Questions: Counterfactual Framework and Potential Outcomes

Question 1

A study finds that patients who voluntarily chose to take a new medication had better outcomes than those who did not. The researchers conclude the medication caused the improvement. Which assumption, if violated, most directly undermines this causal claim?

• A. Positivity — some patient subgroups had zero probability of receiving the medication
• B. Consistency — there are multiple formulations of the medication with different effects
• C. Exchangeability — patients who chose the medication may have been systematically healthier or more health-conscious, making the groups incomparable in their potential outcomes
• D. The sample was too small to estimate the average treatment effect with precision

Answer: (C)

In an observational study where patients self-select into treatment, the most immediate threat is confounding — a violation of exchangeability. Patients who voluntarily take a medication may differ systematically from those who don't (more health-conscious, fewer comorbidities, better access to care). Their potential outcomes under no treatment may already be better, making it impossible to attribute the outcome difference to the medication. Randomization solves this mechanically; observational studies must address it through design and statistical adjustment.

Question 2

Why can't individual causal effects — Y(1) − Y(0) — be directly measured for any single person?

• A. IRB regulations prohibit collecting the same outcome measurement twice from the same participant
• B. Each person experiences only one treatment condition, so one potential outcome is never realized and remains permanently unobserved
• C. The subtraction Y(1) − Y(0) is mathematically undefined when the outcome is a binary variable
• D. Individual effects are too small to detect; only population-level averages are large enough to measure reliably

Answer: (B)

This is the fundamental problem of causal inference: potential outcomes are mutually exclusive in reality. If someone takes the drug, we observe Y(1) and Y(0) — what would have happened without it — is forever unobserved (the counterfactual). If they don't take it, Y(0) is observed and Y(1) is unknown. We cannot run the same person through both conditions simultaneously while holding everything else constant. This is why causal inference requires assumptions about missing potential outcomes — we impute the counterfactual from comparable people, not measure it directly.

Question 3

Randomized controlled trials solve the fundamental problem of causal inference by allowing researchers to observe both potential outcomes Y(1) and Y(0) for the same individual.

• T. True
• F. False

Answer: False

RCTs do not allow observation of both potential outcomes for the same person — the fundamental problem persists. What RCTs accomplish is different: random assignment creates groups that are exchangeable in expectation, so the control group's observed Y(0) is a valid stand-in for the treatment group's counterfactual Y(0). We still observe only one potential outcome per person; we just have a valid design for estimating the average treatment effect without controlling for confounders, because randomization balances both measured and unmeasured variables.

Question 4

Exchangeability requires that, conditional on measured covariates, the distribution of potential outcomes is the same across treatment groups — meaning no unmeasured common cause of treatment assignment and outcome remains.

• T. True
• F. False

Answer: True

Exchangeability (no unmeasured confounding) is the most demanding and least testable of the three key assumptions. It means that after conditioning on measured covariates, treatment assignment is independent of the potential outcomes — the groups are interchangeable in the sense that each could stand in for the other's counterfactual. In DAG terms, this requires that conditioning on measured covariates blocks all backdoor paths. Unlike positivity (detectable from data distributions), exchangeability is a claim about unmeasured variables and cannot be verified from data alone.

Question 5

Explain why the potential outcomes framework defines a causal effect as a contrast between Y(1) and Y(0) rather than as a statistical association between treatment and outcome.

Model answer: Statistical association conflates causation with confounding and selection bias — two variables can correlate strongly because they share a common cause, not because one causes the other. The counterfactual definition forces precision: the causal effect is the difference between what happened under one treatment and what would have happened under the alternative, within the same individual (or population average across individuals). This 'hold everything else constant' requirement is what separates a causal claim from a mere association.

The framework's power is that it makes explicit what a causal claim requires: a comparison between observed fact and a hypothetical counterfactual. By writing the target estimand as E[Y(1) − Y(0)], we can precisely characterize when observational data can validly estimate it (when the three assumptions hold) and when it cannot. This explicitness is what separates modern causal inference from the older tradition of treating association as a proxy for causation.