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Estimands: ATE, ATT, ATO and Estimand Shifts After Matching · Tutorial 48 of 699

ATE and ATT Under Treatment-Effect Heterogeneity

Learn how treatment-effect heterogeneity and the treated group’s covariate distribution determine the difference between the ATE and ATT—and why conditional exchangeability is essential to that relationship.

Intermediate 10 min read

What You'll Learn

  • Distinguish the population targeted by the ATE from the population targeted by the ATT.
  • State the assumptions needed to relate the ATT to conditional average treatment effects.
  • Derive the covariance identity connecting the ATE–ATT difference to treatment-effect heterogeneity and the propensity score.
  • Interpret positive and negative ATE–ATT differences in context.
  • Recognize why constant conditional effects do not guarantee equal estimands without conditional exchangeability.
  • Separate a difference in estimands from bias in an estimate.

Why the ATE and ATT Can Differ

A treatment can have different average effects for people with different pretreatment characteristics. If people who receive treatment also tend to have different characteristics from the population as a whole, averaging those effects over the treated group can produce a different answer from averaging over everyone. This is a difference between causal estimands: it does not, by itself, mean that either estimand is biased.

The distinction depends on two things: how treatment effects vary across covariate profiles, and how the covariate distribution among treated units differs from the population distribution. A precise relationship between these quantities requires an important assumption. In particular, the ATT is not generally the average conditional treatment effect among treated units unless conditional exchangeability holds.

Definition: Let \(A\) indicate treatment, with \(A=1\) for treated and \(A=0\) for untreated, and let \(X\) denote pretreatment covariates. Let \(Y(1)\) and \(Y(0)\) be the potential outcomes under treatment and control. The individual treatment effect is \(Y(1)-Y(0)\). The average treatment effect (ATE) is \(E[Y(1)-Y(0)]\), and the average treatment effect on the treated (ATT) is \(E[Y(1)-Y(0)\mid A=1]\). The conditional average treatment effect is \(\tau(X)=E[Y(1)-Y(0)\mid X]\).

The ATE averages individual effects over the target population; the ATT averages them over the people who actually received treatment. The function \(\tau(X)\) describes the mean effect among people with a given covariate profile. By the law of iterated expectations, the ATE is the population average of this function:

$$ \mathrm{ATE}=E[\tau(X)]. $$

That identity is definitional and does not require treatment to be exchangeable given \(X\). The corresponding expression for the ATT does require an assumption: without it, treatment status can still be associated with individual effects within covariate profiles.

Assumptions for the ATE–ATT Relationship

For a causal interpretation of observed outcomes, use consistency and no interference: a unit’s observed outcome is the potential outcome corresponding to its received treatment, and one unit’s treatment does not change another unit’s outcome. To identify population effects from observed data by adjustment for \(X\), also require conditional exchangeability and positivity.

Assumptions: Conditional exchangeability means \((Y(1),Y(0))\perp A\mid X\): within each covariate profile, treatment assignment is independent of both potential outcomes. Positivity for the ATE requires \(0<P(A=1\mid X=x)<1\) for covariate profiles in the target population. For the ATT, control outcomes must be available in covariate profiles represented among treated units; in particular, \(P(A=1\mid X=x)<1\) on those profiles.

Under conditional exchangeability, the treated people with covariate value \(x\) have the same mean treatment effect as all people with that value. Consequently, \(E[Y(1)-Y(0)\mid A=1]=E[\tau(X)\mid A=1]\). This is the step that permits us to express the ATT using the conditional average effect. Positivity supports identification: if a covariate profile has treated people but no controls, observed data cannot supply the counterfactual control mean for that profile without extrapolation.

$$ \mathrm{ATT}=E[\tau(X)\mid A=1] \quad\text{under conditional exchangeability.} $$

Define the propensity score \(e(X)=P(A=1\mid X)\), and let \(\pi=P(A=1)=E[e(X)]\) be the population treatment prevalence. The next identity shows how the difference between the two estimands depends on treatment-effect variation across \(X\) and the relationship between \(\tau(X)\) and \(e(X)\).

Result: Under conditional exchangeability, $$ \mathrm{ATT}-\mathrm{ATE} =\frac{\operatorname{Cov}(\tau(X),e(X))}{\pi}. $$ The covariance is taken over the target population’s distribution of \(X\), not only over treated units.

