Combining Stratum-Specific Effects for an ATE

Why Stratum Effects Need to Be Combined

Propensity-score subclassification divides a population into strata, then compares treated and untreated outcomes within each stratum. Those comparisons answer a stratum-specific question. To obtain an average treatment effect (ATE) for a target population, the stratum effects must then be combined in a way that reflects how much of that target population belongs to each stratum.

The key is the target population’s composition—not an automatic equal average across strata. If one stratum contains 70% of the target population and another contains 30%, their effects contribute in those proportions. This remains true when the stratum-specific effects differ: the combined ATE averages the differences, not the treatment and control means across strata without regard to their membership.

When the target population is the study sample and that sample represents the target, the usual estimated share is \(\hat q_k=n_k/n\), where \(n_k\) is the number of sample members in stratum \(k\), and \(n\) is the total sample size. If target-population shares are known from an external source, use those shares instead. The choice of \(q_k\) is part of defining the estimand; it is not merely a computational detail.

Identification and the Estimator

A stratum-weighted difference in observed means has a causal interpretation only under conditions that connect observed outcomes to potential outcomes. Let \(Y\) be the observed outcome. The consistency condition says \(Y=Y(A)\): a person’s observed outcome equals the potential outcome under the treatment actually received. Assume also that treatment assignment is conditionally exchangeable within each stratum, \(\{Y(1),Y(0)\}\perp A\mid S\), and that treatment positivity holds: \(0<P(A=1\mid S=k)<1\) for every stratum with positive target share. These conditions, together with an appropriate target-population distribution of \(S\), identify the ATE by stratum-specific comparisons.

For propensity-score strata, conditional exchangeability within a stratum is a substantive requirement, not something guaranteed simply by dividing the score into bins. Stratification may make covariate distributions more comparable, but a coarse grouping can leave residual differences within strata. The earlier material on subclassification provides the setting for forming such groups; here, the central task is to combine their effects correctly.

The arithmetic has two stages: estimate a treated-versus-control contrast separately within each stratum, then average those contrasts using the target shares. A stratum’s share is not its treatment-group proportion, nor the proportion treated within it. The same share \(q_k\) weights that stratum’s estimated treatment effect.

Why the Decomposition Holds

The decomposition follows by conditioning the population average treatment effect on the stratum. First partition the population according to \(S\); the expectation of a variable equals the sum of its conditional expectations multiplied by the probabilities of the corresponding groups. Apply that identity to the individual treatment effect \(Y(1)-Y(0)\):

To see how the observed-data contrast identifies each \(\tau_k\), consider either treatment level \(a\in\{0,1\}\). Conditional exchangeability implies \(E[Y(a)\mid A=a,S=k]=E[Y(a)\mid S=k]\). By consistency, \(Y=Y(a)\) for units with \(A=a\), so \(E[Y\mid A=a,S=k]=E[Y(a)\mid S=k]\). Subtracting the result for \(a=0\) from that for \(a=1\) gives

Positivity ensures both conditional observed means are defined in every stratum that contributes to the target ATE. In a finite sample, \(\hat\tau_k\) estimates that contrast, and the weighted estimator substitutes estimated or known target shares and sample stratum contrasts into the population decomposition.

Target Shares Are Not Always Sample Shares

Using \(n_k/n\) is appropriate when the study sample represents the target population’s stratum distribution. But a sample can deliberately include different numbers of people from different strata—for example, to ensure enough observations in a relatively uncommon group. In that case, the sample shares may not describe the target population. If the target shares are known, retain them when combining the stratum effects. Otherwise, the result describes a different mixture of strata.

Diagnostics and Failure Modes

Before combining estimates, check whether each stratum supplies a credible treated-versus-control comparison and whether the weights correspond to the intended target. These checks do not prove the causal assumptions, but they can reveal when the calculation is unsupported or answers the wrong question.

• Check both treatment groups within every contributing stratum. If a stratum has no treated or no control observations, its observed contrast cannot be estimated. A very small group can also make its estimated mean highly unstable.
• Inspect overlap and covariate balance within strata. Compare the distributions of important pre-treatment covariates between treated and control units within each stratum. Persistent imbalances warn that the stratum may not adequately control confounding. Positivity requires support for both treatment levels; a score bin alone does not ensure it.
• Verify the target shares. Confirm that the \(q_k\) values sum to one and describe the intended population. If they are estimated from the study sample, consider whether the sample composition represents that population.
• Review the stratum-level estimates before aggregation. A large overall estimate may be driven by a stratum with a large effect, a large target share, or both. Report the component effects and shares so the aggregate can be understood.

When treatment positivity fails, do not insert a zero effect for the missing contrast. Nor does a weighted sum repair an unobserved stratum effect. The target ATE is not identified by these within-stratum comparisons unless additional defensible assumptions or information are introduced. Restricting the target population to strata with adequate support changes the estimand and should be stated explicitly.

How This Aggregation Relates to Other Approaches

The stratum-weighted estimator is a form of standardization: it uses within-stratum outcome contrasts and averages them over a specified distribution of strata. A pooled comparison that ignores strata generally does not make the same adjustment, and it need not target the same quantity. Inverse-probability weighting instead uses treatment-probability weights to construct a weighted outcome contrast; it is a different estimator, although under appropriate identification conditions it can target the same ATE. Agreement between methods is not guaranteed in finite samples, especially when overlap, model specification, or target-population representation differs.

Equal weighting of strata is not inherently invalid, but it answers a different question unless the target population gives every stratum equal share. Similarly, using sample shares when the sample was drawn with a different stratum mix changes the population being averaged over. Always identify the target population first, then use its stratum shares.

Common Mistake