Potential Outcomes

Potential outcomes turn “effect” into a comparison of defined alternatives. For each eligible student, $Y(1)$ is the six-week exam score under assignment to Study Sprint access and $Y(0)$ is the score under assignment to usual resources.

Prerequisites

Complete Confounders, Mediators, and Colliders. Revisit Counterfactual Thinking for the observed-outcome equation.

Learning goals

• Define treatment contrasts using potential outcomes.
• State consistency, exchangeability, positivity, interference, and measurement assumptions in plain language.
• Connect those assumptions to an observed-data adjustment formula.

Four assumptions in plain language

Consistency

If a student actually receives the well-defined assignment $A=a$, their observed outcome equals $Y(a)$. This requires the versions grouped under “access” and “usual resources” to be similar enough for the question. Different tutors, start dates, doses, or hidden co-interventions may violate that interpretation.

Exchangeability

The groups being compared can stand in for one another's missing potential outcomes. In a randomized assignment, this holds by design in expectation before losses and deviations. In an observational study, conditional exchangeability means that after conditioning on an adequate set of pre-treatment common causes $L$, assignment is independent of $Y(1)$ and $Y(0)$. It is an untestable causal assumption, not a consequence of model fit.

Positivity

Every type of student in the target population, as defined by $L$, has a nonzero chance of each assignment being compared. If all advanced students receive access and no beginners do, data cannot directly support both potential outcomes in those strata. Near-zero probabilities create unstable extrapolation even when they are technically nonzero.

No interference—or an explicit interference model

One student's potential score depends only on their own assignment. This can fail when participants share notes, compete for tutor time, or change classmates' behavior. The remedy is not to ignore spillovers but to define exposure at a suitable individual, group, or allocation-policy level and target direct or spillover effects.

Measurement assumptions

The recorded assignment must distinguish the relevant treatment versions; the exam score must represent the intended learning outcome; baseline confounders must be measured before assignment with enough validity; and selection into the analytic dataset must not create an open noncausal path. Misclassification and missingness can break the link between the notation and the observed columns.

Assumptions and limits

For a binary treatment, consistency, conditional exchangeability, and positivity imply the adjustment identity

$E[Y(a)] = \sum_l E[Y \mid A=a,L=l]P(L=l).$

This formula averages stratum-specific observed outcomes over a target distribution of $L$. It does not make the assumptions true. With interference, the potential outcome must include others' assignments or an allocation rule, so this simple notation is insufficient.

Common mistakes

• Treating SUTVA as a magic acronym rather than separating treatment versions and interference.
• Claiming exchangeability because observed baseline means are close.
• Ignoring practical positivity when a model produces extreme weights.
• Measuring “treatment” as attendance while defining the intervention as assigned access.
• Assuming a standardized test measures every relevant educational outcome.

Practice

For Study Sprint, write one concrete failure mode for consistency, exchangeability, positivity, no interference, and measurement. State whether changing the design, estimand, or data could address each failure.

Next steps

Estimands: ATE, ATT, and CATE chooses whose potential outcomes to average.

Primary sources

Checked 2026-10-15: Cole and Frangakis, “The Consistency Statement in Causal Inference”, Hudgens and Halloran, “Toward Causal Inference With Interference”, and Hernán and Robins, Causal Inference: What If.