Treatment Effects as Randomized-Study Estimands

From a Study Question to a Target

A randomized study can produce a treatment estimate, but the estimate is meaningful only after the target has been stated. “What is the effect of treatment?” might mean the average effect for the particular people enrolled, the average effect for a broader population, or an effect for a specified subgroup. These are different questions, even when the same regression model is used.

Potential outcomes provide a precise way to define the target. They describe, for each study unit, the outcome that would occur under each treatment condition. Since a unit receives only one condition, the other potential outcome is not observed. Random assignment helps make comparisons informative, but it does not make both outcomes observable for any one unit.

Definition: For unit $i$, let $Y_i(1)$ be the outcome that would be observed if the unit were assigned treatment, and let $Y_i(0)$ be the outcome that would be observed if the unit were assigned control. The unit’s treatment effect is $\tau_i=Y_i(1)-Y_i(0)$. For a defined set of $N$ units, the finite-population average treatment effect (ATE) is $$ \tau_{ATE}=\frac{1}{N}\sum_{i=1}^{N}\bigl[Y_i(1)-Y_i(0)\bigr]. $$

The order of subtraction matters: this definition uses treatment minus control. If the outcome is a score measured in points, the ATE is measured in points; a positive ATE means that the treatment potential outcomes are higher on average. For outcomes such as recovery time, a negative ATE may instead be favorable. The direction of benefit depends on what the outcome measures.

The set of units being averaged over is part of the definition. If the $N$ units are the people enrolled in a trial, $\tau_{ATE}$ is their finite-population ATE. A claim about a larger population requires a target population beyond those enrolled and a reason the study results apply to it. Random assignment within a trial does not, by itself, make the enrolled participants representative of everyone else.

What Random Assignment Does

Let $Z_i=1$ indicate that unit $i$ is assigned to treatment and $Z_i=0$ indicate assignment to control. The observed outcome is the potential outcome corresponding to the assigned condition:

$$ Y_i=Z_iY_i(1)+(1-Z_i)Y_i(0). $$

This equation makes the missing-data problem explicit: when $Z_i=1$, $Y_i(1)$ is observed but $Y_i(0)$ is not; when $Z_i=0$, $Y_i(0)$ is observed but $Y_i(1)$ is not. A comparison between groups therefore does not reveal every individual effect. Its purpose is to estimate an average contrast.

To state the randomization result precisely, consider a fixed set of $N$ study units, with all their potential outcomes fixed. Suppose exactly $n_1$ units are assigned to treatment and $n_0=N-n_1$ to control, with $n_1>0$ and $n_0>0$, and every allocation of that size is equally likely. Each unit then has probability $n_1/N$ of treatment assignment and probability $n_0/N$ of control assignment. Under this design, the average observed outcome among treated units estimates the average treatment potential outcome; the corresponding control average estimates the average control potential outcome. Their contrast is unbiased for the finite-population ATE over repeated assignments of treatment to these same units.

Here “unbiased” describes an average over possible random assignments, not a guarantee about one particular assignment. A realized estimate can be above or below the ATE because random assignment can produce different groups by chance.

Worked Example: Defining an ATE From Potential Outcomes

Suppose four hypothetical participants have the following potential scores on a test, measured in points. The table lists both potential outcomes to make the target visible; in an actual trial, only one outcome per participant would be observed.

Participant	$Y_i(1)$, treatment	$Y_i(0)$, control	$\tau_i=Y_i(1)-Y_i(0)$
1	10	8	2
2	12	11	1
3	14	11	3
4	16	15	1

The finite-population ATE is the mean of the four individual effects:

\tau_{ATE} =\frac{2+1+3+1}{4} =\frac{7}{4} =1.75\text{ points}.

For this defined group of four, the treatment potential scores average $(10+12+14+16)/4=13$ points, and the control potential scores average $(8+11+11+15)/4=11.25$ points. Their difference is $13-11.25=1.75$ points, the same ATE. This equality follows from averaging the individual contrasts: the mean of $Y_i(1)-Y_i(0)$ equals the mean of $Y_i(1)$ minus the mean of $Y_i(0)$.

A Short Randomization Justification

The randomization result can be checked by looking at the expectation of each group mean. Under the fixed-size design above, each unit has the same chance $n_1/N$ of being among the treated. Write $\bar{Y}_{1,\mathrm{obs}}$ for the treated-group observed mean. Its expectation over the random assignment is

E_Z(\bar{Y}_{1,\mathrm{obs}}) =E_Z\left(\frac{1}{n_1}\sum_{i=1}^{N}Z_iY_i(1)\right) =\frac{1}{n_1}\sum_{i=1}^{N}E_Z(Z_i)Y_i(1) =\frac{1}{N}\sum_{i=1}^{N}Y_i(1).

The subscript $Z$ indicates that the expectation is over assignments, holding the units and their potential outcomes fixed. Applying the same reasoning to the control group, where each unit has probability $n_0/N$ of assignment, gives

E_Z(\bar{Y}_{0,\mathrm{obs}}) =\frac{1}{N}\sum_{i=1}^{N}Y_i(0).

Subtracting these expectations shows why the observed group-mean contrast is unbiased for the defined target:

E_Z(\bar{Y}_{1,\mathrm{obs}}-\bar{Y}_{0,\mathrm{obs}}) =\frac{1}{N}\sum_{i=1}^{N}\bigl[Y_i(1)-Y_i(0)\bigr] =\tau_{ATE}.

