Why Define an Overlap Target?
The average treatment effect (ATE) targets the full population of interest. But in observational data, some units may have characteristics that make one treatment almost certain. For those units, the data may contain little information about what would have happened under the less likely treatment. The average treatment effect in the overlap population (ATO) answers a different question: what is the average effect for units whose measured characteristics make either treatment a plausible possibility?
The ATO does not simply discard every unit with an extreme treatment probability. Instead, it gives more weight to covariate profiles where treatment assignment is less predictable and less weight to profiles where it is nearly certain. Its target is therefore related to, but distinct from, the full population, the treated population, and the control population.
The factor \(e(X)\{1-e(X)\}\) is largest at \(e(X)=0.5\) and approaches zero as \(e(X)\) approaches 0 or 1. Thus, profiles compatible with either treatment contribute more to the ATO than profiles strongly associated with one treatment. “Overlap population” describes this weighting target; it is not necessarily a group defined by a single cutoff on the propensity score.
Interpretation and Identification
The ATO is an average of individual causal effects, weighted toward covariate profiles where both treatment options occur. It is not automatically the effect for every person in the study, nor is it necessarily the effect for a clinically eligible subgroup chosen in advance. The target depends on the covariate distribution and propensity score in the population being studied.
To interpret an estimate causally, the usual conditions for identifying effects from observational data are needed. Consistency requires that a unit’s observed outcome equal its potential outcome under the treatment actually received. Conditional exchangeability requires \(\{Y(1),Y(0)\}\) to be independent of \(A\) given \(X\). Positivity requires both treatment probabilities to be nonzero for covariate profiles in the target population. The propensity score must also be correctly specified or estimated adequately for the weighting procedure being used. These conditions are not guaranteed by using overlap weights: in particular, weighting measured covariates cannot remove confounding by unmeasured factors.
Worked Example: ATO Across Three Covariate Profiles
Suppose a hypothetical population has three covariate profiles. Their population shares, propensity scores, and profile-specific average treatment effects are shown below. Assume, for this calculation, that the treatment effect is constant within each profile.
| Profile | Population share | Propensity score | Average treatment effect |
|---|---|---|---|
| Low treatment propensity | 0.40 | 0.10 | 2 outcome units |
| Intermediate propensity | 0.40 | 0.50 | 5 outcome units |
| High treatment propensity | 0.20 | 0.90 | 8 outcome units |
First calculate \(h(X)=e(X)\{1-e(X)\}\): the three profile weights are \(0.10(0.90)=0.09\), \(0.50(0.50)=0.25\), and \(0.90(0.10)=0.09\). The population-share-weighted denominator is
The corresponding numerator, using the profile-specific effects, is
Therefore, \(\tau_{\mathrm{ATO}}=0.716/0.154=4.649\) outcome units. For comparison, the unweighted ATE for this hypothetical population is \(0.40(2)+0.40(5)+0.20(8)=4.4\) outcome units. The difference occurs because the ATO gives the intermediate-propensity profile more influence. Neither result is a calculation error: they average effects over different target populations.
Why the Weighting Targets Overlap
The target distribution can be understood by looking at the covariate distribution among treated and control units after weighting. Let \(f(x)\) be the population density of \(X\). Among treated units, the covariate density is proportional to \(f(x)e(x)\); weighting each treated unit by \(1-e(x)\) makes the resulting density proportional to \(f(x)e(x)\{1-e(x)\}\). Among controls, the density is proportional to \(f(x)\{1-e(x)\}\); weighting each control by \(e(x)\) produces the same proportional density.
After normalization, both treatment groups therefore represent the same covariate distribution: the original distribution reweighted by \(e(x)\{1-e(x)\}\). If exchangeability and consistency hold, their weighted outcome contrast identifies the ATO. This justification also explains the common overlap weights: \(1-e(X)\) for treated units and \(e(X)\) for controls.
Worked Example: Calculating an Overlap-Weighted Contrast
Consider a hypothetical sample with 100 units in each of three propensity-score strata. The table gives the number treated and control, the stratum propensity score, and the observed mean outcome in each treatment group. Assume conditional exchangeability within strata and that these stratum means identify the corresponding potential-outcome means.
| Propensity score | Treated / control | Treated mean | Control mean |
|---|---|---|---|
| 0.20 | 20 / 80 | 12 | 9 |
| 0.50 | 50 / 50 | 15 | 10 |
| 0.80 | 80 / 20 | 20 | 14 |
For the treated group, multiply each treated count by \(1-e\). The resulting weights are \(20(0.80)=16\), \(50(0.50)=25\), and \(80(0.20)=16\). For controls, multiply each control count by \(e\): \(80(0.20)=16\), \(50(0.50)=25\), and \(20(0.80)=16\). Each group has total weight 57.
