Potential Outcomes, Observed Outcomes, and Causal Effects

Why Causal Effects Need Two Outcomes

To ask what a treatment caused for a particular unit is to compare what happened under treatment with what would have happened to that same unit without treatment. The difficulty is immediate: at a given time, a unit is observed in one treatment condition, not both. The potential-outcomes framework makes this comparison precise by defining both outcomes conceptually, even though only one is observed.

This framework separates three ideas that are easy to conflate: the outcomes a unit would have under each treatment condition, the treatment condition actually assigned, and the outcome actually recorded. Keeping these distinct clarifies what a causal effect means and why it cannot be read directly from a single unit’s observed data.

Definition: For unit $i$, let $Y_i(1)$ denote the potential outcome under treatment condition 1 and $Y_i(0)$ the potential outcome under treatment condition 0. The unit’s individual treatment effect is $$ \tau_i = Y_i(1)-Y_i(0). $$ It is a contrast between two potential outcomes for the same unit, measured in the units of the outcome.

The labels 1 and 0 identify the two conditions being compared; they do not mean that the outcome itself is binary. For example, if the outcome is days without symptoms, both potential outcomes may be numbers between 0 and 30. A positive $\tau_i$ means the outcome is larger under condition 1 than under condition 0. Whether that is beneficial depends on what the outcome measures.

The potential outcomes must refer to clearly specified treatment conditions and an outcome measured on a defined scale and time frame. The framework defines a contrast; by itself, it does not establish that an observed association identifies that contrast. Identification and design assumptions are separate matters.

Assignment and the Observed Outcome

Let $W_i$ be the treatment assignment indicator: $W_i=1$ if unit $i$ is assigned to condition 1 and $W_i=0$ if assigned to condition 0. In the basic two-condition setup, the observed outcome $Y_i$ is the potential outcome corresponding to the assigned condition.

Definition: The observed-outcome equation is $$ Y_i = W_iY_i(1)+(1-W_i)Y_i(0). $$ If $W_i=1$, this gives $Y_i=Y_i(1)$. If $W_i=0$, it gives $Y_i=Y_i(0)$. The potential outcome for the other condition remains unobserved for that unit.

This equation describes which potential outcome is observed; it is not a method for calculating both potential outcomes from the data. In particular, the unobserved outcome is not zero and is not necessarily equal to the observed one. It is a counterfactual quantity: the outcome that would have been recorded for that unit under the other condition.

Worked Example: Individual Effects in a Hypothetical Finite Group

Suppose four hypothetical participants have potential outcomes measured as days without symptoms during a 30-day period. The table lists both potential outcomes for teaching purposes; in an actual one-time comparison, both would not be observed for each participant.

Unit	$Y_i(1)$ (days)	$Y_i(0)$ (days)	$\tau_i=Y_i(1)-Y_i(0)$ (days)
A	22	18	4
B	20	21	-1
C	25	22	3
D	19	17	2

For unit A, the individual effect is $22-18=4$ days. Under the stated outcome scale, this means A would have 4 more symptom-free days under condition 1 than under condition 0. For unit B, the effect is $20-21=-1$ day, so the outcome would be one day lower under condition 1. Effects can differ across units, including in sign.

The average individual effect for these four units is $$ \frac{4+(-1)+3+2}{4}=\frac{8}{4}=2\text{ days}. $$ This is an average causal contrast for this particular group, not a claim that every participant gains two days.

Average Treatment Effects

Individual effects describe unit-specific contrasts. Researchers often need a summary across a group or a target population. The relevant average depends on which collection of units is being described.

Definition: For a fixed finite population of $N$ units, the finite-population average treatment effect is the average of their individual effects: $$ \tau_{\mathrm{FP}} = \frac{1}{N}\sum_{i=1}^{N}\left[Y_i(1)-Y_i(0)\right]. $$ It is expressed in the outcome’s units and summarizes the average contrast for those $N$ units.

For a population formulation, consider a unit drawn from a specified target population, with potential outcomes represented by random variables $Y(1)$ and $Y(0)$. The population average treatment effect is

\tau_{\mathrm{P}} = E\left[Y(1)-Y(0)\right],

provided the expectations exist. This is the average individual contrast in that population. By linearity of expectation, it can also be written $E[Y(1)]-E[Y(0)]$. The finite-population and population quantities are defined over different collections: the first averages over a specified set of $N$ units, while the second averages over a target population.

Worked Example: Computing a Finite-Population Average Effect

Return to the four hypothetical participants in the table. Their individual effects are 4, $-1$, 3, and 2 days, and the finite population has $N=4$ units. Therefore,

\tau_{\mathrm{FP}} = \frac{1}{4}\{(22-18)+(20-21)+(25-22)+(19-17)\} = \frac{4-1+3+2}{4} = 2\text{ days}.

