Worked Example Comparing ATE, ATT, and ATC

Why the Same Treatment Can Have Different Average Effects

A treatment can have different effects for different kinds of people. Even when two populations share the same covariate-specific effects, their average effects can differ if they contain different proportions of the covariate groups. This is why an average treatment effect is not fully described by its outcome scale alone: the population over which the effects are averaged matters.

This tutorial compares three summaries on an additive outcome scale: the average treatment effect (ATE) for the full population, the average treatment effect among treated units (ATT), and the average treatment effect among control units (ATC). The calculations use hypothetical data in which both potential outcomes are specified. In real observational data, only one potential outcome is observed for each unit; the examples isolate the estimand calculation from the separate problem of estimating it.

These definitions refer to three different averaging populations. The ATE averages over everyone in the target population; the ATT averages over those who received treatment; and the ATC averages over those who received control. If the outcome is, for example, a score measured in points, all three effects are in points. A positive value means treatment increases the outcome on average for the specified population, while a negative value means it decreases the outcome on average.

For a covariate \(X\) with groups \(x\), write \(\tau(x)=E[Y(1)-Y(0)\mid X=x]\). In a finite dataset with \(n_x\) units in group \(x\), \(n_{1x}\) treated units, and \(n_{0x}\) control units, the corresponding group-weighted calculations are:

Here \(\widehat{\tau}(x)\) is the stipulated or calculated average treatment effect in group \(x\). The hats in these finite-data calculations indicate quantities calculated from the displayed data; they do not by themselves imply that an effect has been estimated without bias from an observational study.

Worked Example: Covariate Composition Changes the Average

Why the Weighting Formula Works

The covariate-group formula is a weighted average: each group effect is multiplied by that group's share of the target population, and the contributions are added. For the ATT, the relevant share is the group's share among treated people, not its share in the full population. The ATC uses the group's share among controls. This is a direct consequence of averaging individual effects within the chosen population.

For example, partition the treated people into covariate groups. Group \(x\) contributes \(n_{1x}\widehat{\tau}(x)\) to the sum of their effects. Dividing the sum across all treated people by their total count \(N_1=\sum_x n_{1x}\) gives:

The same argument with \(n_x\) or \(n_{0x}\) in place of \(n_{1x}\) yields the ATE or ATC. The weights in each formula sum to one, so each quantity is an average rather than a total. If a group's weight is zero in the target population, that group's effect does not contribute to that target's average.

Worked Example: Calculating Directly From Individual Effects

A Useful Identity and a Composition Check

In a single finite population, the full set of units is the disjoint union of treated and control units. The sum of all individual effects is therefore the treated sum plus the control sum. Dividing by the full population size gives an identity linking the three averages:

The weights are the treatment and control proportions in the population. They sum to one because \(N=N_1+N_0\). This identity is a useful arithmetic check, but it applies when the ATE, ATT, and ATC are all defined for the same population and use the same individual effects. It is not a claim that any two of these averages must be equal.

Assumptions, Diagnostics, and Limits

The examples above specify the treatment effects, so their calculations are arithmetic rather than causal estimates from observed outcomes. To interpret these quantities as effects of well-defined treatment strategies, the potential outcomes must correspond to those strategies, and each unit's observed outcome must equal the potential outcome under its received treatment. For estimation from observational data, additional assumptions are needed. In particular, conditional exchangeability requires that, given the measured covariates, treatment assignment carries no further information about the potential outcomes; positivity requires that each treatment option has positive probability at covariate values represented in the target population. These assumptions are not verified just because the weighted averages can be calculated.

A practical diagnostic for a hand calculation is to confirm that the covariate-group counts add correctly: \(n_x=n_{1x}+n_{0x}\), \(N=\sum_xn_x\), \(N_1=\sum_xn_{1x}\), and \(N_0=\sum_xn_{0x}\). Then check that the weights for each target sum to one. For a common finite population, the ATE identity provides an additional check. Disagreement often indicates a denominator error or the use of full-population weights in an ATT or ATC calculation.

A major limitation is that observed data do not reveal both potential outcomes for the same person. Consequently, individual effects generally cannot be directly computed from ordinary observational data. Matching and other adjustment approaches use observed covariates and design or modeling assumptions to estimate a target average; the target population still determines whether the intended summary is an ATE, ATT, ATC, or another effect. Matching does not make these estimands interchangeable.

Common Mistakes and Alternatives

• Using the wrong weights. The ATT weights effects by the covariate distribution among treated people; the ATC uses the distribution among controls; the ATE uses the full target population's distribution.
• Calling the three averages different effects for different people. They are summaries over different populations. If treatment effects vary by covariates, their values can differ because the averaging populations differ.
• Inferring causation from a numerical average alone. Calculating an observed group difference or a weighted mean does not establish the assumptions needed for a causal interpretation.
• Reporting an effect without its scale or units. State, for instance, that an average effect is 4.4 score points, and specify the population whose average is meant.

The direct individual-average calculation is appropriate when both potential outcomes are known in a theoretical example or simulation. The covariate-standardization calculation is useful when group-specific effects and target group proportions are supplied. Matching-based estimates instead depend on the matched design and its induced target population. These approaches answer related but not automatically identical questions; the choice should follow the scientific target, rather than whichever numerical result is easiest to compute.