Monotone Treatment Response Bounds

What Monotone Treatment Response Adds

When an outcome is binary, the treatment effect for a unit can only be \(-1\), \(0\), or \(1\). If treatment is assumed not to lower anyone’s outcome, the value \(-1\) is ruled out for every unit. This is a strong restriction on individual potential outcomes, not a claim that treated units must have better observed outcomes than untreated units. Treatment assignment may still depend on potential outcomes, so the observed treated and untreated groups can differ for many reasons.

Let \(D\in\{0,1\}\) indicate treatment, and let \(Y(1)\) and \(Y(0)\) be a unit’s potential outcomes under treatment and no treatment. The observed outcome is \(Y=DY(1)+(1-D)Y(0)\). Define the population average treatment effect as \(E[Y(1)-Y(0)]\). The goal is to bound this quantity using the observed treatment rate and outcome rates, while allowing treatment assignment to be associated with the potential outcomes.

MTR is the added assumption. Consistency links the observed outcome to the potential outcome under the treatment actually received. No random assignment, exchangeability, or independence between \(D\) and the potential outcomes is assumed here.

Observed Quantities and Potential-Outcome Means

Suppose \(0<P(D=1)<1\), so both treatment groups are present. Write \(p=P(D=1)\), \(q_1=P(Y=1\mid D=1)\), and \(q_0=P(Y=1\mid D=0)\). The observed population masses with outcome one in the two groups are \(p q_1\) and \((1-p)q_0\). These are joint population proportions, rather than conditional outcome rates.

Let \(\mu_1=E[Y(1)]\) and \(\mu_0=E[Y(0)]\). Among treated units, \(Y(1)\) is observed. Under MTR, any treated unit whose observed \(Y(1)=0\) must also have \(Y(0)=0\). Thus the unobserved mass of \(Y(0)=1\) among treated units can range from zero to \(r_1\). Among untreated units, \(Y(0)\) is observed. MTR requires \(Y(1)=1\) whenever \(Y(0)=1\), while units with \(Y(0)=0\) may have either value of \(Y(1)\). These restrictions give the following marginal bounds.

These intervals restrict each potential-outcome mean separately. To bound their difference, it is important to respect which unobserved outcomes belong to which treatment group; subtracting arbitrary endpoints of the two marginal intervals need not give sharp bounds.

Deriving the Sharp Average Treatment Effect Bounds

Let \(x\) be the population mass of treated units whose \(Y(0)=1\). MTR and the observed treated outcomes imply \(0\leq x\leq r_1\). Let \(z\) be the mass of untreated units whose \(Y(1)=1\) and \(Y(0)=0\). The untreated group has mass \(1-p\), of which \(r_0\) has \(Y(0)=1\). Therefore \(0\leq z\leq (1-p)-r_0\). The potential-outcome means can be written as \(\mu_0=r_0+x\) and \(\mu_1=r_1+r_0+z\), so their difference is \(r_1+z-x\).

The smallest value occurs at \(z=0\) and \(x=r_1\), giving zero. The largest occurs at \(z=(1-p)-r_0\) and \(x=0\), giving \(r_1+(1-p)-r_0\). Substituting the observed rates yields the result.

To see why the endpoints are attainable, set \(x=r_1\) and \(z=0\) for the lower endpoint: treated units have \(Y(0)=Y(1)\), and untreated units also have \(Y(1)=Y(0)\). This construction reproduces the observed outcomes and gives zero effect for every unit. For the upper endpoint, set \(x=0\), and set \(Y(1)=1\) for every untreated unit with \(Y(0)=0\). This assigns the largest allowed effect to each group while matching the observed outcomes. The choices are feasible because MTR is imposed within each unit and does not restrict treatment assignment.

Comparison With Worst-Case Bounds

Without MTR, consistency alone gives \(\mu_1\in[r_1,r_1+(1-p)]\) and \(\mu_0\in[r_0,r_0+p]\). Taking the least favorable combinations gives the usual worst-case effect bounds:

Under MTR, the upper endpoint is unchanged, but the lower endpoint becomes zero. The upper endpoint already corresponds to an MTR-compatible construction: treated units can have \(Y(0)=0\), while untreated units with \(Y(0)=0\) can have \(Y(1)=1\). The new restriction rules out negative unit effects and hence removes negative values from the worst-case interval. In contrast, assuming exchangeability or random assignment can identify the average effect from observed group means, but those assumptions address treatment assignment and are not being made here.

Diagnostics, Interpretation, and Limitations

MTR is a substantive, unit-level assumption. For an outcome such as survival, it says treatment cannot cause any individual who would otherwise survive to die. For a higher-is-better test result, it says no individual’s result can be lowered by treatment. Whether such a claim is credible depends on the treatment, outcome, time horizon, and population. If the outcome is coded so that larger values are worse, the inequality must be reversed or the outcome recoded before applying the formulas.

The assumption concerns both potential outcomes for the same unit, but only one is observed. Therefore, the observed data generally cannot verify MTR. A treated unit’s observed outcome of zero does not reveal that unit’s untreated outcome; an untreated unit’s observed outcome of one does not reveal the treated outcome. Observed rates with \(q_1<q_0\) do not by themselves refute MTR, as the preceding example illustrates. Diagnostics should instead examine whether the treatment mechanism and outcome definition make individual-level non-harm plausible, and whether the claim is credible in relevant subgroups. A plausible average benefit is not enough to establish that no individual can be harmed.

The bounds are population statements. If \(p\), \(q_1\), and \(q_0\) are estimated from a sample, substituting sample proportions gives estimated endpoints, not exact population bounds. Sampling uncertainty requires separate inference for the endpoints or the identified set. In addition, if treatment or outcome definitions are inconsistent, the link between observed and potential outcomes may fail. If \(p=0\) or \(p=1\), one of the conditional rates \(q_0,q_1\) is unavailable; the formulas should then be expressed using observed joint masses and the unobserved group’s outcomes will generally leave wider bounds.

A common mistake is to read MTR as saying that observed treated outcomes must be at least as good as observed untreated outcomes. That comparison mixes different units and does not follow from \(Y(1)\geq Y(0)\) unit by unit. The correction is to apply the restriction only to each unit’s pair of potential outcomes, use consistency for the outcome actually observed, and leave treatment assignment unrestricted. This distinction is why MTR can tighten bounds without turning an observed association into a causal effect.