This argument justifies an assignment-based estimate; it does not say that every regression specification will estimate the same quantity, nor does it supply a standard error or confidence interval.

Conditions for a Meaningful Potential-Outcome Target

The potential-outcome definition is useful only when the treatment conditions are sufficiently clear. The assumptions below clarify what the outcomes refer to and why observed outcomes can be linked to them.

Well-defined treatment conditions: “Treatment” and “control” must specify conditions whose outcomes have a coherent interpretation. If treatment includes substantially different versions, the potential outcome under “treatment” may not be uniquely defined without further specifying which version is meant.
Consistency: a unit assigned to a condition has the outcome defined for that condition. In the notation above, $Y_i=Y_i(1)$ when $Z_i=1$, and $Y_i=Y_i(0)$ when $Z_i=0$.
No interference between units: one unit’s outcome under an assignment does not depend on other units’ assignments. If, for example, a participant’s outcome changes because their peers receive treatment, potential outcomes must represent assignment patterns or groups rather than only that participant’s own condition.
Random assignment and a usable design: the assignment mechanism is known and gives the relevant units a chance of receiving each condition. For the fixed-size result derived above, the design assigns a positive number to each arm and treats all fixed-size allocations as equally likely.

These conditions are not interchangeable. Randomization addresses how treatment assignment relates to potential outcomes within the study. It does not define an ambiguous treatment, prevent interference, or establish that the enrolled units represent a separate target population.

The Estimand Is Not a Regression Coefficient

An estimand is the precisely defined quantity a study aims to learn; the ATE above is one such target. An estimate is a value calculated from observed data. A regression coefficient is a model parameter estimated from those data, so it is not automatically identical to the ATE. Whether it estimates the ATE depends on the model, the treatment contrast represented by the coefficient, the population being averaged over, and the analysis design.

For example, a treatment coefficient in a regression model may describe a contrast according to that model’s parameterization and included predictors. The ATE, by contrast, is defined by averaging the individual potential-outcome differences over the target units. The two can align under suitable modeling and design conditions. But a coefficient can also represent a contrast conditional on particular predictor values, or a model-dependent summary, rather than the average contrast over the study population. The word “treatment” in a coefficient’s label does not settle which quantity it represents.

The distinction also separates a target from sampling or assignment variation. The ATE for a fixed set of units does not change when a different valid random assignment is drawn. A regression estimate based on the resulting observed data can change, because the assigned groups and observed outcomes can change. Uncertainty procedures describe variation in an estimate under a stated model or design; they do not redefine the estimand.

Worked Example: A Realized Coefficient Can Differ From the ATE

Use the four hypothetical participants from the first example. Suppose participants 1 and 2 are assigned to treatment, and participants 3 and 4 to control. The observed treatment scores are 10 and 12 points; the observed control scores are 11 and 15 points. Imagine a treatment-only regression reports a treatment-minus-control coefficient of $-2$ points for this assignment. This example uses the reported coefficient to illustrate the distinction; it does not derive the regression calculation.

The finite-population ATE remains $1.75$ points, as calculated from all four pairs of potential outcomes. The reported coefficient is $-2$ points for the realized assignment. In this particular allocation, the observed treatment mean is $(10+12)/2=11$ points and the observed control mean is $(11+15)/2=13$ points, so the realized contrast is $11-13=-2$ points.

There is no contradiction: the ATE is the fixed target for these four participants, while the coefficient is an estimate based on the outcomes revealed by this assignment. A different assignment could yield a different coefficient. In this example the realized estimate is 3.75 points below the ATE, since $-2-1.75=-3.75$. Randomization justifies an average-over-assignments result, not exact equality in every allocation.

Why the Distinction Matters

Stating the estimand before fitting a model makes the analysis easier to evaluate. A clear report identifies the outcome, the treatment and control conditions, the units being averaged over, and the direction of the contrast. It then describes the regression coefficient as an estimator and explains why that coefficient targets the stated quantity under the analysis used.

This prevents two common overclaims. First, a coefficient is not automatically “the treatment effect” merely because it is attached to a treatment term. Second, a randomized study does not automatically identify an average effect for people who were not represented by its target population. The design and the target answer different questions.

Common mistake: “The regression reports a treatment coefficient, so that coefficient is the ATE.” This conclusion skips the definition of the target and the interpretation of the model coefficient. Instead, define the ATE for a specified set of units using their treatment and control potential outcomes, then check whether the fitted coefficient estimates that particular average contrast under the study design and model.

Check Your Understanding

Answer each question using the potential-outcome and estimand distinctions introduced above.

In words, what does $Y_i(1)-Y_i(0)$ represent, and what are its units?
Why can the ATE for enrolled participants differ from an average effect claimed for a broader population?
In a fixed-size randomized study, what does “unbiased over assignments” mean? Does it guarantee that one realized estimate equals the ATE?
Give one reason a treatment regression coefficient might not automatically represent the ATE for the study units.
What additional issue arises if one participant’s outcome can depend on whether other participants receive treatment?

Participant	\(Y_i(1)\), treatment	\(Y_i(0)\), control	\(\tau_i=Y_i(1)-Y_i(0)\)
1	10	8	2
2	12	11	1
3	14	11	3
4	16	15	1