is consistent with the one you are actually answering with your analysis.
• Mathematically: \(Y_{obs} = (X)Y(1) + (1-X)Y(0)\)
• Well defined exposure
• No interference

Well defined exposure

• We assume that for each value of the exposure, there is no difference between subjects in the delivery of that exposure
• Put another way, multiple versions of the treatment do not exist

No interference

• We assume that the outcome (technically all potential outcomes, regardless of whether they are observed) for any subject does not depend on another subject’s exposure

Exchangeability

• We assume that within levels of relevant variables (confounders), exposed and unexposed subjects have an equal likelihood of experiencing any outcome prior to exposure
• i.e. the exposed and unexposed subjects are exchangeable
• This assumption is sometimes referred to as no unmeasured confounding.

Positivity

• We assume that within each level and combination of the study variables used to achieve exchangeability, there are exposed and unexposed subjects.
• Said differently, each individual has some chance of experiencing every available exposure level.
• Sometimes this is referred to as the probabilistic assumption.