Deriving Bounds Under Monotone Treatment Response

From a Monotonicity Assumption to ATE Bounds

The introductory material on Monotone Treatment Response Bounds introduces the idea that treatment does not move anyone’s outcome in the harmful direction. Here the goal is to derive exactly what that restriction implies for the average treatment effect when outcomes are binary, and to check the restriction separately among treated and untreated units.

Let \(D\) indicate treatment, with \(D=1\) for treated and \(D=0\) for untreated. Let \(Y(1)\) and \(Y(0)\) be the binary potential outcomes under treatment and no treatment, respectively, and let the observed outcome be \(Y=DY(1)+(1-D)Y(0)\). The population average treatment effect is \(E[Y(1)-Y(0)]\). The observed data reveal only one potential outcome for each unit.

The assumption is about each unit’s response, not merely about a positive average effect. Its meaning depends on outcome coding: \(Y=1\) must represent the direction treated as no worse under treatment. If a larger value means a worse health outcome, the proposed MTR direction may be inappropriate unless the outcome is recoded.

What Each Treatment Group Can Tell Us

Write \(q=P(D=1)\), \(p_1=P(Y=1\mid D=1)\), and \(p_0=P(Y=1\mid D=0)\), when the corresponding treatment groups occur in the population. These are observed quantities: \(p_1\) is the treated-group outcome rate and \(p_0\) is the untreated-group outcome rate. MTR restricts the unobserved potential outcome differently in each group.

Let \(a=E[Y(0)\mid D=1]\) be the treated group’s mean untreated potential outcome, and let \(b=E[Y(1)\mid D=0]\) be the untreated group’s mean treated potential outcome. In the treated group, a unit with observed \(Y(1)=0\) must have \(Y(0)=0\); only treated units with \(Y(1)=1\), a fraction \(p_1\), can have \(Y(0)=1\). Thus \(0\leq a\leq p_1\). In the untreated group, every unit with \(Y(0)=1\), a fraction \(p_0\), must also have \(Y(1)=1\), while the remaining units may or may not have \(Y(1)=1\). Thus \(p_0\leq b\leq 1\).

These are also feasibility checks. For any observed binary outcome rate between 0 and 1 in either group, the required potential-outcome assignments are possible: set \(Y(0)=0\) for treated units with \(Y(1)=0\), and set \(Y(1)=1\) for untreated units with \(Y(0)=1\). The other missing outcomes can be chosen within the allowed ranges. Because only one potential outcome is observed per unit, the observed group rates alone cannot reveal whether the individual-level restriction is true.

Deriving and Establishing Sharpness

Partition the population into treated and untreated units. The average effect in the treated group is \(p_1-a\), and the average effect in the untreated group is \(b-p_0\). Weighting these conditional effects by the proportions in each group gives

Since \(a\leq p_1\) and \(b\geq p_0\), each group’s average effect is nonnegative. The smallest overall effect is therefore 0. For the largest effect, choose the smallest feasible \(a\) and largest feasible \(b\): \(a=0\) and \(b=1\). This gives the bounds

These bounds are sharp: neither endpoint can be improved using only the observed rates and MTR. To attain the lower endpoint, assign \(Y(0)=Y(1)\) to every unit: in the treated group set the missing \(Y(0)\) equal to observed \(Y(1)\), and in the untreated group set the missing \(Y(1)\) equal to observed \(Y(0)\). To attain the upper endpoint, set \(Y(0)=0\) for every treated unit and \(Y(1)=1\) for every untreated unit. Both constructions respect MTR and preserve the observed outcomes.

Intermediate effect values are feasible as well. Among treated units with observed \(Y(1)=1\), assign \(Y(0)=1\) to a suitable fraction to vary \(a\) from 0 to \(p_1\). Among untreated units with observed \(Y(0)=0\), assign \(Y(1)=1\) to a suitable fraction to vary \(b\) from \(p_0\) to 1. These choices can be made independently across the two groups, so the identified set is the full interval between the bounds, not just a pair of attainable endpoints.

Comparison, Diagnostics, and Limitations

Without MTR, the usual worst-case binary-outcome bounds allow the missing potential outcomes to take either value without a directional restriction. Their lower endpoint is \(q(p_1-1)-(1-q)p_0\), and their upper endpoint is \(q p_1+(1-q)(1-p_0)\). MTR changes the lower endpoint to 0 because it rules out negative individual effects. In this binary-outcome setting, it leaves the upper endpoint unchanged: the worst-case upper-bound assignment already sets treated units’ \(Y(0)\) to 0 and untreated units’ \(Y(1)\) to 1, which satisfies MTR.

A practical check is to verify that \(q,p_1,p_0\) are defined for the observed groups and lie between 0 and 1. With both treatment groups present, the group-by-group assignments above establish compatibility of any such rates with MTR. If \(q=0\), \(p_1\) is unobserved and the bound is \([0,1-p_0]\); if \(q=1\), \(p_0\) is unobserved and the bound is \([0,p_1]\). These formulas follow by applying MTR to the only observed group, rather than treating an undefined rate as observed.

The principal failure risk is substantive rather than arithmetic: MTR may be implausible if treatment can harm some units. The data generally cannot test individual-level MTR because both potential outcomes are never observed for the same unit. Also, MTR is stronger than assuming only that the average effect is nonnegative. The bounds describe uncertainty remaining under the stated restriction; they are not a confidence interval and do not account for sampling error.

• Do not compare group rates as if they were individual responses. The treated and untreated groups may differ in who receives treatment, so \(p_1<p_0\) does not by itself refute MTR.
• Do not confuse an endpoint with a point estimate. The upper bound is attainable, but the observed data and MTR do not establish that the ATE equals it.
• Check outcome direction. If treatment can make the event coded 1 more likely in a harmful way, the chosen monotonicity direction may be unsuitable.