Why Covariates Can Tighten Worst-Case Bounds

Why Conditioning Can Help Without Guaranteeing a Narrower Interval

Observed covariates divide a population into groups with potentially different treatment rates and outcome distributions. An analysis that uses those groups retains information that a pooled analysis discards. The central result is a set-inclusion result: when the conditional identified sets are the actual feasible sets, their population-weighted aggregate is contained in the corresponding pooled identified set. Thus conditioning cannot make the identified set wider under the stated comparison.

The result does not promise strict tightening. With unrestricted binary potential outcomes and a common support of zero to one, the usual worst-case bounds for the average treatment effect can be identical before and after stratification. Nor does the result apply automatically to any intervals someone labels “valid bounds.” A loose outer interval may contain values that no data-compatible model can produce.

For each stratum, let \(C_z\) be the sharp identified set for the conditional effect \(E[Y(1)-Y(0)\mid Z=z]\), using the observed data and assumptions specified for that stratum. If the model imposes no restrictions linking potential outcomes across different strata, the conditional identified set for the population effect is the weighted sum of those sets:

If \(C_z=[L_z,U_z]\), this is the interval \([\sum_z p_zL_z,\sum_z p_zU_z]\). This endpoint calculation relies on the fact that the stratum-specific choices can be jointly realized: each collection of feasible stratum models can be combined into a population model by assigning stratum \(z\) its population weight \(p_z\). If the model includes cross-stratum restrictions, the feasible choices may not be independent; then the global conditional identified set must be derived under those restrictions rather than obtained from this simple weighted sum.

The Inclusion Result

Define the corresponding pooled model by using the marginal observed distribution of \(D\) and \(Y\), rather than their distribution within each \(Z\) stratum. It must retain the common outcome support and assumptions used in the comparison, as well as the pooled restrictions implied by the conditional model. Let \(C_{\mathrm{pool}}\) be its sharp identified set. The pooled model may lose the allocation of observed treatment and outcome patterns across strata, but it cannot contradict the pooled data generated by a conditional model.

The theorem compares identified sets, not merely reported intervals. It also requires a coherent comparison: if the conditional analysis uses restrictions that the pooled model is allowed to discard, then the pooled set may be wider precisely because it is based on fewer restrictions. The inclusion remains valid, but one should state which information and assumptions the pooled model retains.

The proof uses feasibility, not an assumption that every numerical value between two reported endpoints is compatible with the data. When the conditional sets are intervals, sharpness ensures that their endpoints are attained or can be approached arbitrarily closely under the model. An arbitrary outer interval does not provide that guarantee.

Worked Binary Bounds and a Case With No Tightening

For binary outcomes, suppose the only restrictions are consistency and \(Y(1),Y(0)\in\{0,1\}\). There is no assumption that treatment is independent of either potential outcome. In stratum \(z\), write \(q_z=P(D=1\mid Z=z)\), \(m_{1z}=E[Y\mid D=1,Z=z]\), and \(m_{0z}=E[Y\mid D=0,Z=z]\). The observed treated outcomes identify the treated group’s mean \(Y(1)\), while the untreated group’s missing \(Y(1)\) mean can be anywhere from zero to one. The reverse applies to \(Y(0)\).

With no restriction linking the missing potential outcomes, the sharp conditional effect interval is \([q_zm_{1z}-(1-q_z)m_{0z}-q_z,\ q_zm_{1z}+1-q_z-(1-q_z)m_{0z}]\). The lower endpoint pairs the smallest feasible \(E[Y(1)\mid z]\) with the largest feasible \(E[Y(0)\mid z]\); the upper endpoint pairs the largest feasible first mean with the smallest feasible second mean.

When Inclusion Is Strict, and Why Sharpness Matters

The set result allows strict tightening when the conditional data and assumptions rule out some models that remain feasible after pooling. At the level of set arithmetic, suppose two equally weighted strata have actual sharp effect sets \(C_1=[0,0.1]\) and \(C_2=[0.2,0.3]\). Their aggregate is \([0.1,0.2]\). If the pooled model permits effects throughout \([-0.1,0.4]\), then the conditional set is strictly narrower and is contained in the pooled set. The reason is not the act of averaging by itself: it is that the conditional analysis has excluded pooled-feasible values by using information that pooling does not preserve.

A crucial limitation follows. “Valid bounds” sometimes means only that the interval contains the true parameter. Such an interval may be wider than the sharp identified set. For example, consider one stratum in which everyone is treated, \(Y=0\) for everyone, and both potential outcomes are binary. The observed data fix \(Y(1)=0\); the unobserved \(Y(0)\) mean can range from zero to one. Therefore the sharp ATE set is \([-1,0]\). The interval \([-1,1]\) contains every feasible value, so it is a valid but loose outer bound. It is not contained in the sharp pooled set.

A practical report should say whether an interval is sharp or is a conservative outer bound. If only outer bounds are available, aggregation can still give a valid outer bound when its construction is justified, but the resulting interval need not be contained in another sharp identified set. This distinction is especially important when comparing analyses that use different information or assumptions.

Diagnostics, Alternatives, and Common Pitfalls

Before interpreting tighter conditional bounds, check that the comparison is well defined. The strata must partition the target population, the weights \(p_z\) must refer to that target population, and each conditional set must use the observed data and assumptions claimed for its stratum. If a stratum has zero population weight, it contributes nothing to the population effect; if its conditional effect is undefined because the target is conditional on an event with zero probability, do not assign it an arbitrary effect and multiply by zero.

• Check the estimand. Conditional effects weighted by population stratum shares target the population ATE. Weighting by treated-group shares or another distribution targets a different quantity.
• Check compatibility. A conditional model that cannot be combined with the other strata under a cross-stratum assumption cannot be aggregated by independently choosing endpoints.
• Check what pooling retains. The pooled model must use the same target, outcome support, and common restrictions. State whether it retains the pooled implications of conditional restrictions.
• Check sharpness. If conditional intervals are outer bounds rather than exact feasible sets, do not infer set inclusion from their endpoints alone.
• Check for equality. For unrestricted binary-outcome ATE bounds, stratification can leave the interval unchanged, as the numerical example showed. A narrower reported interval is not guaranteed merely because covariates were measured.

The main alternative is to pool immediately and calculate worst-case bounds from marginal treatment and outcome information. This is simpler, but it may discard conditional restrictions. A conditional analysis preserves the stratum-specific observed distributions and assumptions, then aggregates effects using target-population weights. The trade-off is added modeling and data demands: sparse strata can yield wide or unstable estimates in a sample, while the population identified-set argument itself is not a sampling-uncertainty calculation. Neither approach substitutes for stating the assumptions that define the feasible models.