What ATT Weighting Estimates
The average treatment effect among the treated (ATT) asks how outcomes for the people who received treatment compare with the outcomes those same people would have had without treatment. The latter outcomes are not observed, so an analysis uses untreated people to estimate that missing counterfactual. ATT weighting does so by reweighting untreated observations to resemble the treated population in measured covariates.
For a binary treatment, define the propensity score \(e(X_i)=P(T_i=1\mid X_i)\), where \(X_i\) is the set of measured pre-treatment covariates. ATT weights leave treated observations at weight 1 and weight untreated observations by the propensity-score odds, \(e(X_i)/(1-e(X_i))\). These odds give greater weight to untreated people whose measured covariates make them more like the treated.
These are not the ATE weights. For the ATT, treated outcomes are averaged over the treated group without reweighting; only the untreated group is reweighted to serve as the treated group's comparison. The estimand is therefore not the average effect for the whole population unless the treated population and whole population happen to have the same average effect.
Assumptions and the Weighted-Mean Formula
The calculation is meaningful as a causal estimate under assumptions. First, consistency requires that each person's observed outcome equals the potential outcome under the treatment actually received. The treatment must be defined clearly enough that “receiving treatment” corresponds to the potential outcome being discussed. Second, conditional exchangeability requires that, given the measured pre-treatment covariates \(X\), treatment assignment is independent of the potential outcomes. In particular, no unmeasured confounder may remain that jointly predicts treatment and outcome after conditioning on \(X\). Third, positivity for the ATT requires that treated people’s covariate patterns also have a chance of appearing among untreated people: \(P(T=0\mid X=x)>0\) for covariate values represented in the treated population.
The propensity-score model must also be suitable for estimating treatment probabilities given the measured covariates. A misspecified model can produce weights that fail to balance those covariates. The assumptions about exchangeability and correct measurement cannot be verified from the weights alone; balance checks assess measured covariates, not the absence of unmeasured confounding.
With estimated scores, the usual normalized weighted means (also called Hájek means) are
The ATT estimate is \(\hat\tau_{\mathrm{ATT}}=\hat\mu_1^{\mathrm{ATT}}-\hat\mu_0^{\mathrm{ATT}}\). The first mean is the observed outcome mean among treated people. The second is a weighted mean among untreated people; its denominator normalizes their weights so they sum to one. Using unnormalized weights in both numerator and denominator gives the same weighted mean.
Why Propensity-Score Odds Reweight Controls
The weighting choice follows from comparing the covariate distribution among treated people with that among untreated people. Conditional on \(X\), the probability of treatment is \(e(X)\), while the probability of no treatment is \(1-e(X)\). Weighting the untreated group by their odds multiplies their conditional representation by exactly the factor needed to match the treated group.
This identity holds for any function \(f(X)\) for which the expectations exist. Taking \(f(X)=1\) shows that the expected total weighted control mass matches the expected treated mass. Taking \(f(X)\) to be a covariate, or a function of covariates, shows why the weighted controls reproduce treated-group covariate averages in expectation. In a finite sample, the match need not be exact.
Under consistency and conditional exchangeability, outcomes among untreated people with covariates \(X\) provide information about \(Y(0)\) for treated people with those same covariates. The odds weighting aligns the untreated covariate distribution with the treated distribution; consequently, the weighted untreated outcome mean targets \(E[Y(0)\mid T=1]\). Subtracting it from the treated outcome mean targets the ATT. This argument depends on both the causal assumptions and adequate score estimation, not merely on applying a formula.
Worked Calculation
Worked Example: Calculating an ATT From Individual Data
Consider a hypothetical dataset with estimated propensity scores and observed outcomes shown below. The outcome is a symptom score, for which higher values indicate more symptoms. The covariate patterns fall into two groups, with estimated scores of 0.80 and 0.50. Assume consistency, conditional exchangeability given the measured covariate group, positivity for the treated groups, and an adequate propensity-score model for this illustration.
| Person | Treatment \(T_i\) | \(\hat e(X_i)\) | Observed outcome \(Y_i\) | ATT weight |
|---|---|---|---|---|
| A | 1 | 0.80 | 10 | 1 |
| B | 1 | 0.80 | 12 | 1 |
| C | 1 | 0.80 | 9 | 1 |
| D | 1 | 0.80 | 11 | 1 |
| E | 1 | 0.50 | 8 | 1 |
| F | 1 | 0.50 | 10 | 1 |
| G | 0 | 0.80 | 7 | 4 |
| H | 0 | 0.50 | 5 | 1 |
| I | 0 | 0.50 | 7 | 1 |
For a treated person, the weight is 1. For person G, who is untreated with score 0.80, the weight is \(0.80/(1-0.80)=0.80/0.20=4\). For each untreated person with score 0.50, the weight is \(0.50/0.50=1\). These scores are below 1, as required for untreated observations to have positive probability under the stated positivity condition.
