A Complete Overlap-Weighting Calculation

Why Use Overlap Weights?

Inverse probability weighting can target different populations by changing the weights assigned to treated and untreated individuals. Overlap weights focus on individuals whose measured baseline characteristics make either treatment plausible. They give less influence to individuals whose propensity scores are near zero or one, where the observed treatment group offers little information about what would have happened under the alternative treatment.

The resulting effect is not automatically the average treatment effect (ATE) for everyone, or the average treatment effect for the treated (ATT). It is an effect for a particular target population: one that emphasizes covariate profiles with greater treatment overlap. The distinction matters because effect estimates can differ across target populations, especially when treatment effects vary across individuals.

Because \(0\leq e(X)\leq1\), the overlap function is largest when \(e(X)=0.5\), and it approaches zero as the score approaches either extreme. The weights are bounded between zero and one: treated individuals receive \(1-e(X)\), while untreated individuals receive \(e(X)\). In the target population, the estimand is the mean difference in potential outcomes, \(E_h[Y(1)-Y(0)]\), where the subscript \(h\) indicates averaging over the overlap covariate distribution.

Assumptions and the Weighted Estimator

The causal interpretation requires the usual assumptions for identifying treatment effects from observational data. Treatment and outcome definitions must be consistent: an individual's observed outcome equals the potential outcome under the treatment actually received. There must be no interference between individuals, so one person's treatment does not change another person's outcome. Conditional exchangeability requires \(\{Y(1),Y(0)\}\perp A\mid X\): after conditioning on the measured pretreatment covariates, treatment assignment is independent of the potential outcomes. This assumption is substantive and cannot be established from the observed data alone.

The propensity score must also be correctly specified or consistently estimated, unless it is known by design. For covariate profiles included in the overlap target, both treatment options must be represented: \(0<e(X)<1\). Profiles with a score exactly zero or one have \(h(X)=0\) and contribute no mass to the overlap target. This does not recover their treatment effects; it defines a target that excludes them.

For observed data on \(n\) individuals, the overlap-weighted means are Hájek ratio estimators: each weighted outcome sum is divided by the sum of weights in that treatment group. The estimated treatment-effect contrast is the treated weighted mean minus the untreated weighted mean.

Here \(\hat e(X_i)\) is the estimated propensity score for individual \(i\). The denominators normalize the weighted outcomes within their treatment groups. In a finite sample, the weighted covariate distributions in the two groups need not match perfectly, so the estimate is not guaranteed to equal a simple average of individual treatment effects in a mechanically constructed sample. Its target-population interpretation follows from the weighting scheme under the assumptions above.

Why These Weights Target the Overlap Population

The weighting scheme has a direct justification. Write \(f(x)\) for the covariate distribution in the original population. Among treated individuals, the covariate density is proportional to \(e(x)f(x)\). Multiplying by the treated overlap weight \(1-e(x)\) gives a density proportional to \(e(x)\{1-e(x)\}f(x)\). Among untreated individuals, the covariate density is proportional to \(\{1-e(x)\}f(x)\); multiplying by their weight \(e(x)\) produces the same target density.

The proportionality constants are removed when each group is normalized to calculate its weighted mean. Therefore, in the population, both weighted groups represent the same overlap covariate distribution. Under consistency and conditional exchangeability, their outcome mean difference identifies the average causal effect in that population.

With an unpenalized logistic propensity model that includes an intercept and has converged, overlap weights also have a useful finite-sample balance property for covariate terms included in the model. The logistic score equations give \(\sum_i(A_i-\hat e(X_i))g(X_i)=0\) for each included model term \(g(X_i)\). Since

the weighted totals of each such term agree between the treatment groups. The intercept equation also makes the two total overlap weights equal. This exact balance applies to the model terms under those fitting conditions; it does not guarantee balance on omitted, misspecified, or otherwise unrepresented covariate features.

Diagnostics, Alternatives, and Limitations

After calculating weights, inspect whether they were formed from pretreatment covariates and whether the estimated propensity model is plausible. Check weighted covariate balance, including important nonlinear terms or interactions when relevant. For logistic-model overlap weights, exact balance on fitted terms can be a useful check, but it is not evidence that unmeasured confounding is absent. Also inspect the weight distribution and the amount of weighted information in each treatment group.

One compact measure of weight concentration is the effective sample size, \(ESS=(\sum_i w_i)^2/\sum_i w_i^2\), calculated separately for each treatment group. It equals the actual sample size when all weights are equal and decreases when a few observations dominate. It is a diagnostic, not a substitute for a variance estimate or a guarantee of adequate precision.

Overlap weighting differs from two common alternatives in its target and its handling of extreme scores. ATE weighting uses \(1/e(X)\) for treated individuals and \(1/\{1-e(X)\}\) for untreated individuals to target the original population. ATT weighting gives treated individuals weight 1 and untreated individuals weight \(e(X)/\{1-e(X)\}\), targeting the treated population. Those alternatives answer different questions and can assign very large weights near propensity-score extremes. Overlap weights are bounded, but bounded weights do not fix confounding, model misspecification, or a lack of treatment information for excluded profiles. Matching and subclassification are other approaches with their own target and balance considerations; they should not be treated as interchangeable calculations of the same estimand.

A common mistake is to call the overlap-weighted contrast “the ATE” merely because both treatment groups are weighted. The correction is to identify the target before interpreting the contrast: overlap weights define a population emphasizing covariate profiles where either treatment was plausible. A second mistake is to treat a low weight as proof that an individual was safely “adjusted away.” The estimate is instead about the overlap target, and it may not answer a decision-maker's question about the full population or the treated group.

If propensity scores are estimated, uncertainty in their estimation should be accounted for in inference; treating fitted scores as known can understate uncertainty. The weighting calculation also cannot establish that all confounders were measured. Report the target, model and balance checks, estimate and uncertainty in context, and limitations. A sound balance diagnostic supports the calculation but does not prove its causal assumptions.