Deriving the Covariance Identity

Since the ATT is \(E[\tau(X)\mid A=1]\) under conditional exchangeability, write the conditional expectation as a ratio. The law of iterated expectations gives \(E[\tau(X)A]=E[\tau(X)e(X)]\), because \(E[A\mid X]=e(X)\). Also, \(E[A]=\pi=E[e(X)]\).

$$ \begin{aligned} \mathrm{ATT} &=\frac{E[\tau(X)A]}{P(A=1)}\\ &=\frac{E[\tau(X)e(X)]}{\pi}\\ &=\frac{E[\tau(X)]E[e(X)]+\operatorname{Cov}(\tau(X),e(X))}{\pi}\\ &=\mathrm{ATE}+\frac{\operatorname{Cov}(\tau(X),e(X))}{\pi}. \end{aligned} $$

The sign has a direct interpretation. If higher-effect covariate profiles tend to have higher treatment probabilities, the covariance is positive and the ATT exceeds the ATE. If treatment is more common among profiles with lower effects, the covariance is negative and the ATT is below the ATE. If the covariance is zero, the estimands are equal under these assumptions; that can happen even when effects vary, for example if treatment probability and effect are uncorrelated across profiles.

Worked Example: Treated Units Have Higher-Effect Profiles

Consider a hypothetical population split equally between two covariate profiles, \(H\) and \(L\). Suppose the conditional average treatment effects are 4 and 1 outcome units, respectively. The treatment probabilities are 0.8 for \(H\) and 0.2 for \(L\). Assume consistency, no interference, conditional exchangeability, and positivity in both profiles.

The ATE averages the two conditional effects using the population proportions:

$$ \mathrm{ATE}=0.5(4)+0.5(1)=2.5\text{ outcome units}. $$

The probability of treatment is \(\pi=0.5(0.8)+0.5(0.2)=0.5\). Among treated units, the profile proportions are \[ P(H\mid A=1)=\frac{P(H)P(A=1\mid H)}{\pi} =\frac{0.5(0.8)}{0.5}=0.8 \] and \(P(L\mid A=1)=0.2\). Therefore,

$$ \mathrm{ATT}=0.8(4)+0.2(1)=3.4\text{ outcome units}. $$

The difference is \(3.4-2.5=0.9\) outcome units. Check the covariance identity: \(E[\tau(X)]=2.5\), \(E[e(X)]=0.5\), and \(E[\tau(X)e(X)]=0.5(4)(0.8)+0.5(1)(0.2)=1.7\). Thus \(\operatorname{Cov}(\tau(X),e(X))=1.7-(2.5)(0.5)=0.45\), and \(0.45/0.5=0.9\). The treated group has a higher ATT because the higher-effect profile is more likely to receive treatment.

Worked Example: A Negative Difference

In another hypothetical population, half the units have profile \(H\) and half have profile \(L\). Let their conditional average treatment effects be 5 and 1 outcome units, while their treatment probabilities are 0.2 and 0.8. Assume conditional exchangeability, positivity, consistency, and no interference.

The population ATE is \(0.5(5)+0.5(1)=3\) outcome units. The treatment prevalence is \(0.5(0.2)+0.5(0.8)=0.5\). Among treated units, the profile proportions are \(P(H\mid A=1)=0.5(0.2)/0.5=0.2\) and \(P(L\mid A=1)=0.8\). Hence,

$$ \mathrm{ATT}=0.2(5)+0.8(1)=1.8\text{ outcome units}. $$

Here the ATT is \(1.8-3=-1.2\) outcome units below the ATE. The covariance is \(E[\tau(X)e(X)]-E[\tau(X)]E[e(X)]=[0.5(5)(0.2)+0.5(1)(0.8)]-(3)(0.5)=0.9-1.5=-0.6\). Dividing by \(\pi=0.5\) gives \(-1.2\). Treatment is more common in the lower-effect profile, so the treated group’s average effect is lower.

Why Conditional Exchangeability Matters

The covariance identity is not an unconditional fact about the ATE and ATT. If people with different individual effects are selected into treatment even after conditioning on \(X\), then the ATT need not equal \(E[\tau(X)\mid A=1]\). In that case, the ATE is still \(E[\tau(X)]\), but the ATT averages the individual effects among treated people, and the difference cannot generally be summarized by the stated covariance.

Worked Example: Constant \(\tau(X)\) Does Not Ensure Equal Effects

Suppose \(X\) is constant for everyone. In a hypothetical population, half the people have \(Y(0)=0\) and \(Y(1)=0\), while the other half have \(Y(0)=0\) and \(Y(1)=2\). Treatment is assigned exactly to the second half. Thus half the population is treated, and every treated person has an individual effect of 2.

