From Potential Outcomes to an Individual Effect
A treatment can help one person, have little effect on another, and harm a third. A single average may summarize the overall pattern, but it cannot by itself describe these differences. To define variation in causal effects, begin with the two potential outcomes for the same unit: the outcome that would occur under treatment and the outcome that would occur without it.
Consider a binary treatment, with treatment states coded 1 and 0. For unit \(i\), let \(Y_i(1)\) be the potential outcome under treatment and \(Y_i(0)\) the potential outcome without treatment. Both outcomes must refer to the same outcome measure and the same follow-up time. The individual treatment effect compares these two potential outcomes for that one unit.
The subtraction order matters: the convention here is treated potential outcome minus untreated potential outcome. If the outcome is a test score for which higher is better, a positive \(\tau_i\) indicates improvement under treatment. If the outcome is pain severity for which lower is better, a negative \(\tau_i\) indicates improvement. An effect of 3 points is not inherently beneficial or harmful without knowing how the outcome is coded.
The individual effect is defined even when the treatment is not assigned randomly. Randomization and other design or modeling assumptions matter for learning about effects from data; they are not needed to write down the contrast. The treatment states must, however, be meaningful comparisons, and both potential outcomes must describe the same unit on the same outcome scale. The introductory material on the missing counterfactual problem explains why, for a given unit, only one of these two outcomes is observed.
Calculating and Interpreting Effects
Worked Example: A Treatment That Reduces Pain
Suppose, hypothetically, that a patient’s pain score at a specified follow-up would be 7 on a 0-to-10 scale without a new treatment and 4 with it. Treat the stated values as that patient’s two potential outcomes for this illustration.
The individual effect is \(\tau_i=Y_i(1)-Y_i(0)=4-7=-3\) pain-score points.
The negative sign means the score is lower under treatment. Because lower pain scores are better, the treatment would reduce this patient’s pain score by 3 points relative to the untreated potential outcome. This is a comparison of potential outcomes, not a claim that a before-and-after change of 3 points would necessarily be caused by treatment.
An effect can also be zero: if \(Y_i(1)=Y_i(0)\), then \(\tau_i=0\) for that unit. A positive or negative effect is not a probability, and the effect’s units are the outcome’s units. If the outcome is measured in kilograms, for example, the difference is in kilograms; if it is a binary outcome coded 0 or 1, the individual difference can only be \(-1\), 0, or 1.
How Effects Can Vary Across Units
Effects are homogeneous across a set of units if their individual treatment effects are the same. They are heterogeneous if those effects differ in size or direction. Heterogeneity is a property of the collection of unit-specific contrasts; it does not require that the units have different observed outcomes before treatment.
Worked Example: Different Effects on the Same Outcome Scale
In a hypothetical study of a tutoring program, suppose three students have the following potential test scores, all measured in points out of 100.
| Student | \(Y_i(0)\): no tutoring | \(Y_i(1)\): tutoring | \(\tau_i=Y_i(1)-Y_i(0)\) |
|---|---|---|---|
| A | 60 | 68 | 8 points |
| B | 75 | 77 | 2 points |
| C | 82 | 79 | -3 points |
For student A, \(68-60=8\), so the treatment effect is an increase of 8 points. For student B, \(77-75=2\), an increase of 2 points. For student C, \(79-82=-3\), a decrease of 3 points. The effects vary in both size and direction, even though all three students have scores on the same scale.
These calculations use both potential outcomes for each student. In real data, the table’s two outcome columns cannot both be observed for the same student under a single treatment assignment. The entries here are hypothetical values that make the definition and the pattern of variation visible; they are not a recipe for recovering each student’s effect from an observed score.
Variation can arise even when all individual effects have the same sign. For example, effects of 1, 2, and 5 points are heterogeneous in size but all positive. Conversely, effects can differ in direction, as in the tutoring illustration. These distinctions matter: a summary that reports only whether an effect is generally positive can conceal important differences in magnitude or the presence of units with effects in the opposite direction.
Why Averages Do Not Reveal Individual Variation
Suppose a researcher knows the average outcome under treatment and the average outcome without treatment for a population. Those two averages describe the marginal distributions of potential outcomes, but they do not say which treated potential outcome belongs with which untreated potential outcome for each unit. Since an individual effect depends on that pairing, the averages alone cannot show how effects vary across units.
Worked Example: Identical Averages, Different Effect Patterns
Consider four hypothetical units. In both scenarios below, the untreated potential outcomes are 2, 4, 6, and 8. The treated potential outcomes are the same set of values in each scenario, namely 4, 6, 8, and 10; only the pairing of treated and untreated outcomes across units changes.
