Combining Monotone Response And Selection

Why Combine the Two Restrictions?

When treatment is not randomly assigned, the observed outcome difference between treated and untreated units need not equal the average treatment effect. Bounds describe a set of causal effects compatible with the observed data and stated assumptions. Monotone treatment response restricts how treatment can affect each unit; monotone treatment selection restricts how the potential outcomes of treated and untreated units compare. Used together, these restrictions can tighten the bounds, but they can also be incompatible with the observed group means.

Consider a binary treatment \(D\), with \(D=1\) for treated units, and a binary outcome \(Y\). Let \(Y_1\) and \(Y_0\) be the potential outcomes under treatment and no treatment. The observed outcome is \(Y=DY_1+(1-D)Y_0\). Assume consistency and no interference, so the observed outcome for each unit equals its potential outcome under the treatment actually received, and one unit's treatment does not alter another unit's outcome. The target is the population average treatment effect, \(E[Y_1-Y_0]\).

For binary outcomes, MTR says treatment cannot change an individual outcome from 1 to 0. MTS is a pair of comparisons of conditional means; it does not say that every treated unit has higher potential outcomes than every untreated unit. The two restrictions concern different comparisons: MTR compares potential outcomes within a unit, while MTS compares groups under the same potential treatment state.

Observed and Missing Potential-Outcome Means

Let \(q=P(D=1)\), and suppose \(0<q<1\), so both treatment groups occur in the population. Write \(a=E[Y\mid D=1]\) and \(b=E[Y\mid D=0]\). By consistency, \(a=E[Y_1\mid D=1]\) and \(b=E[Y_0\mid D=0]\) are observed. Define the unobserved group means \(x=E[Y_0\mid D=1]\) and \(y=E[Y_1\mid D=0]\). Because the outcome is binary, all four means lie between 0 and 1.

Partitioning the population into treated and untreated groups gives an expression for the target in terms of those observed and missing means:

Here \(a-x\) is the average effect among treated units, and \(y-b\) is the average effect among untreated units. The population effect is their treatment-prevalence-weighted average. The key task is to restrict \(x\) and \(y\) using the assumptions.

How Each Assumption Restricts the Endpoints

Under MTR, treated units have \(Y_1\geq Y_0\), so their group means satisfy \(x\leq a\). Untreated units also have \(Y_1\geq Y_0\), so \(y\geq b\). Binary outcomes add \(x\geq0\) and \(y\leq1\). Thus MTR alone permits \(0\leq x\leq a\) and \(b\leq y\leq1\). Substituting the largest and smallest allowable values into the expression for \(\tau\) gives the MTR bounds.

Under MTS, the restriction for \(Y_0\) is \(x\geq b\), and the restriction for \(Y_1\) is \(a\geq y\). With the binary-outcome limits, this gives \(b\leq x\leq1\) and \(0\leq y\leq a\). MTS alone therefore gives:

The lower endpoint uses the largest allowed \(x\) and smallest allowed \(y\); the upper endpoint uses the smallest allowed \(x\) and largest allowed \(y\). Without MTR, MTS alone does not require either group’s treatment effect to be nonnegative.

When both assumptions hold, their restrictions intersect: \(b\leq x\leq a\) and \(b\leq y\leq a\). In particular, compatibility requires \(a\geq b\). The smallest possible effect sets \(x=a\) and \(y=b\); the largest sets \(x=b\) and \(y=a\). This yields the combined bounds:

Why These Bounds Are Sharp

An interval is sharp when every value in it is compatible with the observed data and assumptions, and no value outside it is compatible. To see why the combined endpoints are attainable, first take \(x=a\) and \(y=b\). In both groups, the two potential-outcome means are then equal. Binary potential outcomes can be coupled so that \(Y_1=Y_0\) for each unit in each group. MTR holds, MTS holds because both group means for each treatment state are ordered equally when \(a\geq b\), and the resulting average effect is 0.

For the upper endpoint, take \(x=b\) and \(y=a\). In each group, the mean under treatment is \(a\) and the mean without treatment is \(b\). A valid binary construction assigns potential outcomes \((Y_0,Y_1)=(1,1)\) to a fraction \(b\), \((0,1)\) to a fraction \(a-b\), and \((0,0)\) to a fraction \(1-a\). These nonnegative fractions sum to 1 when \(0\leq b\leq a\leq1\). This construction satisfies MTR and gives group means \(b\) and \(a\). MTS holds with equal group means under each treatment state, and the average effect is \(a-b\).

Mixing these endpoint constructions in any proportion preserves the observed means and assumptions and produces every intermediate average effect. Thus the combined interval is not merely a conservative envelope: it is the full set of effects allowed by these restrictions.

Diagnostics, Interpretation, and Limitations

A practical check is to calculate the observed group means before reporting any combined interval. If \(a<b\), inspect the outcome coding, treatment-group definitions, and data summaries. If those are correct, the population versions of MTR and MTS cannot both hold for these means. If \(a\geq b\), that necessary compatibility check passes, but it does not establish that either substantive assumption is credible.

The bounds are identification bounds, not a point estimate and not a confidence interval. An interval such as \([0,0.30]\) says that the stated population assumptions and observed means permit average effects from 0 to 0.30; it does not say that the effect is equally likely to be any value in that interval. In data analysis, estimated means are subject to sampling uncertainty, which must be handled separately.

The restrictions are also deliberately substantive. MTR rules out treatment harm for every unit; MTS asserts favorable selection on both potential outcomes. Either may be implausible in some applications. Violations can make the resulting interval uninformative or misleading, and passing the compatibility check cannot verify the assumptions. Without these restrictions, the worst-case bounds for a binary outcome are generally wider. With only one restriction, the bounds may also be wider, as the examples demonstrate. The central advantage of combining the assumptions is tighter endpoints when both are defensible and jointly compatible.