The ATE is \(0.5(0)+0.5(2)=1\) outcome unit, while the ATT is 2 outcome units. Because \(X\) is constant, \(\tau(X)=E[Y(1)-Y(0)\mid X]=1\) for everyone, and \(e(X)=0.5\). The covariance of these two constant functions is zero, which would incorrectly suggest no difference if the covariance identity were used without its assumption.

The missing condition is conditional exchangeability: treatment perfectly selects people with effect 2, so potential outcomes are not independent of treatment given \(X\). Positivity holds because \(e(X)=0.5\), but positivity alone is not enough. This example shows why a constant conditional average effect does not guarantee that the ATE and ATT coincide without conditional exchangeability.

Interpretation, Diagnostics, and Limits

A difference between ATE and ATT can be a real consequence of targeting different populations. It is not automatically evidence of confounding, nor does observing a difference establish treatment-effect heterogeneity. Estimation error, model misspecification, violations of exchangeability, and changes in the population being analyzed can also affect estimated contrasts. State the target population and assumptions before interpreting an estimated effect.

In practice, covariate summaries can reveal whether treated units differ from the broader population, and estimated propensity scores can help describe that difference. Outcome models or other substantive knowledge may inform whether effects plausibly vary across covariate profiles. These checks cannot verify conditional exchangeability, which concerns unobserved potential outcomes. Diagnostics can flag poor overlap or influential extrapolation; they cannot prove that all confounding has been removed.

  • Limited overlap: If treated units occupy profiles rarely or never observed among controls, the ATT may require extrapolating control outcomes. For the ATE, lack of treatment or control observations in parts of the target population raises the same identification problem.
  • Unmeasured effect modifiers: If treatment selection depends on effect-related characteristics not included in \(X\), conditional exchangeability may fail, and the covariance identity need not hold.
  • Weakly supported heterogeneity models: Estimated effect differences can depend on modeling choices when some profiles have few observations. A numerical ATE–ATT gap should not be treated as proof of a particular mechanism.
  • Target-population ambiguity: An estimate for treated units answers a different question from an estimate for the full population. Choosing between them is a scientific decision, not merely a choice of calculation.

The principal alternative to reporting both quantities is to choose one target in advance and use an analysis designed to estimate it. The ATE describes average effects in the defined population; the ATT describes average effects among those treated. Regression adjustment and propensity-score methods can be used for either target, but the target and identifying assumptions must be stated. Matching or weighting does not by itself make the ATE and ATT interchangeable, and changing which units contribute can change the population being represented.

Key takeaway: Under conditional exchangeability, \(\mathrm{ATT}-\mathrm{ATE}=\operatorname{Cov}(\tau(X),e(X))/P(A=1)\). The identity links effect variation to differences in covariate distributions. Without conditional exchangeability, the ATT remains the average individual effect among treated units, but this covariance formula need not describe its difference from the ATE.

Check Your Understanding

Answer each question, then compare with the solutions below.

  1. Under conditional exchangeability, what determines whether the ATT is above or below the ATE in the covariance identity?
  2. A population is split equally between two profiles with effects 3 and 1 outcome units and treatment probabilities 0.6 and 0.2. Calculate the ATE, treatment prevalence, treated profile proportions, and ATT.
  3. Does positivity alone justify writing \(\mathrm{ATT}=E[\tau(X)\mid A=1]\)? Explain.
  4. If \(\tau(X)\) is constant, must the ATE and ATT be equal? State the relevant qualification.
  5. Why is an estimated ATE–ATT difference not, by itself, proof of treatment-effect heterogeneity?

Solutions. (1) The sign of \(\operatorname{Cov}(\tau(X),e(X))\): positive means ATT exceeds ATE; negative means ATT is below ATE. (2) \(\mathrm{ATE}=0.5(3)+0.5(1)=2\) outcome units, and \(\pi=0.5(0.6)+0.5(0.2)=0.4\). The treated profile proportions are \(0.5(0.6)/0.4=0.75\) and \(0.5(0.2)/0.4=0.25\). Thus \(\mathrm{ATT}=0.75(3)+0.25(1)=2.5\) outcome units. (3) No. Positivity concerns the availability of both treatment conditions; conditional exchangeability is needed to equate the treated group’s mean individual effect with its mean \(\tau(X)\). (4) Not in general. Under conditional exchangeability, constant \(\tau(X)\) implies the two averages of \(\tau(X)\) are equal; without exchangeability, treatment can select people with different individual effects within the same \(X\). (5) The gap can also reflect estimation error, model choices, confounding, or a change in target population. Interpretation requires the assumptions and target to be made explicit.

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