The Average Treatment Effect on the Controls

Whose Treatment Effect Does the ATC Describe?

A causal effect is not fully specified by saying what treatment is compared with what control condition. The target population matters too. The average treatment effect on the controls, or ATC, asks: among the units that actually received control, what would their average outcome difference have been if they had instead received treatment? It is a question about a particular group, not an effect that automatically applies to everyone.

This target differs from the ATT, which averages effects among treated units. If treatment effects vary across people, or treatment assignment is associated with characteristics that predict different effects, the ATC and ATT can differ. Even when the two are numerically equal, they remain distinct estimands because they refer to different groups.

Definition: Let $T_i$ equal 1 if unit $i$ receives treatment and 0 if it receives control. Let $Y_i(1)$ and $Y_i(0)$ denote that unit’s potential outcomes under treatment and control. The individual treatment effect is $\tau_i=Y_i(1)-Y_i(0)$. The average treatment effect on the controls is $$ \operatorname{ATC}=E\big[Y(1)-Y(0)\mid T=0\big]. $$ It is the mean individual treatment effect in the population of units with $T=0$. By contrast, $\operatorname{ATT}=E\big[Y(1)-Y(0)\mid T=1\big]$, which averages effects among treated units.

Only one potential outcome is observed for each unit: $Y_i=Y_i(T_i)$. Thus, for a control unit, $Y_i(0)$ is observed but $Y_i(1)$ is not. The ATC includes the unobserved treated outcome those control units would have had. The difference between the observed mean outcome of treated and control units is not generally the ATC: the groups may differ before treatment, and the observed difference mixes outcome differences with treatment effects.

The Target Population in a Small Example

Worked Example: Computing ATC and ATT From Potential Outcomes

Consider a hypothetical finite population of six units. Outcomes are measured in points on a scale where a larger value is better. The table shows both potential outcomes for teaching purposes; in an actual study, both outcomes would not be observed for the same unit.

Unit	T	Y(0)	Y(1)	Effect Y(1) − Y(0)
A	1	40	48	8
B	1	50	60	10
C	1	55	61	6
D	0	45	47	2
E	0	52	56	4
F	0	58	64	6

The controls are D, E, and F. Their treatment effects are 2, 4, and 6 points, so $$ \operatorname{ATC}=\frac{2+4+6}{3}=4\text{ points}. $$ The treated units are A, B, and C. Their effects are 8, 10, and 6 points, giving $$ \operatorname{ATT}=\frac{8+10+6}{3}=8\text{ points}. $$ The two averages differ because this hypothetical population has treatment-effect heterogeneity, and the two groups contain units with different effects. The ATC answers a question about the controls; it is not a second estimate of the ATT.

The observed treated mean is $(48+60+61)/3=56.33$ points, while the observed control mean is $(45+52+58)/3=51.67$ points. Their difference is 4.667 points, which is neither the ATC of 4 points nor the ATT of 8 points. The observed difference compares treated units’ $Y(1)$ with control units’ $Y(0)$; it does not compare the two potential outcomes within either group.

Identification: Connecting the Estimand to Observed Data

Defining the ATC does not by itself show that observed data identify it. Let $X$ denote measured pretreatment covariates, and let $e(X)=P(T=1\mid X)$ be the treatment propensity. A common identification strategy assumes consistency, conditional exchangeability, and positivity for the covariate values represented among controls. These assumptions concern the causal interpretation of the data; they are not guaranteed by a particular calculation or adjustment method.

Assumptions for identifying the ATC:

Consistency and no interference: the observed outcome satisfies $Y=Y(T)$, and one unit’s treatment does not alter another unit’s outcome. The treatment and control conditions must be defined clearly enough for these potential outcomes to be meaningful.
Conditional exchangeability: $\big(Y(1),Y(0)\big)\perp T\mid X$. Given the measured covariates, treatment assignment carries no further information about the potential outcomes.
Positivity for the control target: for covariate values $x$ occurring among controls, $P(T=1\mid X=x)>0$ as well as $P(T=0\mid X=x)>0$. Both treatment conditions must be represented to learn the treated outcome that controls with those covariates would have had.

Under these assumptions, define $m_t(x)=E[Y\mid T=t,X=x]$, for $t\in\{0,1\}$. Conditional exchangeability and consistency imply that $m_t(x)=E[Y(t)\mid X=x]$ wherever both treatment conditions are possible. Averaging the conditional difference over the covariate distribution of controls identifies the ATC:

\operatorname{ATC} =E\big[m_1(X)-m_0(X)\mid T=0\big].

The conditioning on $T=0$ is essential. It determines how much weight each covariate profile receives. Averaging conditional effects over the full population instead targets the ATE; averaging over treated units targets the ATT.

Worked Example: Standardizing to the Control Population

Suppose a hypothetical population has two covariate profiles, low risk and high risk, each making up half the population. Among low-risk units, 25% receive treatment; among high-risk units, 75% receive treatment. Assume the conditional outcome means are estimated as follows, in symptom points improved:

Profile	Mean under treatment, $m_1(x)$	Mean under control, $m_0(x)$	Conditional effect
Low risk	12	10	2
High risk	20	14	6

To find the covariate distribution among controls, multiply each profile’s population share by its probability of control. The unnormalized shares are $0.5(1-0.25)=0.375$ for low risk and $0.5(1-0.75)=0.125$ for high risk. Their total is 0.5, so among controls the shares are $0.375/0.5=0.75$ low risk and $0.125/0.5=0.25$ high risk. Therefore, $$ \operatorname{ATC}=0.75(12-10)+0.25(20-14)=1.5+1.5=3\text{ symptom points}. $$

For comparison, treated units are 25% low risk and 75% high risk: their unnormalized shares are $0.5(0.25)=0.125$ and $0.5(0.75)=0.375$. Standardizing to that group would give the ATT, $0.25(2)+0.75(6)=5$ symptom points. The conditional effects are the same in both calculations; the target-population weights differ. This calculation identifies the ATC only if the stated assumptions hold and both treatment conditions are possible within the relevant profiles.

