Average Causal Effects as Potential-Outcome Contrasts

From Individual Effects to a Population Average

An individual causal effect compares what would happen to one unit under treatment with what would happen to that same unit under control. A population-level question asks for a summary across a defined set of units: what is the average of those individual contrasts? The answer depends on which population is being averaged over and on the outcome scale.

Let \(Y(1)\) denote a unit’s potential outcome under treatment and \(Y(0)\) its potential outcome under control. These are defined for each unit in the target population. The individual effect is \(Y(1)-Y(0)\). Averaging that contrast over the target population gives the population average treatment effect, usually abbreviated ATE.

The first expression emphasizes that the ATE averages individual effects. The second expresses it as a difference between the population mean potential outcome under treatment and the population mean potential outcome under control. Both refer to the same target population and the same outcome definition. The ATE is an average contrast, not a claim that every individual has that effect.

The target population is part of the definition, not a detail to add afterward. For example, an average effect among all eligible patients need not equal the average effect among patients who actually received treatment. Changing who is included changes the distribution over which the potential-outcome contrast is averaged.

Defining the Estimand Carefully

Before calculating an ATE, specify the population and treatment contrast. The treatment states must be clear enough to define \(Y(1)\) and \(Y(0)\); the outcome must use the same measurement scale and time point under both states; and the expectations in the definition must exist. These are requirements for the estimand to be meaningful and mathematically defined. They do not, by themselves, ensure that observed data identify or estimate it.

For a finite population of \(N\) units, the corresponding average is an arithmetic mean:

Here \(i\) indexes units in the target population. This finite-population expression makes the weighting explicit: each unit contributes equally. In a population formulation, \(E[\cdot]\) denotes the average under the distribution of units in the target population. Other weighting choices are possible, but they define a different target average and must be stated.

The Population Average Is Sensitive to Its Weights

An overall average combines subgroup effects according to how much of the target population each subgroup contributes. It is generally not the simple average of subgroup averages unless the subgroup sizes or the chosen weights make it so. This matters when results are summarized across populations with different compositions.

Why an Observed Group Difference Is Not the ATE

An observed treated-versus-control difference compares outcomes recorded for two groups. If \(T\) indicates treatment status and \(Y\) is the recorded outcome, the population version of that descriptive contrast is

This quantity compares the mean recorded outcome among units with \(T=1\) to the mean among units with \(T=0\). By contrast, the ATE compares the two potential-outcome means for the same specified target population. The definitions are different: the observed contrast averages over treatment groups, while the ATE averages two potential outcomes over the target population.

The distinction is not merely that one quantity is called “observed” and the other “causal.” An observed group difference can reflect both treatment-related outcome differences and differences in which units are represented in the groups. The group contrast alone does not establish that it equals the target-population ATE. Whether and how observed data can support an estimate of the ATE depends on additional conditions and a suitable design or analysis.

A useful way to keep the quantities separate is to write down what each averages before interpreting a number:

Derivation and Interpretation

The equivalence of the two ATE expressions follows directly from linearity of expectation. Provided \(Y(1)\) and \(Y(0)\) have finite means,

The same calculation applies to the finite-population mean because sums distribute over subtraction: $$ \frac{1}{N}\sum_{i=1}^{N}\{Y_i(1)-Y_i(0)\} =\frac{1}{N}\sum_{i=1}^{N}Y_i(1) -\frac{1}{N}\sum_{i=1}^{N}Y_i(0). $$ This equality is an algebraic property of the estimand. It does not say that both potential outcomes are observed for each unit, nor that a particular observed statistic equals the ATE.

The ATE uses the outcome’s units. If \(Y\) is measured in points, the ATE is measured in points; if \(Y\) is measured in days, it is measured in days. Its sign also follows the order of subtraction: a positive ATE means the mean potential outcome under treatment is higher than under control, while a negative ATE means it is lower. Whether a higher value is desirable depends on what the outcome measures.

Common Pitfalls and Limits

• Treating a group difference as a causal effect. A difference between observed means is descriptive by definition. Do not label it the ATE unless the required link between the observed data and the target potential-outcome means has been justified.
• Leaving the target population implicit. An ATE is always an average for some population. State who is included and what weights define that average.
• Assuming the mean describes everyone. An ATE of 2 points does not imply that each individual effect is 2 points. Effects may vary around the average.
• Changing the outcome or treatment definition. Potential outcomes must refer to the same outcome scale and clearly specified treatment states. Changing the follow-up time, outcome measure, or target population changes the estimand.
• Overstating what the definition provides. Defining an ATE specifies a causal quantity of interest; it does not show that available data identify it or estimate it precisely.

The ATE is often a useful summary, but it can conceal important variation. It also may not be the most relevant average for every decision: a question focused on a particular subgroup or on people who received treatment has a different target. The first task is to define the contrast and population that match the question; estimation comes afterward.