The same result follows by averaging each potential-outcome column first:

\frac{22+20+25+19}{4} - \frac{18+21+22+17}{4} = \frac{86}{4}-\frac{78}{4} = 21.5-19.5 = 2\text{ days}.

This calculation summarizes the four units’ defined potential outcomes. It does not show that those outcomes would all be available in observed data.

The equality used above is a simple but useful identity. For a fixed finite population, expand the average of the individual differences:

\begin{aligned} \tau_{\mathrm{FP}} &= \frac{1}{N}\sum_{i=1}^{N}\left[Y_i(1)-Y_i(0)\right]\\ &= \frac{1}{N}\sum_{i=1}^{N}Y_i(1) - \frac{1}{N}\sum_{i=1}^{N}Y_i(0)\\ &= \bar{Y}(1)-\bar{Y}(0), \end{aligned}

where $\bar{Y}(1)$ and $\bar{Y}(0)$ here denote the averages of the $N$ units’ potential outcomes under conditions 1 and 0, respectively. The second line follows because a sum of differences equals the difference of the sums. This identity is about the defined potential outcomes; it does not imply that both averages can be calculated from the observed outcomes.

Worked Example: A Population Average Treatment Effect

Consider a hypothetical target population with three types of units. The listed proportions describe the population, and each type has the same potential outcomes within that type. Outcomes are measured in points.

Type	Population proportion	$Y(1)$ (points)	$Y(0)$ (points)	Individual effect (points)
A	0.50	9	5	4
B	0.30	3	5	-2
C	0.20	7	6	1

The population proportions sum to $0.50+0.30+0.20=1.00$. Weight each type’s effect by its population proportion:

\tau_{\mathrm{P}} = (0.50)(4)+(0.30)(-2)+(0.20)(1) = 2.00-0.60+0.20 = 1.60\text{ points}.

Thus, the population average contrast is 1.60 points. This does not mean that every type has a positive effect: type B has an effect of $-2$ points. As a check, the population average under condition 1 is $(0.50)(9)+(0.30)(3)+(0.20)(7)=6.80$ points, and under condition 0 it is $(0.50)(5)+(0.30)(5)+(0.20)(6)=5.20$ points. Their difference is $6.80-5.20=1.60$ points.

The Missing Counterfactual

For one unit at one defined occasion, assignment selects one of the two conditions. If $W_i=1$, the observed outcome reveals $Y_i(1)$, while $Y_i(0)$ is unobserved. If $W_i=0$, the observed outcome reveals $Y_i(0)$, while $Y_i(1)$ is unobserved. This is why an individual treatment effect is well-defined in the framework but cannot generally be calculated from that unit’s single observed outcome.

The limitation is not solved merely by collecting more measurements of the same observed outcome under the same condition. To know the individual contrast, one would need information about the outcome for that same unit under the alternative condition, under the same relevant circumstances. The unobserved potential outcome is therefore often called the missing counterfactual.

Across units, both treatment conditions may be represented in the data, but each unit still contributes only its outcome under its own assignment. Comparing the average observed outcome among treated units with the average among untreated units compares outcomes from different sets of units. That difference is not, by definition alone, the average of the individual effects. Whether it can identify an average causal effect depends on the assignment process and additional conditions.

Common Mistake: Treating an Observed Difference as the Effect

Incorrect reasoning: “The treated group’s observed mean is 3 points higher, so treatment caused a 3-point average increase.”
Correction: The observed mean difference is a comparison between observed outcomes in two groups. The causal effect is defined using contrasts between potential outcomes for the same units, or an average of those contrasts. The observed difference equals a causal estimand only when suitable design and identification conditions justify that interpretation.
Another error: Filling in an unobserved potential outcome with zero, or with the unit’s observed outcome. Neither substitution follows from the definition. The missing potential outcome is unknown, not a recorded zero and not automatically equal to the observed value.

Key takeaway: Potential outcomes define the causal comparison; assignment determines which potential outcome is observed. Individual and average treatment effects are contrasts of potential outcomes, while observed data reveal only one potential outcome per unit in the basic two-condition setup.

Check Your Understanding

Use the definitions and distinctions in this tutorial to answer the following questions.

If a unit has $Y_i(1)=12$ hours and $Y_i(0)=9$ hours, what is its individual treatment effect, and how should its sign be interpreted?
For $N$ units, explain what collection of units the finite-population average treatment effect summarizes.
A unit is assigned to condition 0. Which potential outcome is observed, and which remains counterfactual?
Why does observing a unit’s outcome under treatment not, by itself, reveal that unit’s individual treatment effect?
In what sense is a treated-versus-untreated observed mean difference different from an average of individual treatment effects?