Potential outcomes and causal estimands
Define the intervention, population, and contrast before asking whether observed data identify an effect.
On this page
The question for today
What you’ll understand
- Distinguish observed outcomes from treatment-specific potential outcomes.
- Compare population, finite-sample, and treated-population effects.
- Separate the definition of an estimand from assumptions identifying it.
A causal effect compares outcomes under different interventions for the same target units. An observed treated-versus-untreated contrast instead compares different groups of units, whose underlying prospects may differ. Potential outcomes make that distinction explicit. We begin with a binary treatment A, a specified outcome measured at a specified time, and two well-defined interventions. The first task is to state what effect is wanted; an estimator or a fitted model cannot supply that missing scientific definition.
One observed outcome and two intervention-specific quantities
Write Y(1) and Y(0) for the outcome that would occur under the two treatment assignments. Under consistency, the observed outcome equals the potential outcome indexed by the treatment actually received. This notation also assumes that the treatment labels represent well-defined versions and that another unit’s treatment does not change this unit’s outcome. If spillovers matter, potential outcomes must instead depend on a relevant assignment vector or exposure mapping. Such an extension changes the estimand and cannot be repaired by adjusting a regression standard error.
The individual contrast is defined even though both potential outcomes are not jointly observed for an ordinary unit at one occasion. This missing-counterfactual problem does not make average effects meaningless; it means their identification requires additional information or assumptions. Repeated measurements are not automatically both potential outcomes either, because calendar time, carryover, and changing unit characteristics can differ. A hypothetical full potential-outcome table is useful for reasoning, provided it is not presented as something an observational dataset actually reveals.
The target population determines the averaging weights
The population average treatment effect averages over a defined population distribution. A finite-sample average instead averages the individual effects of the particular enrolled units. The average effect on the treated conditions on treatment actually received in the observational population, changing the weighting when effects vary with characteristics related to treatment selection. These quantities can differ without any estimator being wrong. Reporting a single coefficient as the effect while leaving the averaging population unnamed hides an essential part of the question.
A conditional average effect describes the expected contrast at a baseline covariate value. Averaging that function over the overall covariate distribution gives the population effect; averaging over the treated covariate distribution gives the treated effect under appropriate identification assumptions. A policy that changes treatment probabilities may target a different weighted contrast again. Outcome scale matters as well: a mean difference, risk ratio, and odds ratio are distinct estimands and need not rank effects the same way. Here we use mean differences throughout.
Definition, identification, and estimation are different steps
An estimand is identified if the maintained assumptions express it uniquely in terms of the observed-data distribution. Only after that question is settled do sampling variability, model fitting, and numerical error enter estimation. Under unconditional random treatment assignment with both arms possible, consistency and independence of assignment from potential outcomes identify the average effect by a difference of observed conditional means. In observational data that same algebraic difference generally mixes the causal contrast with differences in untreated prospects.
No-interference and well-defined-treatment conditions are substantive features of a design, not conclusions of a significance test. Likewise, random assignment within enrolled units does not by itself make those units representative of a broader population. A careful analysis names the intervention, outcome, population, and contrast, then lists assumptions linking that contrast to observable quantities. This sequence prevents an impressive predictive model from being mistaken for evidence about what would happen under a different intervention.
Work it through
Read a full illustrative table
Four hypothetical units have untreated outcomes (1,2,4,5) and treated outcomes (3,3,4,8). Units one and three receive treatment. Compute the finite average effect, the treated-unit average effect, and the observed mean difference.
Subtract within each row of the full hypothetical table to get effects (2,1,0,3). Their average is six divided by four, or 1.5.
Among the treated units, the effects are two and zero, averaging one. The observed outcome vector is (3,2,4,5), because each unit reveals only its assigned column.
The treated observed mean is (3+4)/2=3.5 and the untreated observed mean is (2+5)/2=3.5. Their difference is zero, even though the finite average effect is positive.
The full table distinguishes the estimands. Its missing entries would not ordinarily be available from the observed vector, so this calculation illustrates the identification problem rather than solving it from data.
The living notebook
Explore the model
Interactive lab · function
Intervention comparisons require a declared model
Use the toy structural rule Y=x² with coefficient a=1. Reset and set both controls to a=1,x=1; predict Y=1. Reset and set a=1,x=2; predict Y=4, so the model-implied contrast is three. The instrument computes a stipulated deterministic quadratic; it cannot establish that this is a real causal mechanism or display both potential outcomes of a measured unit. Control changes pause and retain time; Reset restores both controls and zero time.
- y at chosen x
- 0.608
- Tangent slope · 2ax
- 1.35
- Signed area · ax³/3
- 0.182
This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.
Look forA model can define counterfactual contrasts only after its intervention interpretation has been justified externally.
Pause & reason
Check your understanding
Your turn
Try it yourself
Three hypothetical units have Y(0)=(0,2,1), Y(1)=(1,2,4), and assignment A=(1,1,0). Compute their finite average effect, treated-unit average effect, and observed outcomes.
Take a moment to make a prediction or work it out on paper. Open a hint when you need one.
Give me a hint
Compute within-unit contrasts first; apply the assignment only when constructing the observed vector.
Compare with the solution
The unit effects are (1,0,3), giving a finite average of four thirds. The first two units are treated, so their average effect is one half. Observed outcomes are (1,2,1). The treated observed mean is three halves and the untreated mean is one, giving an observed difference one half. Its agreement with the treated-unit effect in this illustration does not establish a general identity or make the finite overall effect equal to one half.
Keep the idea
Take it with you
- A causal contrast compares intervention-specific outcomes under explicit consistency and interference assumptions.
- ATE, finite-sample effects, and ATT differ through their target populations.
- Defining an effect precedes identifying it and estimating it.
Read further
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