Why the Estimand Comes First
An experiment can produce several defensible answers to the question “What was the effect?” The answer depends on the treatment contrast and on whose outcomes are being averaged. An effect for the people enrolled in a particular trial, an effect for a wider target population, and an effect for a prespecified subgroup are different estimands—even if they are discussed using the same treatment and outcome.
An estimand is the precise causal quantity an analysis aims to learn. It specifies the outcome, the treatment contrast, and the target units or population. Defining the estimand is a conceptual step: it does not by itself show that the quantity can be identified from the observed data or say how to estimate it. Those questions depend on the design and analysis.
Use the potential-outcomes notation \(Y_i(1)\) and \(Y_i(0)\) for unit \(i\)'s outcome under treatment and control, respectively. The individual causal effect is \(Y_i(1)-Y_i(0)\). These two potential outcomes describe a specified pair of treatment conditions. As required by SUTVA, the conditions must be sufficiently well-defined, and one unit’s outcome must not depend on other units’ treatment assignments.
Sample and Population Average Treatment Effects
The sample average treatment effect (SATE) averages individual effects over the particular \(n\) units in an experiment. It is a finite-sample quantity: its target is the enrolled set itself, not automatically everyone represented by that set.
The population average treatment effect (PATE) instead averages effects over a specified target population. If units in that population are represented by a probability distribution \(\mathcal P\), the PATE is the population expectation of their individual effects. Its meaning therefore depends on which population \(\mathcal P\) is intended: for example, all eligible patients in a region, rather than only those who enrolled.
Both quantities compare the same treatment and control conditions, but average over different units. If the enrolled sample is itself the target, the SATE may be the desired estimand. If the goal is to describe effects in a larger population, that population must be named and the target becomes a PATE. A sample average is not automatically a population average just because the sample was drawn from a broader population.
Worked Example: Calculating a Sample Average Treatment Effect
Suppose a hypothetical experiment enrolls four people. Their outcomes are measured in points on a symptom scale, where higher values mean more severe symptoms. For illustration, both potential outcomes are shown, although in an actual experiment only one is observed for each person.
| Person | \(Y_i(1)\), treatment | \(Y_i(0)\), control | Effect \(Y_i(1)-Y_i(0)\) |
|---|---|---|---|
| A | 6 | 10 | -4 |
| B | 9 | 8 | 1 |
| C | 5 | 11 | -6 |
| D | 7 | 8 | -1 |
The stated target is these four enrolled people, and the contrast is treatment versus control. Thus the target is the SATE:
For this hypothetical finite sample, treatment reduces the symptom score by an average of 2.5 points relative to control. The positive effect for person B does not contradict the negative average; the SATE summarizes the four individual effects. This calculation defines the estimand from the potential outcomes. It does not imply that both outcomes are observed for any person.
Effects for a Target Subgroup
Sometimes the main question concerns a subgroup rather than everyone in the sample or population. Let \(G_i=1\) indicate membership in a subgroup defined using characteristics measured before treatment—for example, people with a particular baseline severity level. The subgroup average treatment effect for the enrolled sample averages effects among sample members with \(G_i=1\). Its population counterpart averages among members of the named target population who meet that subgroup definition.
Here \(n_{G=1}\) is the number of sample units in the subgroup. The subgroup definition is part of the estimand: “people with high baseline severity” and “people who improved after treatment” are not interchangeable descriptions. In particular, defining groups by a post-treatment outcome can make group membership depend on treatment itself, changing the question and potentially making a comparison difficult to interpret causally.
Worked Example: Defining an Effect for a Baseline Subgroup
Consider a hypothetical sample of five patients. The subgroup of interest is patients whose baseline severity score is at least 7. Outcomes are on the same symptom scale as before; lower scores are better. The potential outcomes and effects are:
| Patient | Baseline severity | \(Y_i(1)\) | \(Y_i(0)\) | Effect |
|---|---|---|---|---|
| A | 8 | 6 | 10 | -4 |
| B | 9 | 8 | 9 | -1 |
| C | 7 | 5 | 8 | -3 |
| D | 5 | 4 | 6 | -2 |
| E | 4 | 5 | 7 | -2 |
The baseline rule includes A, B, and C, so \(n_{G=1}=3\). The subgroup SATE is:
For these three sample members with baseline severity of at least 7, treatment reduces the symptom score by an average of about 2.667 points relative to control. This is not the SATE for all five patients, which would be \((-4-1-3-2-2)/5=-2.4\) points. Nor is it automatically the subgroup PATE for all patients in a wider population: that would name a population and average effects over all its members meeting the baseline rule.
