Choosing Between ATE, ATT, ATU, and Overlap Weights

One Treatment, Different Target Populations

A treatment-effect estimate is incomplete unless it states which population the effect describes. The average treatment effect (ATE) concerns the full population; the average treatment effect on the treated (ATT) concerns those who received treatment; the average treatment effect on the untreated (ATU) concerns those who did not; and the overlap effect concerns a population weighted toward people whose treatment status is comparatively uncertain. These are different causal questions, not merely alternative formulas for the same target.

Propensity-score weighting makes the distinction visible. Changing the weighting scheme changes the covariate distribution represented by the weighted sample, and therefore can change the effect when treatment effects vary across covariate profiles. The introductory material on inverse probability weighting and overlap weights develops the individual estimators; here the focus is how their targets and weights compare.

A Common Form for the Four Weighting Schemes

A useful way to compare these estimands is to define a nonnegative tilting function \(h(e)\). It determines how much each covariate profile contributes to the target population. The target distribution is the original covariate distribution reweighted in proportion to \(h\), and normalized to sum to one. If \(h\) depends on \(X\) only through \(e(X)\), the target share of a profile with original population share \(q\) is proportional to \(q h(e)\).

The corresponding treatment-arm weights are \(h\{e(X)\}/e(X)\) for treated units and \(h\{e(X)\}/[1-e(X)]\) for control units. The reason is that, within a covariate profile, the treated fraction is \(e(X)\) and the control fraction is \(1-e(X)\). Multiplying each fraction by its arm-specific weight gives the same tilted contribution \(h\{e(X)\}\) in both arms.

Thus, the two weighted arms represent the same target covariate distribution. Under the identification assumptions stated below, their weighted outcome means estimate \(E_h[Y(1)]\) and \(E_h[Y(0)]\), and their difference estimates the causal effect in that target population.

These formulas describe the target as well as the weights. For example, the ATT tilting function \(h(e)=e\) gives profiles more representation when they are more common among treated units. The ATU tilt \(1-e\) instead follows the untreated covariate distribution. The overlap tilt is largest at \(e=0.5\) and approaches zero as \(e\) approaches either boundary.

Assumptions for a Causal Interpretation

Weights alone do not establish that a weighted contrast is causal. For the target effect to be identified from observed outcomes, the standard conditions are needed:

• Consistency: the observed outcome is \(Y=Y(A)\), so a person’s observed outcome equals the potential outcome under the treatment actually received. The treatment and outcome must be defined clearly enough for this statement to be meaningful.
• Conditional exchangeability: \(\{Y(1),Y(0)\}\) is independent of \(A\) conditional on \(X\). In context, measured covariates must suffice to control confounding for the comparison being made.
• Positivity on the target support: where \(h\{e(X)\}>0\), both treatment probabilities must be positive for the target effect to be identified by these arm comparisons. For the ATE, this requires \(0<e(X)<1\) throughout the population of interest. The ATT and ATU have their own target support, but still require outcome information from both arms wherever that target assigns positive mass.

In practice, the propensity score is estimated, not known. The formulas then yield estimated weights, and their usefulness depends on propensity-model adequacy and the observed covariate balance. Weighting cannot remove unmeasured confounding, and it cannot create data in covariate regions where one treatment arm is absent.

How the Arm Weights Produce Those Shares

For the same hypothetical profiles, the weights are obtained by substituting each propensity score into the arm-specific formulas. The table below shows one unit’s weight in each arm for each profile. These individual weights need not equal the target-population shares; it is the weighted distribution of each arm, after accounting for how many units of each profile occur in that arm, that represents the target.

Effect Heterogeneity Can Change the Answer

When effects differ across profiles, different target populations generally have different average effects. This is not an inconsistency: each estimand answers a different question. If the effect is constant for everyone, the four population averages coincide, assuming they are identified. With heterogeneity, they need not.

Choosing and Diagnosing an Estimand

The choice should start with the question, then be checked against the data. If the aim is the effect for the full eligible population, ATE weighting is conceptually aligned, but can be unstable when some propensities are near zero or one. If the question concerns people who received treatment, ATT targets their covariate distribution; if it concerns those who remained untreated, ATU targets theirs. Overlap weighting instead describes a population with emphasis on profiles represented in both arms. It can reduce the influence of extreme propensities, but it does not estimate the full-population effect unless additional conditions make the targets coincide.

Useful diagnostics include examining propensity-score distributions by treatment arm, checking covariate balance after weighting, and assessing weight concentration. A common summary is the effective sample size, \(ESS=(\sum_i w_i)^2/\sum_i w_i^2\), calculated separately by arm where appropriate. A small effective sample size signals that a few observations carry much of the weighted information. For ATE weights, propensities near either boundary can create large weights; ATT control weights can be large when \(e\) is near one, and ATU treated weights can be large when \(e\) is near zero. Overlap weights are bounded between zero and one for both arms, but they deliberately downweight boundary profiles and change the target population.

A failure to balance measured covariates after weighting may indicate propensity-model misspecification, insufficient overlap, or an implementation error. Balance diagnostics do not prove exchangeability, which concerns potential outcomes and cannot be verified from observed data alone. Trimming or changing the target to improve stability may be reasonable in some applications, but it changes the population being described and must be made explicit.

Common Mistake

The main alternatives are not interchangeable shortcuts. ATE, ATT, ATU, and overlap weighting each define a target population; matching or outcome-model standardization may address related causal questions but also require explicit target definitions and their own assumptions. The practical comparison is therefore not “which weights are universally best?” but “which population is relevant, and can the available data support a credible comparison for it?”