Question
A voluntary coaching course is offered to 100 applicants. Forty choose to attend: 30 pass a later qualification test and 10 fail. Of the 60 who do not attend, 24 pass and 36 fail. Everyone's attendance and outcome are observed, but attendance was not randomly assigned.
Suppose coaching can never turn a person who would have passed without coaching into someone who fails with coaching. For each applicant, define the causal benefit as their outcome with coaching minus their outcome without it, where \(\text{pass}=1\) and \(\text{fail}=0\).
What are the sharpest possible lower and upper bounds on the average causal benefit across all 100 applicants? Exhibit two hypothetical assignments of missing counterfactual outcomes that attain the bounds. How do these bounds compare with simply subtracting the observed pass rates?
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Hint 1
Write each person's two potential outcomes as \((Y(0),Y(1))\). Monotonicity rules out \((1,0)\), leaving only \((0,0)\), \((0,1)\) and \((1,1)\).
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Solution
Let \(D\in\{0,1\}\) denote attendance, \(Y(0)\) the pass indicator without coaching, and \(Y(1)\) the pass indicator with coaching. Everyone reveals only \(Y(D)\). The monotonicity assumption \(Y(1)\ge Y(0)\) permits exactly three potential-outcome types: never pass \((0,0)\), helped \((0,1)\), and always pass \((1,1)\). The individual causal effect equals one only for the helped type, so the population average is the fraction of that type.
An attending applicant who failed must be \((0,0)\), and a non-attending applicant who passed must be \((1,1)\). Neither can have been helped. But each of the 30 attending passers might be either \((0,1)\) or \((1,1)\); each of the 36 non-attending failures might be either \((0,0)\) or \((0,1)\). Therefore the helped count lies between \(0\) and \(30+36=66\), so
\[\boxed{0\le E[Y(1)-Y(0)]\le66/100=0.66}.\]
Both endpoints are feasible. For the lower bound, assign \((1,1)\) to every observed passer and \((0,0)\) to every observed failure. Everyone's unobserved outcome agrees with their observed outcome, so nobody benefits and all four observed counts remain unchanged. For the upper bound, assign \((0,1)\) to all 30 attending passers and all 36 non-attending failures; assign \((0,0)\) to the 10 attending failures and \((1,1)\) to the 24 non-attending passers. Exactly 66 people benefit, the observed data are again identical, and monotonicity holds for every person. Thus the bounds are sharp, not merely conservative.
The treated pass rate is \(30/40=0.75\) and the untreated pass rate is \(24/60=0.40\). Their difference of 0.35 compares two different, self-selected groups. Since we did not observe the counterfactual result for any person, that difference need not equal the causal effect, even though monotonicity ensures the effect cannot be negative.
Assumptions
Binary potential outcomes are well-defined and consistent with the observed outcome under the received treatment.
No interference between applicants; the effect is measured for these 100 applicants, not a newly sampled population.
Monotone treatment response holds person by person: \(Y(1)\ge Y(0)\), with no assumption of random or ignorable participation.
Final Answer
Sharp average causal benefit bounds: 0 to 0.66 (0–66 percentage points). The observed pass-rate difference is \(0.75-0.40=0.35\), an association rather than an identified causal effect.