Average Treatment Effects on the Treated

• ATT is defined as the average causal effect on treated units, expressed as E[Y(1) - Y(0) | Z=1].
• Inverse probability weighting can yield unstable estimates due to extreme weights when there is poor overlap between treated and control groups.
• Balancing weights improve covariate balance and reduce bias and variance, providing more reliable inference for ATT estimation.

Average treatment effects on the treated (ATT) quantify the average causal impact of an intervention among units that actually received the treatment. Formally, when potential outcomes are defined in the Neyman–Rubin framework, the ATT is expressed as

$\text{ATT} = \mathbb{E}[Y(1) - Y(0) \mid Z=1],$

where $Y(1)$ and $Y(0)$ denote the potential outcomes under treatment and control, respectively, and $Z=1$ indicates treatment status. In settings with observational data and especially when there is poor overlap between treated and control groups, estimation of the ATT is challenging because standard methods, like inverse probability weighting (IPW), may yield extreme weights, leading to unstable and biased estimates. Recent methodological advancements have introduced alternative weighting schemes—most notably balancing weights—that directly target covariate balance among control units to obtain more reliable inference even when overlap is limited.

1. Conceptual Framework and Definitions

The ATT focuses exclusively on the subgroup of units that received the treatment. Unlike the average treatment effect (ATE), which is defined over the entire population, ATT is defined via conditioning on treatment:

$\text{ATT} = \mathbb{E}[Y(1) - Y(0) \mid Z=1].$

This conditional estimand is of central importance in policy evaluation and other settings where the effect on the treated is most relevant. Identification of the ATT requires that treated and control units be comparable on observed covariates—a requirement often formalized as the positivity (or overlap) assumption. When this assumption is violated—meaning some treated observations have few or no similar controls—standard estimators may suffer from extreme inverse probability weights and consequent high variances or bias.

2. Inverse Probability Weighting and the Overlap Challenge

Inverse probability weighting (IPW) is a widely used method in causal inference. In the context of ATT estimation, the IPW estimator takes the form

$$\hat{\Delta}_{IPW, ATT} = \frac{1}{n_1}\sum_{i:Z_i=1} Y_i - \frac{\sum_{i:Z_i=0} Y_i \cdot \frac{\hat{e}(X_i)}{1-\hat{e}(X_i)}}{\sum_{i:Z_i=0} \frac{\hat{e}(X_i)}{1-\hat{e}(X_i)}},$$

where $\hat{e}(X_i) = P(Z_i=1\mid X_i)$ is the estimated propensity score and $n_1$ is the number of treated units. When treated and control groups overlap well, IPW can recover the ATT; however, if some controls have $\hat e(X)$ values near 1 (reflecting poor overlap with the treated), the weights $\hat{e}(X_i)/[1-\hat{e}(X_i)]$ can become exceedingly large. This instability leads to inflated variances and potential biases, rendering IPW-based inference unreliable in these settings.

3. Balancing Weights as an Alternative Approach

Balancing weights address the shortcomings of IPW by directly targeting the balance of covariate distributions between treated units and reweighted controls. Rather than relying on a correctly specified or well-estimated propensity score model to provide weights, balancing weights are constructed via constrained or penalized optimization. They minimize a loss function (such as a variance or entropy penalty) subject to the constraint that the weighted control covariate moments match those observed in the treated sample. In compact form, a typical formulation is

$$\min_{w_i \geq 0} \sum_{i:Z_i=0} f(w_i) \quad \text{subject to} \quad \sum_{i:Z_i=0} w_i X_i = \frac{1}{n_1} \sum_{i:Z_i=1} X_i, \quad \sum_{i:Z_i=0} w_i = 1.$$

This approach retains the familiar causal estimand—the ATT—while offering several advantages over IPW:

• It directly enforces covariate balance, which can lead to bounded weights even when overlap is poor.
• It reduces bias arising from model misspecification, as the weighting procedure is less reliant on the propensity score’s functional form.
• It provides a flexible framework where weight regularization can be tuned to trade off bias and variance.

For ATT estimation via balancing weights, the estimator is defined as

$$\hat{\Delta}_{BW, ATT} = \frac{1}{n_1} \sum_{i:Z_i=1} Y_i - \sum_{i:Z_i=0} w_i Y_i,$$

with the weights $w_i$ chosen to satisfy the balancing constraints.

4. Empirical Evidence and Simulation Studies

Simulation studies have been conducted to compare IPW, overlap weights, and balancing weights under various scenarios of covariate overlap and treatment effect heterogeneity. These studies typically demonstrate that:

• Under poor overlap, the IPW estimator exhibits high bias and large root mean squared error (RMSE) as extreme weights dominate the estimation.
• Overlap weights, which target the population with the greatest covariate overlap (often corresponding to the average treatment effect for the overlap population, ATO), are more stable; however, their estimand is less familiar to practitioners than the ATT.
• Balancing weights, by contrast, maintain low bias and low variance across a broad range of overlap scenarios, thereby effectively recovering the ATT. In many cases, balancing weights achieve similar point estimates to overlap weights but allow interpretation in terms of the ATT—specifically, the effect on those treated.

For example, in an empirical paper of right heart catheterization, balancing weights achieved near-perfect covariate balance and a higher bias reduction (100% vs. 81% for the IPW approach). Simulation results consistently show that—even when overlap is poor—balancing weights provide stable estimation of the ATT with narrower confidence intervals and improved inferential properties.

A summary comparison is provided in the following table:

Estimator	Target	Sensitivity to Overlap	Bias/Variance Under Poor Overlap
IPW	ATT (or ATE)	High	High bias/variance
Balancing Weights	ATT	Moderate	Low bias/variance (unless extremely poor)
Overlap Weights	ATO	Low	Low bias/variance

5. Practical Implications for Causal Inference

For researchers conducting causal analyses in observational studies—particularly when the overlap between treated and control units is limited—the choice of estimator has critical implications. Balancing weights offer a robust and interpretable means to target the ATT even when standard IPW fails because of extreme weights. A recommended workflow is as follows:

1. Assess covariate balance and the degree of overlap between treated and control groups.
2. If overlap is less than ideal and IPW weights are extreme, implement balancing weights via convex optimization to directly target the covariate balance.
3. Evaluate the performance of the chosen estimator using simulation studies or sensitivity analyses, and report covariate balance diagnostics together with confidence intervals.
4. In scenarios of extreme non-overlap where no weighting scheme achieves sufficient covariate similarity, consider redefining the target estimand toward the overlap population (ATO) while clearly reporting the associated interpretability trade-offs.

The key advantage is that balancing weights preserve the familiar causal estimand (the ATT) while accommodating practical limitations in empirical data, thereby enhancing the scientific relevance of the results.

6. Conclusion

Average treatment effects on the treated remain a central causal estimand in observational studies. The challenges posed by poor overlap and extreme IPW weights have motivated the development of balancing weights, which directly enforce covariate balance and yield more reliable and interpretable estimates. Balancing weights not only reduce instability by bounding weight variability but also allow for more efficient inference when treatment effect heterogeneity is present. As demonstrated through extensive simulation studies and empirical applications, incorporating balancing weights into ATT estimation expands the range of applied settings in which causal interpretations remain valid. The methodological advancements now allow practitioners to obtain sharper confidence intervals and improved statistical power without requiring additional assumptions beyond standard unconfoundedness.

This framework is implemented in the R package estCI (Ben-Michael et al., 2022), providing researchers with directly applicable tools for robust ATT estimation even under challenging overlap conditions.