Monotone Treatment Selection Bounds

What Monotone Treatment Selection Restricts

Treatment groups may differ even in what their outcomes would have been without treatment. For example, people who choose a program might have had better untreated outcomes, on average, than people who do not choose it. Monotone treatment selection (MTS) formalizes one such directional claim: it orders the mean untreated potential outcome across the two treatment groups. Because the untreated outcome is observed only for the control group, this restriction concerns a partly unobserved comparison.

Let \(D\in\{0,1\}\) indicate treatment, and let \(Y(1)\) and \(Y(0)\) be the potential outcomes under treatment and no treatment. The observed outcome is \(Y=DY(1)+(1-D)Y(0)\). Write \(p=P(D=1)\), and assume \(0<p<1\), so both treatment groups occur in the population. The average treatment effect is \(ATE=E[Y(1)-Y(0)]\).

To obtain finite bounds from this restriction, suppose the outcome is known to lie in a common interval \([L,U]\), where \(L<U\). Thus \(L\le Y(0),Y(1)\le U\) for everyone. This assumption is important: if outcomes are unbounded, the missing potential-outcome means need not have finite bounds. The observed data identify \(\mu_{1T}=E[Y\mid D=1]=E[Y(1)\mid D=1]\) and \(\mu_{0C}=E[Y\mid D=0]=E[Y(0)\mid D=0]\). Define the missing group means \(\alpha=E[Y(0)\mid D=1]\) and \(\beta=E[Y(1)\mid D=0]\). These are not observed directly.

Deriving the Bounds

Split the population into treated and control groups. In the treated group, the mean effect is \(\mu_{1T}-\alpha\); in the control group, it is \(\beta-\mu_{0C}\). Weighting those conditional means by the group proportions gives

The outcome range implies \(L\le\beta\le U\) and \(L\le\alpha\le U\). Positive MTS adds \(\alpha\ge\mu_{0C}\), so \(\mu_{0C}\le\alpha\le U\). Negative MTS instead adds \(\alpha\le\mu_{0C}\), so \(L\le\alpha\le\mu_{0C}\). Since \(p\) and \(1-p\) are positive, the ATE expression decreases as \(\alpha\) increases and increases as \(\beta\) increases. Therefore its minimum uses the largest permitted \(\alpha\) and smallest permitted \(\beta\), while its maximum uses the smallest permitted \(\alpha\) and largest permitted \(\beta\).

These are sharp bounds given the stated information and assumptions: every value in the interval is compatible with some distribution of the unobserved potential outcomes, and both endpoints can be attained. For positive MTS, the lower endpoint sets \(\alpha=U,\ \beta=L\); the upper endpoint sets \(\alpha=\mu_{0C},\ \beta=U\). For negative MTS, the lower endpoint sets \(\alpha=\mu_{0C},\ \beta=L\); the upper endpoint sets \(\alpha=L,\ \beta=U\). Each choice respects the outcome range and the relevant ordering. The unobserved outcomes belong to different treatment groups, so these choices can be made independently without changing the observed outcomes. Intermediate values are attainable by choosing intermediate missing means. This establishes sharpness for the mean restrictions used here.

Assumptions, Diagnostics, and Interpretation

MTS is a substantive assumption about selection, not a pattern that can generally be verified from the observed treatment and outcome data alone. The control group reveals \(\mu_{0C}\), but the comparison mean \(\alpha=E[Y(0)\mid D=1]\) is counterfactual for treated people. Consequently, observing that \(\mu_{1T}\) is above or below \(\mu_{0C}\) does not establish the direction of selection: \(\mu_{1T}\) concerns treated potential outcomes, whereas MTS orders untreated potential outcomes. Confusing these quantities is a common error.

A credible analysis should explain why the proposed ordering is plausible in the application. Subject-matter knowledge, institutional rules, or auxiliary information may support a direction, but the observed difference in outcomes by treatment status is not itself a test of MTS. It is useful to report how the bounds change under the reverse ordering and under no selection restriction. If the conclusions differ substantially, the substantive assumption is consequential and should be made explicit.

Check the inputs before calculating: \(p\) must be between zero and one; both observed group means must lie in the asserted outcome range; and the direction of the inequality must match the stated selection claim. A range that is not genuinely known can yield misleadingly tight bounds. For an outcome with no defensible finite limits, these formulas do not provide finite bounds. MTS also restricts only one missing mean, \(\alpha\); it leaves \(\beta\) bounded only by the outcome scale. This is why the resulting interval can remain wide.

MTS differs from monotone treatment response, which orders a person's two potential outcomes, \(Y(1)\) and \(Y(0)\). MTS instead orders group means of \(Y(0)\) across treatment status. Neither claim implies the other. Worst-case bounds impose no ordering on missing outcomes and are usually wider; MTS can tighten one endpoint or both depending on the direction and the quantities restricted. Unlike a point-identifying approach, MTS does not generally supply a single ATE. Its conclusion is a set of effects compatible with the assumptions and observed means, not a probability distribution over effects.