Conditional Bounds Within Covariate Strata

Why Calculate Bounds Within Strata?

Treatment and outcomes can have different distributions across groups defined by measured covariates. A single calculation that ignores those groups may conceal how much the data leave unknown within each one. Conditional worst-case bounds make that uncertainty explicit: within each stratum, they use the treatment prevalence and the outcomes observed in each treatment arm, while leaving unobserved potential outcomes unrestricted except for a known outcome range.

The target here is a treatment effect within a stratum, not an effect obtained by combining strata. Let \(D\) indicate treatment, with \(D=1\) for treated and \(D=0\) for untreated. Let \(Y(1)\) and \(Y(0)\) be the potential outcomes under treatment and no treatment. For a covariate stratum \(X=x\), the conditional average treatment effect is \(\tau(x)=E[Y(1)-Y(0)\mid X=x]\). The observed outcome is \(Y=DY(1)+(1-D)Y(0)\), which states consistency: each person’s observed outcome is the potential outcome corresponding to their actual treatment.

These are identification bounds: under the stated information and assumptions, they describe a range of possible values for the conditional effect. They are not a confidence interval, and they do not rely on treatment being as-if randomly assigned within a stratum. The assumptions are consistency, a correctly specified outcome range applying to both potential outcomes, and a well-defined stratum with positive probability. No restriction is imposed on how treatment assignment relates to potential outcomes within that stratum.

Deriving the Conditional Bounds

Within stratum \(x\), the observed treated outcomes reveal \(Y(1)\) for the proportion \(p_x\), but not for the untreated proportion \(1-p_x\). The observed untreated outcomes reveal \(Y(0)\) for proportion \(1-p_x\), but not for the treated proportion \(p_x\). Because the unobserved potential outcomes are only known to lie between \(L\) and \(U\), their means can be bounded by those endpoints.

To obtain a lower bound for the difference \(E[Y(1)\mid X=x]-E[Y(0)\mid X=x]\), use the smallest allowed treated-potential-outcome mean and the largest allowed untreated-potential-outcome mean. For the upper bound, use the largest allowed treated-potential-outcome mean and the smallest allowed untreated-potential-outcome mean. Subtracting gives the formula in the definition.

The endpoints are sharp under these assumptions. For the lower endpoint, assign \(Y(1)=L\) to untreated individuals, whose treated potential outcomes are missing, and assign \(Y(0)=U\) to treated individuals, whose untreated potential outcomes are missing. This preserves every observed outcome and satisfies the outcome limits, while attaining both adverse extremes simultaneously. Reversing these assignments attains the upper endpoint. Intermediate choices of the missing potential-outcome means yield intermediate treatment-effect values. Thus the bounds do not merely contain the possible effects: without additional information or assumptions, their endpoints cannot be improved.

Using Observed Outcome Distributions

The conditional means in the formula come from the observed outcome distributions within each treatment arm and stratum. For a binary outcome, each mean is the observed proportion with outcome 1. For a bounded quantitative outcome, it is the average of the recorded values. The full observed distributions can be useful for checking data quality and the assumed range, but for these mean-effect bounds the distribution enters through its mean; no other feature of that observed distribution tightens the bounds without an additional assumption.

When a Treatment Arm Is Missing

The displayed formula uses both \(\mu_{1x}\) and \(\mu_{0x}\), so it cannot be applied literally if one treatment arm is absent from a stratum. The logic still applies, but the observed potential outcome for the absent arm cannot be estimated from that stratum’s data. If \(p_x=0\), then every individual in the stratum is untreated, and \(E[Y(0)\mid X=x]=\mu_{0x}\) is observed. The treated potential outcome mean is only known to lie in \([L,U]\), so the effect lies in \([L-\mu_{0x},\,U-\mu_{0x}]\). If \(p_x=1\), the effect instead lies in \([\mu_{1x}-U,\,\mu_{1x}-L]\).

Diagnostics, Comparisons, and Limitations

Before interpreting the results, check that the strata and outcome scale correspond to the target question. The limits \(L\) and \(U\) must be defensible for both potential outcomes, not merely for the values that happen to appear in the observed data. If a binary outcome is coded as 0 and 1, those are its limits; for a quantitative outcome, limits should come from the outcome’s meaningful or prespecified scale, not be chosen after seeing the results to produce narrower bounds.

• Check the strata and treatment prevalence. Confirm that every reported \(p_x\) is calculated among the individuals in that stratum. If \(p_x=0\) or \(p_x=1\), use the one-arm logic above rather than inserting a nonexistent observed mean into the two-arm formula.
• Check observed means against the support. Each observed mean must lie in \([L,U]\). A value outside the range can indicate a coding error, a mismatch in units, or an invalid outcome limit.
• Check sparse cells. A mean based on very few treated or untreated observations may be highly variable. The plug-in bounds calculated from sample means do not account for this sampling uncertainty.
• Keep identification separate from estimation. The formulas describe population bounds. Substituting sample prevalences and sample means gives estimated endpoints; uncertainty in those estimates requires separate statistical inference.

A point-identifying analysis based on conditional exchangeability, for example, can produce a more specific causal estimate, but it does so by adding assumptions not used here. Conversely, these worst-case bounds avoid such assumptions but may be wide. Comparing the observed treated and untreated means alone is simpler still, but it does not account for the unobserved potential outcomes and is not, by itself, a causal effect. The strength of the bounds is their transparency about what the data and outcome limits alone establish; their limitation is that they can leave a broad range of effects possible.

A common mistake is to pool observations across strata first, calculate one overall treatment prevalence and two overall outcome means, and then call the resulting interval a set of within-stratum bounds. That calculation uses a different population mixture and does not answer the conditional question. Another mistake is to treat the observed outcome distributions as if they revealed outcomes under the treatment not received. They do not: consistency reveals only the factual potential outcome for each person. Conditional bounds preserve that distinction and apply the support limits to the missing outcomes within each stratum.