The weighted treated mean is
The weighted control mean is
The estimated overlap-weighted contrast is \(15.56-10.84=4.719\) outcome units. The calculation gives greater influence to the middle stratum than to either extreme stratum. A causal interpretation still depends on the stated identification assumptions; the arithmetic alone does not establish them.
How the ATO Differs From Other Average Effects
The ATE, average treatment effect on the treated (ATT), and average treatment effect on the controls (ATC) differ in which population they average over. The ATO uses a fourth target: a population tilted toward covariate profiles with treatment overlap. This distinction matters because treatment effects may vary across profiles, so changing the target can change the numerical effect and the population to which it applies.
| Effect | Target population | Covariate weighting idea |
|---|---|---|
| ATE | Full population | Original covariate distribution |
| ATT | Units who received treatment | Covariate distribution tilted toward \(e(X)\) |
| ATC | Units who received control | Covariate distribution tilted toward \(1-e(X)\) |
| ATO | Units with treatment overlap | Covariate distribution tilted toward \(e(X)\{1-e(X)\}\) |
Overlap weighting is one way to estimate an effect for the overlap target. It is not a relabeling of the ATE or ATT, and it does not make those other estimands interchangeable. A restriction to a chosen subgroup or common-support region can also define a target, but that target is determined by the restriction rule rather than by the continuous overlap weighting described here.
Diagnostics, Limitations, and a Common Mistake
Before interpreting an ATO estimate, examine the propensity-score distributions by treatment group and assess covariate balance after weighting. The weighted groups should represent similar measured covariate distributions. Inspect whether the estimated propensity scores are plausible and whether conclusions are unduly sensitive to reasonable changes in the propensity model. If a covariate remains imbalanced, or if treatment assignment is nearly deterministic in important regions, the intended comparison may not be well supported.
Overlap weights are bounded between zero and one, which avoids the very large individual weights that can arise with inverse-probability weights when estimated propensity scores are near zero or one. This is a stability advantage, not a guarantee of validity. Poor propensity-model specification can still leave imbalance; a small effective weighted sample can still make estimates imprecise; and no measured-data diagnostic can verify exchangeability with respect to unmeasured confounders. Standard errors and confidence intervals should reflect the estimation procedure and, where relevant, clustering or other dependence in the data.
- Common mistake: “Overlap weighting removes low- and high-propensity units, so its estimate is the ATE among everyone who remains.” This treats a continuous weighting scheme as if it were a hard inclusion rule.
- Correction: The ATO averages effects under a covariate distribution reweighted by \(e(X)\{1-e(X)\}\). Extreme-propensity profiles receive less influence, but are not necessarily removed. The ATO is not generally the full-population ATE.
- Another limitation: The overlap population depends on the treatment mechanism and measured covariates. Its clinical or policy relevance should be explained rather than assumed.
Worked Example: Individual Overlap Weights
In a small hypothetical illustration, two treated units have propensity scores 0.20 and 0.80 and outcomes 8 and 16. Two control units have the same respective propensity scores and outcomes 5 and 11. Apply weights \(1-e\) to treated units and \(e\) to controls.
The treated weighted mean is \(\{0.80(8)+0.20(16)\}/(0.80+0.20)=9.6\) outcome units. The control weighted mean is \(\{0.20(5)+0.80(11)\}/(0.20+0.80)=9.8\) outcome units. The illustrative weighted contrast is \(9.6-9.8=-0.2\) outcome units.
This tiny sample is not adequate for a dependable causal estimate; it only shows how the weights operate. The two units with propensity 0.20 and 0.80 receive smaller weights within their respective treatment groups than units with propensity 0.50 would receive. In an actual analysis, one would assess balance, uncertainty, sample size, and the causal assumptions before drawing a conclusion.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(e(X)=0.10\), what is \(h(X)\), and how does its influence compare with a profile having \(e(X)=0.50\)?
- In a hypothetical population, 60% have propensity score 0.50 and effect 4 units; 40% have propensity score 0.10 and effect 10 units. Calculate the ATO.
- Does overlap weighting alone establish conditional exchangeability? Why or why not?
- Explain why the ATO need not equal the ATE even when both are calculated from the same population.
Solutions. (1) \(h=0.10(0.90)=0.09\), compared with \(0.50(0.50)=0.25\), so the first profile receives less target weight. (2) The denominator is \(0.60(0.25)+0.40(0.09)=0.186\); the numerator is \(0.60(0.25)(4)+0.40(0.09)(10)=0.96\). Thus the ATO is \(0.96/0.186=5.161\) outcome units. (3) No. Weighting measured covariates cannot guarantee the absence of unmeasured confounding. (4) The ATO reweights covariate profiles by \(e(X)\{1-e(X)\}\), while the ATE uses the original population distribution; if effects vary across profiles, the averages can differ.