A potential outcome is the result a unit would have under one specified treatment level. For patient i, Y₀(i) is the recovery outcome without treatment and Y₁(i) is the outcome with treatment. Framing both possibilities turns a practical question—“Would this treatment help?”—into a comparison between alternative interventions rather than a description of who recovered. section 10.1 section 10.1.1
Only one potential outcome becomes visible for each unit. Once treatment T(i) is assigned, the observed, or factual, outcome Y(i) equals the potential outcome associated with that assignment; the other is the counterfactual outcome. Thus, treating a patient reveals Y₁(i) but leaves Y₀(i) hidden, while not treating reveals Y₀(i) but leaves Y₁(i) hidden. section 10.1.1
Table 10.3 POs are (in principle) unobserved; we only have data about measured variables.
|
Potential outcomes (in principle unobserved)
|
Measured variables (observed)
|
|---|---|
|
•
Y
0(
i): the outcome of patient
i if they
were not given the treatment
•
Y
1(
i): the outcome of patient
i if they
were given the treatment
|
•
Y(
i) observed outcome of patient
i
•
T(
i) treatment given to patient
i
|
Compare the two in-principle unobserved potential outcomes with the treatment and outcome variables that are actually measured. section 10.1.1
The observed outcome can be written as a switch: Y(i) is Y₁(i) when T(i)=1 and Y₀(i) when T(i)=0. This missing alternative is the central obstacle in causal inference: individual treatment effects cannot be directly observed, so causal estimates rely on comparisons and assumptions that make unobserved outcomes informative. section 10.1.1 section 10.3
Potential outcomes can be used at several levels:
- For an individual, represent the possible response under each treatment value, such as recovery with and without treatment. section 10.1.1
- For a population, express averages such as E[Y₁] as the average outcome if sampled individuals were treated; these quantities are not directly observable. section 10.1.2
- For a time-related event, represent outcome series with and without the event; real data normally expose only one series at each timepoint. section 11.2.2
- For treated and control groups over time, distinguish observed post-event outcomes from unobserved alternatives used in difference-in-differences reasoning. section 11.4.2
Potential-outcome notation and causal DAGs describe the same intervention idea. Giving treatment t corresponds to Y(i)|do(T(i)=t), so Yt(i)=Y(i)|do(T(i)=t). Their notation and working style differ, but they can yield the same causal conclusions; choose the framework that best fits the problem. section 10.2.1
A movie release over time
To study a movie release at time T, define Y₀(t) as views without the release and Y₁(t) as views with it. Before the event, the outcomes can overlap; after it, observed views follow the factual series. A synthetic dataset can display both series, but real observations reveal only the one selected by the event. section 11.2.1 section 11.2.2
Potential outcomes do not by themselves identify a causal effect. In randomized assignment, treatment groups are expected to be balanced; in observational settings, adjustment requires an ignorability condition: conditional on covariates, treatment must be independent of the potential outcomes. Positivity may also be included in how ignorability is defined. section 10.3
