Conditional Average Treatment Effects and Their Aggregation

Why Conditional Effects Must Be Aggregated

Treatment effects can vary with pre-treatment characteristics. A treatment may have a small average effect for one covariate profile and a larger average effect for another. A conditional average treatment effect describes the mean contrast at a given covariate value; an average treatment effect for a population combines those conditional contrasts using that population’s distribution of covariates. The weights matter: the same conditional effects can imply different overall averages in populations with different covariate compositions.

Let $D\in\{0,1\}$ denote treatment status, $X$ a vector of pre-treatment covariates, and $Y(1)$ and $Y(0)$ the potential outcomes under treatment and control. The observed outcome is $Y=DY(1)+(1-D)Y(0)$. These definitions specify the causal quantities; additional assumptions are needed to identify them from observed data.

Definition: The conditional average treatment effect (CATE) at covariate value $x$ is $$ \tau(x)=E[Y(1)-Y(0)\mid X=x] =E[Y(1)\mid X=x]-E[Y(0)\mid X=x]. $$ It is the mean potential-outcome contrast among units with $X=x$, not an observed difference for a particular unit.

Identification From Observed Outcomes

The CATE is defined in terms of potential outcomes, only one of which is observed for each unit. A standard identification result uses consistency, conditional exchangeability, and positivity. Consistency means the observed outcome equals the potential outcome corresponding to the treatment actually received. Conditional exchangeability means treatment assignment is independent of both potential outcomes given $X$: $(Y(1),Y(0))\perp D\mid X$. Positivity requires $0<P(D=1\mid X=x)<1$ for covariate values in the target population. In addition, the usual potential-outcomes setup rules out interference between units.

Under these conditions, the conditional potential-outcome means can be recovered from observed conditional means. For $d\in\{0,1\}$, exchangeability allows conditioning additionally on $D=d$, and consistency then gives

E[Y(d)\mid X=x] =E[Y(d)\mid D=d,X=x] =E[Y\mid D=d,X=x].

Consequently, on covariate values satisfying positivity,

\tau(x)=E[Y\mid D=1,X=x]-E[Y\mid D=0,X=x].

In a randomized experiment, conditional exchangeability holds by random assignment (and positivity must still hold for the covariate values being considered). In observational data, exchangeability is a substantive assumption: adjusting for recorded covariates does not itself establish that all relevant confounding has been removed. The aggregation identities below are properties of the estimands; they do not by themselves guarantee identification.

Worked Example: CATEs for a Binary Covariate

Suppose a hypothetical randomized study has a binary pre-treatment covariate $X$, with $P(X=1)=0.40$. The outcomes are measured in points. The conditional mean outcomes are:

Covariate value	$E[Y\mid D=1,X=x]$	$E[Y\mid D=0,X=x]$	$\tau(x)$
$X=0$	10	8	2
$X=1$	20	12	8

Randomization supports exchangeability, and both treatment arms occur for each covariate value, so the conditional contrasts identify the CATEs in this hypothetical population. For $X=0$, the estimated conditional contrast from the stated means is $10-8=2$ points; for $X=1$, it is $20-12=8$ points. Thus $\tau(0)=2$ and $\tau(1)=8$.

The population average treatment effect weights these CATEs by the proportions of the two covariate groups:

E[\tau(X)] =P(X=0)\tau(0)+P(X=1)\tau(1) =(0.60)(2)+(0.40)(8) =1.20+3.20 =4.40\text{ points}.

The corresponding population mean contrast is 4.40 points. The subgroup with $X=1$, by contrast, has an average effect of 8 points because all members of that subgroup have $X=1$. These are different target averages, even though they are constructed from the same conditional effects.

How Conditional Effects Aggregate

The key result follows from the law of iterated expectations. The population average treatment effect (ATE) is the mean potential-outcome contrast in the target population. Conditioning that contrast on $X$ and then averaging over the distribution of $X$ gives:

Theorem: If $E[|Y(1)|+|Y(0)|]$ is finite, then $$ \operatorname{ATE} =E[Y(1)-Y(0)] =E[\tau(X)]. $$ If $X$ is discrete, this is a probability-weighted sum over its values. If $X$ has density $f_X$, it is an integral with respect to that density.

The derivation is direct. Let $\Delta=Y(1)-Y(0)$. Because $\tau(X)=E[\Delta\mid X]$, the law of iterated expectations gives $E[\Delta]=E[E[\Delta\mid X]]$. Substituting the definition of $\tau(X)$ yields the result:

\begin{aligned} \operatorname{ATE} &=E[Y(1)-Y(0)]\\ &=E\!\left[E[Y(1)-Y(0)\mid X]\right]\\ &=E[\tau(X)]. \end{aligned}

For a subgroup defined by a pre-treatment event $S$ that is determined by $X$, the corresponding subgroup average treatment effect is

\operatorname{ATE}_{S} =E[Y(1)-Y(0)\mid S] =E[\tau(X)\mid S],

provided the conditional expectations exist and $P(S)>0$. In a discrete covariate setting, this conditional average weights each covariate value in the subgroup by its probability within that subgroup. For a continuous covariate, it averages $\tau(x)$ against the subgroup’s conditional density of $X$. A subgroup definition based on post-treatment variables would require a different causal interpretation; here, subgroups are formed from pre-treatment covariates.