Scenario A: effects are paired in the same order. Pair untreated outcomes \(2,4,6,8\) with treated outcomes \(4,6,8,10\). The four effects are \(4-2=2\), \(6-4=2\), \(8-6=2\), and \(10-8=2\) points. Each unit has an effect of 2 points.
Scenario B: effects are paired in reverse order. Pair untreated outcomes \(2,4,6,8\) with treated outcomes \(10,8,6,4\). The four effects are \(10-2=8\), \(8-4=4\), \(6-6=0\), and \(4-8=-4\) points. The effects now range from an increase of 8 points to a decrease of 4 points.
In both scenarios the average untreated outcome is \((2+4+6+8)/4=5\) points, and the average treated outcome is \((4+6+8+10)/4=7\) points. Thus, knowing those two averages does not distinguish a constant effect from a pattern with substantial variation and even a negative effect for one unit. This is why an average summary should not be interpreted as a description of every individual.
The example isolates the information that is missing: the joint pairing of each unit’s two potential outcomes. Data generally reveal only one potential outcome per unit, so even knowing the separate distributions of outcomes under the two treatment states would not automatically reveal the distribution of individual differences. The individual effect is a well-defined causal quantity, but it is usually not directly observed or identified unit by unit.
What Can and Cannot Be Learned from Observed Outcomes
For an individual, the observed outcome under the assigned treatment is only one of \(Y_i(1)\) and \(Y_i(0)\). The other is the counterfactual outcome for that same unit. Consequently, subtracting an observed treated unit’s outcome from an observed untreated unit’s outcome does not calculate either person’s individual treatment effect: it compares outcomes from two different units.
A useful distinction is between three claims. The individual effect concerns one unit’s pair of potential outcomes. A group or population summary describes a collection of units and may conceal individual variation. An observed outcome difference compares what was recorded in groups and, without appropriate causal design or assumptions, need not equal a causal effect. The next material on average causal effects develops the population-level contrasts more formally.
This distinction also guards against over-interpreting subgroup patterns. If estimated effects differ across age groups, for example, that can suggest effect variation across those groups, but it does not show that every older person has one effect and every younger person another. A group-level pattern and a unit-level effect are different claims, and the data and assumptions must support the level of the claim being made.
Common Pitfalls and Checks
- Reversing the subtraction. Using \(Y_i(0)-Y_i(1)\) changes the sign. Use the course convention \(Y_i(1)-Y_i(0)\), then interpret the sign in light of the outcome scale.
- Treating an observed change as an individual causal effect. A person’s score before and after treatment is not automatically the pair \(Y_i(1)\) and \(Y_i(0)\). Time, events, and other changes can affect the later score; the untreated potential outcome at the same follow-up is still counterfactual.
- Assuming the average applies to everyone. A positive average does not imply that every unit benefits. The pairing example shows that identical outcome averages can be consistent with both constant and mixed effects.
- Claiming to observe an individual effect. One unit does not simultaneously reveal both potential outcomes under a single treatment assignment. Hypothetical calculations can define or illustrate an effect, but observed data alone do not supply the missing counterfactual.
- Ignoring outcome direction or units. A negative effect is not necessarily harmful. State what higher and lower outcomes mean and report the effect in the outcome’s units.
When interpreting a result, check that the two potential outcomes concern the same unit, treatment contrast, outcome, and follow-up time; verify the subtraction order; and separate a unit-specific claim from a group-level summary. These checks clarify what the definition says without implying that an individual effect can be directly read from an observed record.
Check Your Understanding
Answer each question using the definition \(\tau_i=Y_i(1)-Y_i(0)\). Solutions follow within the box.
- A hypothetical patient has a pain score of 6 under treatment and 9 without treatment. Calculate the individual effect and interpret it.
- For an outcome where higher is better, one unit has \(Y_i(1)=14\) and \(Y_i(0)=14\). What is the individual effect, and what does it mean?
- Can two potential-outcome averages determine whether every unit has the same individual effect? Explain briefly.
- Why is the observed difference between a treated person’s outcome and an untreated person’s outcome not either person’s individual treatment effect?
Solutions. (1) \(\tau_i=6-9=-3\) pain-score points; the treated potential outcome is 3 points lower, which indicates less pain on this scale. (2) \(\tau_i=14-14=0\) points; the two potential outcomes are equal for this unit. (3) No. The averages do not show how each unit’s two potential outcomes are paired; the worked pairing example has the same averages but different effects. (4) The two observed outcomes belong to different units, whereas an individual effect compares both potential outcomes for the same unit.