Why Define the Causal Question First?
A regression analysis can compare outcomes across exposure groups, but the comparison is only meaningful once the causal question is clear. “What is the effect of treatment?” is incomplete unless treatment is specified: which treatment strategy, compared with which alternative, on what outcome scale, and for whom? A precisely stated causal question is an estimand: the target quantity an analysis aims to learn about.
This tutorial sets out a potential-outcomes framework for defining such quantities. It focuses on the causal contrast itself, not on whether observational data and a particular statistical method can identify or estimate it. Those are separate questions: defining a causal estimand does not, by itself, show that the available data can recover it.
Exposure Contrasts and Potential Outcomes
An exposure contrast names the two strategies or exposure levels to compare. For a binary exposure, label the strategies \(a=1\) and \(a=0\). For example, these might mean initiating a specified medication regimen and not initiating it. The labels alone are not enough: the strategies need substantive descriptions so that the comparison has a clear interpretation.
Potential outcomes describe a comparison, not two measurements that can ordinarily be collected from the same person at the same time. For a person who follows strategy 1, the observed outcome is \(Y_i(1)\), while \(Y_i(0)\) is unobserved. The unobserved outcome is often called the counterfactual for that person. This missing counterfactual is central to causal questions.
The outcome scale matters too. If \(Y\) is systolic blood pressure measured in mmHg, a difference in potential outcomes is measured in mmHg. If \(Y\) indicates whether a person experienced an event, the relevant causal contrast may instead be a difference or ratio of risks. The same exposure contrast can therefore have more than one valid estimand, each answering a different question.
Average Causal Effects for a Population
For a defined population, the average treatment effect (ATE) is the average of the individual effects for the contrast 1 versus 0. The term “treatment” is often used even when the exposure is not assigned by a researcher.
By linearity of averages, this can also be written as the difference between the two population mean potential outcomes:
This identity follows because the mean of a difference is the difference of the means. It does not require observing both potential outcomes for each person, nor does it establish that either mean can be calculated from the observed data alone.
Worked Example: Average Effect on a Continuous Outcome
Suppose a hypothetical finite population consists of four adults. The exposure contrast is initiating a specified treatment regimen (\(1\)) versus not initiating it (\(0\)); the outcome is systolic blood pressure at a stated follow-up, in mmHg. For illustration, suppose all eight potential outcomes are known:
| Adult | \(Y_i(1)\) (mmHg) | \(Y_i(0)\) (mmHg) | \(Y_i(1)-Y_i(0)\) (mmHg) |
|---|---|---|---|
| 1 | 138 | 145 | -7 |
| 2 | 132 | 136 | -4 |
| 3 | 150 | 154 | -4 |
| 4 | 142 | 143 | -1 |
The finite-population ATE is the mean of the four individual contrasts:
For this hypothetical population and these specified strategies, the average potential systolic blood pressure under initiation is 4 mmHg lower than under non-initiation. This is a causal interpretation of the defined estimand, not a claim that every adult’s pressure would fall by 4 mmHg. In an actual study, each adult’s two potential outcomes would not both be observed.
Binary Outcomes: Differences and Ratios of Risks
For a binary outcome, let \(Y(a)=1\) mean that the event occurs under strategy \(a\), and \(Y(a)=0\) mean that it does not. The risk under strategy \(a\) is \(P(Y(a)=1)\). Two common population estimands compare these risks: the risk difference and the risk ratio. Their numerical values depend on the outcome definition, exposure contrast, and population.
Worked Example: A Risk Difference and Risk Ratio
Consider six hypothetical people and the binary outcome of experiencing an event during a defined follow-up. The exposure contrast is receiving strategy 1 versus strategy 0. The table lists both potential outcomes for each person only to illustrate the estimand; it is not a claim that both outcomes would be observed in practice.
| Person | \(Y_i(1)\) | \(Y_i(0)\) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 1 | 1 |
| 3 | 0 | 1 |
| 4 | 1 | 0 |
| 5 | 0 | 0 |
| 6 | 1 | 0 |
Under strategy 1, three of the six potential outcomes equal 1, so the risk is \(3/6=0.500\). Under strategy 0, two of the six equal 1, so the risk is \(2/6=0.333\). Thus:
In this hypothetical population, the risk under strategy 1 is 0.167 higher, or 16.7 percentage points higher, than under strategy 0. The risk under strategy 1 is 1.50 times the risk under strategy 0. These describe two scales for the same contrast; they are not interchangeable, and neither statement says that every person’s outcome changes in the same way.