Effects of Treatment Assignment
In an experiment, assignment to a treatment condition and receipt of the treatment need not be the same. A participant assigned to an offer may decline it; a participant assigned to control may obtain a similar service elsewhere. When this distinction matters, define the assignment itself as the intervention.
Let \(Z_i\) denote assignment, with \(Z_i=1\) for assignment to the offer and \(Z_i=0\) for assignment to control. Write \(Y_i^{Z}(z)\) for the outcome that would occur for unit \(i\) if assigned to condition \(z\), allowing actual treatment receipt to follow as it would under that assignment. The average effect of assignment, often called the intention-to-treat (ITT) effect, is an average contrast between these assignment potential outcomes over a stated target set.
This estimand answers a policy-relevant question such as “What is the average effect of offering this program rather than assigning control?” It includes the consequences of assignment, including any changes in actual treatment receipt that assignment induces. By contrast, the treatment-receipt effect compares outcomes under receiving treatment versus not receiving it. When assignment and receipt differ, that is a different causal contrast and cannot be substituted for the assignment effect without additional assumptions and analysis.
Worked Example: The Effect of an Offer When Receipt Is Incomplete
Suppose a hypothetical program is offered to one group and not offered to a control group. Some people offered the program do not attend, while some control-group members obtain similar help elsewhere. For each of six people, the table shows potential outcomes under assignment to the offer and control, and the potential receipt under each assignment. The outcome is a score in points, with higher values representing better performance.
| Person | Receipt if offered | Receipt if control | \(Y_i^Z(1)\) | \(Y_i^Z(0)\) | Assignment effect |
|---|---|---|---|---|---|
| A | 1 | 0 | 12 | 10 | 2 |
| B | 0 | 0 | 9 | 8 | 1 |
| C | 1 | 1 | 11 | 10 | 1 |
| D | 1 | 0 | 10 | 9 | 1 |
| E | 0 | 0 | 7 | 8 | -1 |
| F | 1 | 0 | 13 | 11 | 2 |
The target is these six people and the contrast is assignment to an offer versus control. The average assignment effect is:
For this hypothetical sample, offering the program increases the score by an average of 1 point relative to assigning control. The result is about the effect of the offer, not the effect of actually receiving the program. For example, person B has an assignment effect of 1 point even though the table indicates no receipt under either assignment; person C receives treatment under both assignments, so their assignment effect is not a simple treated-versus-untreated comparison.
Why Averages Depend on the Target
When subgroups differ in size and in their effects, the overall average depends on how those subgroups are represented in the target population. This is not a technical detail: it is part of the question being answered. The relationship can be seen directly by partitioning a population into two mutually exclusive groups, \(G=1\) and \(G=0\). Let \(q=P_{\mathcal P}(G=1)\). The overall PATE is the weighted average of the two subgroup PATEs:
The second equality follows by splitting the expectation over the two exhaustive groups: the effect is averaged within each group, then weighted by that group’s share of the target population. Consequently, an overall effect for a population with many high-baseline-severity patients may differ from the effect for a population with few such patients, even when the two subgroup effects are unchanged. Reporting an effect without saying who is represented can conceal this dependence.
Common Pitfall: Treating Estimands as Interchangeable
- Wrong reasoning: “The trial enrolled a sample from a large population, so its average effect is the population effect.” Correction: The sample average targets the enrolled units. A population estimand must name a target population, and transport from the sample to that target is a separate issue.
- Wrong reasoning: “An effect in a subgroup is just the overall effect with a more detailed label.” Correction: The subgroup effect averages over a different set of units and can have a different value. Define subgroup membership precisely, preferably using pre-treatment characteristics.
- Wrong reasoning: “The effect of being assigned an offer is the effect of receiving treatment.” Correction: Assignment and receipt are distinct interventions when adherence is imperfect or control participants can obtain treatment. State which contrast the estimand uses.
- Wrong reasoning: “If the estimand is written down, the experiment has identified it.” Correction: Defining the target quantity is not the same as establishing that the design and observed data identify it. Identification and estimation require their own arguments.
Check Your Understanding
For each question, identify the target units and treatment contrast before giving an answer.
- In one sentence, distinguish the SATE from the PATE.
- A study targets adults with a baseline score above a prespecified threshold. What information is needed to define its subgroup estimand clearly?
- Why can an effect of assignment to a program differ from an effect of receiving the program?
- A sample contains 40% of subgroup A, but the target population contains 70% of subgroup A. If subgroup effects differ, why might the sample and population average effects differ?
- Does defining a PATE establish that an experiment identifies it? Explain briefly.