Conditional Average Treatment Effects and Subgroup Estimands

Why Define the Target Before Comparing Effects?

A treatment may have different effects for people with different baseline characteristics. A useful analysis therefore needs to specify not just which treatment contrast is of interest, but for whom the contrast is being averaged. The effect for people with a particular covariate profile and the average effect for everyone in a prespecified subgroup are related, but they are not the same estimand.

This tutorial uses a binary treatment and a fixed outcome scale. A potential outcome is the outcome an individual would have under a specified treatment condition. The contrasts below are causal estimands: they are defined in terms of potential outcomes, whether or not those outcomes can all be observed in a study.

The individual effect is a contrast between two potential outcomes for the same person. In ordinary parallel-group studies, only one of those outcomes is observed for each person. Thus, individual effects are useful for defining population targets, but they are generally not directly calculable person by person from observed data.

Conditional Average Treatment Effects

Let \(X\) denote baseline characteristics that may describe an individual’s covariate profile. For a profile \(x\), the conditional average treatment effect (CATE) is the mean individual effect among people with that profile. If \(X\) is continuous, the profile \(x\) may represent a value or specified set of values; in practice, the definition also requires a meaningful target population with those characteristics.

The CATE is an average, not a claim that every person with profile \(x\) has the same individual effect. For example, \(\tau(x)=5\) points means that the average contrast between treatment and control potential outcomes among the target individuals with profile \(x\) is 5 points. Individual effects within that profile could differ.

Average Treatment Effects Within Prespecified Subgroups

A subgroup is defined by a baseline rule, such as belonging to an age category or having a particular diagnosis before treatment. Let \(S=g\) mean that an individual belongs to subgroup \(g\). The average treatment effect within that subgroup averages individual potential-outcome contrasts over the members of that subgroup.

A prespecified subgroup rule is set using baseline information, rather than chosen after examining which groups appear to have the largest observed effects. Prespecification helps make the target clear and limits outcome-driven selection; it does not, by itself, guarantee that the effect is identified or that an estimate is precise.

If \(X\) distinguishes profiles within subgroup \(g\), the subgroup effect averages the corresponding conditional effects using the profile distribution in that subgroup. For discrete profiles, the relationship follows from conditioning on \(X\): $$ \tau_g =\sum_x E[Y(1)-Y(0)\mid X=x,S=g]P(X=x\mid S=g). $$ When subgroup membership is determined by \(X\), this can be written using \(\tau(x)\) for the profiles included in \(g\). With continuous covariates, the corresponding expression is an expectation, or integral, over the distribution of profiles in the subgroup. The weights are the proportions of the subgroup at each profile, not automatically equal weights across profile categories.

Subgroup Weights Determine the Population Target

When subgroups partition a population, the population average treatment effect (ATE) is a weighted average of the subgroup effects. The weights are the proportions of the target population in each subgroup. Changing the target population can therefore change the overall average, even if each subgroup-specific effect remains the same.

This result is a direct application of the law of total expectation. Let \(D=Y(1)-Y(0)\). For a partition into subgroups, $$ E[D]=\sum_g E[D\,\mathbf{1}(S=g)] =\sum_g P(S=g)E[D\mid S=g], $$ where \(\mathbf{1}(S=g)\) equals 1 for members of subgroup \(g\) and 0 otherwise. Thus, each subgroup’s mean contrast contributes in proportion to the subgroup’s share of the target population.

Estimands and What Observed Data Can Establish

The definitions above specify targets in terms of potential outcomes. Estimating those targets from observed data requires additional design or identification assumptions. In the usual no-interference setup, consistency means that the observed outcome is \(Y=Y(A)\): a person assigned treatment has observed outcome \(Y(1)\), and a person assigned control has observed outcome \(Y(0)\). No interference means one person’s potential outcome is not changed by other individuals’ treatment assignments. If treatment conditions are not well defined or outcomes depend on others’ assignments, the potential-outcome definition must be adapted to the setting.

For an observational comparison, a common sufficient set of conditions for identifying conditional effects given baseline covariates \(X\) is conditional exchangeability, consistency, and positivity. Conditional exchangeability states that, given \(X\), treatment assignment is independent of the pair of potential outcomes. Positivity requires a positive probability of receiving either treatment at each covariate profile in the target population. Under these conditions, the conditional effect can be represented as the difference in conditional observed-outcome means: $$ \tau(x)=E[Y\mid A=1,X=x]-E[Y\mid A=0,X=x]. $$ Random assignment can support exchangeability by design, but it does not make every subgroup estimate precise, and it does not change the definition of the estimand. The subgroup target still averages over the subgroup population of interest.

These assumptions clarify a crucial distinction: defining an estimand is not the same as identifying it from observed data, and identifying it is not the same as estimating it accurately in a finite sample. The examples used complete hypothetical potential-outcome schedules so that the target calculations could be seen directly; real studies generally require design-based or model-based methods to learn about those targets.

Common Mistakes

• Treating a CATE as an individual effect. A CATE is the mean contrast among people with a profile, not a guarantee that each person with that profile experiences that contrast.
• Replacing a subgroup average with an effect at one profile. A subgroup can contain several profiles. Its effect averages their contrasts according to how common each profile is in that subgroup.
• Using equal weights without defining an equal-weight target. Population subgroup weights come from the target population’s subgroup proportions. Equal weighting defines a different target when subgroup sizes differ.
• Choosing subgroups after inspecting apparent effects. This can make the reported target depend on outcome results. Prespecifying the subgroup rule makes the intended population clearer, although it does not eliminate all inferential concerns.
• Assuming both potential outcomes are observed for each person. The contrast \(Y_i(1)-Y_i(0)\) defines an individual effect, but ordinary studies observe only the outcome under the treatment actually received.