Inverse-Probability Weighting for the ATC

The same target can be expressed with weights based on the propensity $e(X)$. In the control group, the observed outcome directly represents $Y(0)$. To estimate the missing mean $Y(1)$ for controls, reweight treated units so their covariate distribution resembles that of controls. The relevant treated-unit weight is the odds of control relative to treatment, $(1-e(X))/e(X)$.

A population identity explains the weight. Under conditional exchangeability, write $m_1(X)=E[Y(1)\mid X]$. Conditional on $X$, the expected weighted treated outcome is $E[T\{1-e(X)\}Y/e(X)\mid X]=\{1-e(X)\}m_1(X)$. The expected contribution from controls to the corresponding potential-outcome total is $E[(1-T)Y(1)\mid X]=\{1-e(X)\}m_1(X)$. Thus the two totals agree after averaging over $X$. Also, $E[T\{1-e(X)\}/e(X)]=E[1-e(X)]=P(T=0)$. Dividing by this total weight gives the mean treated potential outcome in the control population. Subtracting the observed control mean identifies the ATC.

\operatorname{ATC} = \frac{E\left[T\frac{1-e(X)}{e(X)}Y\right]} {E\left[T\frac{1-e(X)}{e(X)}\right]} - \frac{E[(1-T)Y]}{E[1-T]}.

Worked Example: A Simple Weighted Estimate

Consider hypothetical data with one low-risk treated unit whose outcome is 10 points, three high-risk treated units whose outcomes are 16, 15, and 17 points, and four controls whose outcomes are 7, 8, 9, and 13 points. Suppose the estimated treatment propensities are 0.25 for low-risk units and 0.75 for high-risk units. In this sample, the controls are three-quarters low risk and one-quarter high risk, while the treated are one-quarter low risk and three-quarters high risk.

The treated-unit weights for the ATC are $(1-e)/e$: the low-risk treated unit receives $(1-0.25)/0.25=3$, and each high-risk treated unit receives $(1-0.75)/0.75=1/3$. The weighted mean treated outcome is $$ \frac{3(10)+\frac13(16+15+17)} {3+\frac13+\frac13+\frac13} =\frac{30+16}{4}=11.5\text{ points}. $$

The control mean is $(7+8+9+13)/4=9.25$ points. The resulting ATC estimate is $11.5-9.25=2.25$ points. This calculation estimates the treatment-versus-control difference for a population with the controls’ covariate distribution. It relies on the propensity model and causal assumptions; the arithmetic alone does not establish that the estimate is unbiased. In particular, these small hypothetical data do not support a reliable uncertainty assessment.

Diagnostics, Alternatives, and Limitations

A central diagnostic is whether the covariate profiles of controls are represented among treated units. If controls have profiles with estimated $e(X)$ near zero, the odds $(1-e(X))/e(X)$ become large. A few treated observations can then dominate the weighted estimate, making it sensitive to small changes in the data or propensity model. Inspect the propensity distributions by treatment group and the resulting weights; also examine whether weighting brings relevant covariate distributions into alignment. Poor overlap or persistent imbalance signals that the estimate may depend heavily on extrapolation or modeling choices.

Standardization and weighting are alternative ways to use the same conditional-exchangeability idea. Standardization estimates outcome means under treatment and control across covariate profiles, then averages their difference over controls. Weighting uses treated outcomes reweighted toward the control covariate distribution and subtracts the control mean. A well-specified outcome model may make standardization useful; weighting makes the target-population reweighting explicit. Either approach can fail if confounders are unmeasured, if required treatment profiles are absent, or if the relevant models are misspecified.

The ATC is useful when the question concerns people currently receiving control—for example, the average outcome change they would experience under treatment. It does not automatically answer what treatment would do for treated units, for the entire population, or for a different eligible population. Nor does identifying the ATC from observational data remove the need to justify the assumptions. Those assumptions cannot generally be verified from observed outcomes alone.

Common mistake: “The ATC is the observed mean outcome among controls minus the observed mean outcome among treated.” This is wrong for two reasons: the sign may be reversed relative to $Y(1)-Y(0)$, and the observed groups contribute different potential outcomes. The ATC is the mean of $Y(1)-Y(0)$ among controls. Because $Y(1)$ is missing for them, identification requires assumptions and an appropriate adjustment; the raw group difference is not a substitute.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

In words, which units define the target population for the ATC, and which define it for the ATT?
Suppose conditional effects are 3 points for profile A and 7 points for profile B. Controls are 60% profile A and 40% profile B. Calculate the ATC.
Why does a control profile with an estimated propensity near zero create a problem for the ATC weighting formula?
State the conditional exchangeability assumption and explain why it cannot be established simply by observing similar measured covariates.

Solutions. (1) The ATC averages individual treatment effects among units with $T=0$; the ATT averages them among units with $T=1$. (2) Standardize to the control shares: $0.60(3)+0.40(7)=1.8+2.8=4.6$ points. (3) Its treated-unit weight $(1-e)/e$ is large when $e$ is near zero, so a small number of treated observations may dominate the estimate and increase its sensitivity. (4) Conditional exchangeability is $\big(Y(1),Y(0)\big)\perp T\mid X$. Similarity in measured covariates does not rule out unmeasured confounding or prove that, conditional on $X$, assignment is independent of both potential outcomes.