Worked Example: Averaging a CATE Function Over a Continuous Covariate

Consider a hypothetical population with a continuous covariate $X\sim\operatorname{Uniform}(0,1)$. Suppose the conditional mean outcome under treatment is $E[Y(1)\mid X=x]=5+3x$, and under control it is $E[Y(0)\mid X=x]=4+x$, with outcomes measured in points. Assume these conditional potential-outcome means are identified under the study’s design and assumptions.

Subtracting the two conditional means gives the CATE function:

\tau(x)=(5+3x)-(4+x)=1+2x\text{ points}.

Because $X$ is uniform on $[0,1]$, its density is $f_X(x)=1$ over that interval. The population ATE is therefore the integral of the CATE function against this density:

\begin{aligned} \operatorname{ATE} &=\int_0^1 \tau(x)f_X(x)\,dx\\ &=\int_0^1(1+2x)(1)\,dx\\ &=[x+x^2]_0^1\\ &=2\text{ points}. \end{aligned}

Now consider the subgroup with $X\geq 0.5$. Within that subgroup, $X$ is uniform on $[0.5,1]$, so its conditional mean is $(0.5+1)/2=0.75$. Since $\tau(x)=1+2x$, the subgroup average is

\operatorname{ATE}_{X\geq0.5} =E[1+2X\mid X\geq0.5] =1+2(0.75) =2.50\text{ points}.

The subgroup average exceeds the population average because the subgroup places more weight on larger values of $X$, where the stated CATE function is larger. The calculation averages across the covariate distribution; it does not assert that every unit in the subgroup has an effect of 2.50 points.

Target Populations Determine the Weights

A conditional effect function and an average effect answer different questions. The function $\tau(x)$ describes the conditional mean contrast across covariate values. To obtain a single average, one must also specify the population over which the averaging occurs. If two populations have different distributions of $X$, they can have different ATEs even when the conditional effects are the same in both populations. This is why reporting an average without stating its target population can be incomplete.

Worked Example: The Same CATEs, Different Population Averages

Suppose the conditional effects are 2 points for $X=0$ and 8 points for $X=1$, and suppose these conditional effects apply in each of two hypothetical target populations. In Population A, $P(X=1)=0.20$; in Population B, $P(X=1)=0.70$. The effects are the same across populations, but the covariate weights differ.

For Population A, $P(X=0)=0.80$, so

\operatorname{ATE}_{A}=(0.80)(2)+(0.20)(8)=1.60+1.60=3.20\text{ points}.

For Population B, $P(X=0)=0.30$, so

\operatorname{ATE}_{B}=(0.30)(2)+(0.70)(8)=0.60+5.60=6.20\text{ points}.

The 3.20-point and 6.20-point averages do not conflict: each describes a different target population. The calculation assumes the specified conditional effects are applicable to both populations; that assumption is separate from the arithmetic of aggregation.

Common Mistakes and Practical Interpretation

Aggregation is simple once the estimand and target population are fixed, but several errors can change the question being answered:

Giving every covariate group equal weight. If the groups have different population proportions, their CATEs must not be averaged equally unless equal weighting is actually the target. Use the population proportions, or the relevant conditional proportions for a subgroup.
Using the wrong population’s distribution. Weights from a study sample, a policy target population, and a specified subgroup need not match. Make clear which distribution defines the average.
Treating an observed conditional contrast as automatically causal. The identity using observed conditional means requires identification assumptions such as consistency, conditional exchangeability, and positivity. A conditional difference alone does not establish those assumptions.
Confusing an average with an effect for every unit. An ATE or subgroup average summarizes a mean contrast. It does not say that each member experiences that amount of change.

For a discrete covariate, a useful check is to verify that the weights are nonnegative and sum to one. For a continuous covariate, the weighting density must integrate to one over the target support. In either case, the resulting average should be interpreted in the outcome’s units and for the stated target population. Changing the weights changes the target average, not the definition of the conditional effects themselves.

Key takeaway: A CATE is a covariate-specific difference in mean potential outcomes. The population ATE is the expectation of that function under the target population’s covariate distribution; a subgroup average uses the covariate distribution within that subgroup.

Check Your Understanding

Use the definitions and aggregation rules above to answer the following.

If $P(X=1)=0.25$, $\tau(0)=3$ points, and $\tau(1)=7$ points, what is the population ATE?
State the three central identification conditions used to express a CATE as a contrast of observed conditional means.
If $X\sim\operatorname{Uniform}(0,2)$ and $\tau(x)=2+x$, write the integral for the population ATE and evaluate it.
Why can two populations with the same conditional effects have different ATEs?
For a subgroup $S$ defined using pre-treatment covariates, write the formula for its average treatment effect in terms of $\tau(X)$.

Covariate value	\(E[Y\mid D=1,X=x]\)	\(E[Y\mid D=0,X=x]\)	\(\tau(x)\)
\(X=0\)	10	8	2
\(X=1\)	20	12	8