Which People Are Averaged?
An estimand also specifies which people contribute to the average. The ATE averages individual effects over the full population of interest. Sometimes the question instead concerns people who actually received the exposure. The average treatment effect among the treated (ATT) averages their effects, comparing their outcomes under exposure with the outcomes those same people would have had under the alternative.
The ATE and ATT need not be equal: the average effect among exposed people may differ from the average effect in the full population. Choosing between them is a substantive decision about the question, not simply a choice of regression coefficient. Neither definition is the observed difference in mean outcomes between exposure groups.
Worked Example: Comparing the ATE, ATT, and Observed Difference
Suppose six hypothetical people have the potential earnings, in thousands of dollars, shown below under participation (\(1\)) or non-participation (\(0\)) in a training program. The actual exposure \(A_i\) is also shown.
| Person | \(A_i\) | \(Y_i(1)\) | \(Y_i(0)\) | Individual effect |
|---|---|---|---|---|
| 1 | 1 | 30 | 20 | 10 |
| 2 | 1 | 25 | 20 | 5 |
| 3 | 1 | 40 | 35 | 5 |
| 4 | 0 | 50 | 50 | 0 |
| 5 | 0 | 40 | 25 | 15 |
| 6 | 0 | 25 | 15 | 10 |
The ATE averages all six individual effects:
The ATT averages only the effects for the three people with \(A_i=1\):
The observed outcome is \(Y_i(1)\) for the three participants and \(Y_i(0)\) for the three non-participants. The observed participant mean is \((30+25+40)/3=31.67\) thousand dollars; the observed non-participant mean is \((50+25+15)/3=30.00\) thousand dollars. Their difference is \(31.67-30.00=1.67\) thousand dollars, not the ATE of 7.50 thousand dollars. The observed group difference compares different people; the ATE compares the two potential outcomes averaged over the same population. This example defines the quantities and illustrates their difference; it does not establish what an analysis of real observational data could identify.
State an Estimand Precisely
Before fitting a model, write down the causal quantity in words and, when useful, as a formula. A clear statement identifies the exposure strategies being compared, the outcome and its scale, the population whose outcomes are averaged, and the summary measure. For example, “the mean difference in systolic blood pressure, in mmHg, under initiating this regimen versus not initiating it, averaged over the stated population” makes the contrast and scale explicit.
A compact checklist helps keep these pieces aligned:
- Exposure contrast: What does each exposure strategy mean?
- Outcome: What is measured, and when is it defined?
- Population: For which people is the average intended?
- Effect measure: A mean difference, risk difference, risk ratio, or another explicitly named quantity?
- Average: Is the target the full population average or an average for a subgroup such as those exposed?
These are parts of the question being asked, not conclusions supplied by a regression model. A model can produce a numerical estimate, but interpreting it causally requires further justification about the design, data, and assumptions. Keep the estimand distinct from both the observed association and the method used to estimate it.
Common Mistake: Treating an Observed Difference as the Causal Effect
A tempting argument is: “The exposed group has a mean outcome 5 units higher, so exposure caused an increase of 5 units.” That conclusion does not follow from the group means alone. The groups may differ in who received exposure, and the observed comparison uses each group’s outcome under its actual exposure rather than both strategies’ outcomes for the same people. The potential-outcomes definition makes the missing comparison explicit.
Check Your Understanding
For each question, identify the relevant contrast or explain the distinction using potential outcomes.
- For a binary exposure, write the individual causal effect and the population ATE using potential-outcome notation.
- A study compares a medication strategy with usual care and measures a continuous outcome in points. What information should be stated to make the exposure contrast and effect measure interpretable?
- For a binary event, the risks under two strategies are 0.10 and 0.15. Calculate the risk difference for strategy 1 versus strategy 0 and state its interpretation in percentage points.
- In words, distinguish the ATE from the ATT.
- Why does a difference in observed mean outcomes between exposed and unexposed groups not, by itself, equal the ATE?