The treated outcome mean is
For the untreated group, the weighted outcome total is \(4(7)+1(5)+1(7)=40\), and the total weight is \(4+1+1=6\). Therefore,
The estimated ATT is \(10-6.667=3.333\) symptom-score points. Under the stated assumptions, this estimates the mean effect of treatment for the treated population: their mean symptom score is estimated to be 3.333 points higher under treatment than it would have been without treatment. This is an estimate, not a claim that every treated person’s score increased by that amount.
Worked Example: Seeing Why the Control Weights Matter
In another hypothetical example, two treated people have estimated scores of 0.75 and observed outcomes of 10 and 14. Two untreated people have scores and outcomes \((0.75,4)\) and \((0.50,8)\). Assume the causal identification conditions hold for these measured covariates and the scores are adequately estimated.
The treated weights are both 1, so the treated mean is \((10+14)/2=12\). The first untreated person receives weight \(0.75/(1-0.75)=3\); the second receives weight \(0.50/(1-0.50)=1\). Thus,
The untreated outcome of 4 contributes three times as much weighted mass as the outcome of 8 because its score indicates greater similarity to the treated covariate profile. If both controls were instead given weight 1, their mean would be 6, not 5, and the resulting difference would be 6 rather than 7. Equal weighting would target the observed untreated group’s composition, not the treated group’s composition.
Diagnostics, Failure Modes, and Alternatives
ATT weights can become large when an untreated person has an estimated propensity score close to 1. Such a person is unusual among controls but resembles people who were very likely to be treated. The formula assigns that person a large weight, which can make the weighted control mean sensitive to one or a few outcomes. A score of exactly 1 for a treated covariate pattern violates the required positivity condition: there are no untreated observations at that pattern to represent the treated counterfactual.
After calculating weights, inspect their distribution and maximum, and check whether weighted controls resemble treated people in measured pre-treatment covariates. Also check whether the estimate changes substantially when influential observations are examined. A common descriptive diagnostic is the weighted control effective sample size,
This quantity equals the actual control sample size when all control weights are equal and is smaller when weights are unequal. It signals how concentrated the weighted information is; it is not a replacement for a variance estimator or a guarantee that the model is valid.
Worked Example: Detecting a Concentrated Control Estimate
Suppose a hypothetical analysis has three treated outcomes of 10, 12, and 11, with estimated scores 0.50, 0.90, and 0.90. Three untreated outcomes are 6 at score 0.50, 5 at score 0.90, and 8 at score 0.50. Assume the propensity model and causal assumptions are appropriate, while recognizing that the large weight warrants scrutiny.
The treated mean is \((10+12+11)/3=11\). The untreated weights are \(1\), \(0.90/0.10=9\), and \(1\). Therefore,
The control effective sample size is \((1+9+1)^2/(1^2+9^2+1^2)=121/83\approx1.458\), although there are three controls. The score-0.90 control contributes \(9/11\) of the weighted control mass. The estimate is therefore highly sensitive to that person’s outcome. This does not prove that the estimate is wrong, but it is a warning to examine score-model specification, covariate balance, and the observations carrying the largest weights. Simply truncating a large weight changes the weighting scheme and may change the target; it does not automatically repair positivity or confounding.
ATT weighting is one option among several. ATE weighting targets the whole population and weights both treatment groups differently; it therefore answers a different question. Subclassification and direct standardization combine outcome information across covariate strata rather than using the individual control odds weights shown here. Matching is another approach to constructing a comparison group. These alternatives can differ in finite samples and in the population to which their estimates apply. The ATT calculation is most useful when the treated population itself is the target and the untreated observations can credibly represent its untreated counterfactual.
A common mistake is to use \(1/(1-\hat e_i)\) for untreated observations because it appears in an ATE weighting formula, then describe the result as an ATT. That weight is not the ATT control weight. The correction is to use \(\hat e_i/(1-\hat e_i)\) for controls and 1 for treated observations, then calculate the normalized weighted control mean. A second mistake is to interpret good measured-covariate balance as proof of no confounding. Balance is useful evidence about the variables measured and modeled; it cannot establish exchangeability with respect to unmeasured factors.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A treated person has estimated propensity score 0.60. What is the ATT weight? A control has the same score. What is that person's ATT weight?
- There are two treated outcomes, 8 and 12. The control outcomes are 4 and 10, with ATT weights 3 and 1. Calculate the ATT estimate.
- What population does the ATT describe, and why are treated observations assigned weight 1?
- A control has an estimated propensity score of 0.98. What is its ATT weight, and what diagnostic concern does this raise?
- Does close balance on measured covariates prove that conditional exchangeability holds? Explain briefly.
Solutions. 1. The treated person's weight is 1. The control's weight is \(0.60/(1-0.60)=1.5\). 2. The treated mean is \((8+12)/2=10\); the weighted control mean is \((3(4)+1(10))/(3+1)=22/4=5.5\). The ATT estimate is \(10-5.5=4.5\) outcome units. 3. It describes the population that received treatment. Treated outcomes already represent that target population, so they are averaged without reweighting. 4. Its weight is \(0.98/0.02=49\). This is a large, potentially influential weight; check weight concentration, covariate balance, and positivity. 5. No. Balance assesses measured covariates, but cannot rule out unmeasured confounding or establish conditional exchangeability.