Potential Outcomes for Two Treatment States

Why Index Outcomes by Treatment State?

In a causal study, the treatment state describes what happens to a unit, while the outcome describes a measurement for that unit. The earlier material on units, treatments, and outcomes distinguishes these roles. For a binary treatment, there are two states to name: control, coded \(0\), and treatment, coded \(1\). Potential-outcome notation attaches an outcome to each state for each unit.

For example, suppose the treatment is access to a tutoring program and the outcome is a student’s score on a specified exam. The notation records the score associated with control for a particular student and the score associated with tutoring for that same student. These are unit-level quantities: they are not, by definition, averages for the control and treatment groups.

Definition: For unit \(i\), let \(Y_i(0)\) denote that unit’s potential outcome under the control state, and let \(Y_i(1)\) denote that unit’s potential outcome under the treatment state. The treatment state is binary, with \(0\) denoting control and \(1\) denoting treatment. Both potential outcomes refer to the same defined unit and the same outcome scale and measurement target.

The subscript \(i\) identifies the unit; the argument in parentheses identifies the treatment state. Thus, \(Y_{12}(1)\) means the potential outcome for unit 12 under treatment, not the twelfth outcome measurement and not an outcome averaged across 12 units. The numbers \(0\) and \(1\) label the treatment states; they are not outcome values.

A potential outcome must be interpreted in context. “Treatment” needs to refer to a specified state, such as receiving a defined program, rather than to a vague label. The outcome also needs a clear scale and measurement target: for example, exam points at the end of a term or symptom-free days during a particular week. If the outcome or treatment state is not specified clearly enough to say what is being measured or assigned, the symbols alone cannot resolve that ambiguity.

A Compact Notation for the Two States

It is sometimes useful to write \(Y_i(a)\), where \(a\) is the treatment state. In the binary case, \(a\) can take only the values 0 and 1. The notation can be summarized by a single expression:

Y_i(a)=(1-a)Y_i(0)+aY_i(1), \qquad a\in\{0,1\}.

This expression selects the potential outcome for the specified state. When \(a=0\), the coefficient of \(Y_i(0)\) is \(1\) and the coefficient of \(Y_i(1)\) is \(0\). When \(a=1\), those coefficients reverse. It is a notational shortcut for choosing one of two indexed outcomes, not a model that predicts outcomes for intermediate treatment values.

\begin{aligned} a=0:\quad Y_i(0)&=(1-0)Y_i(0)+0Y_i(1)=Y_i(0),\\ a=1:\quad Y_i(1)&=(1-1)Y_i(0)+1Y_i(1)=Y_i(1). \end{aligned}

The calculation establishes the selector’s meaning directly: it returns the control potential outcome for \(a=0\) and the treatment potential outcome for \(a=1\). No assumption about random assignment, covariates, or a statistical model is needed to define these symbols. Those issues concern how potential outcomes relate to study data or how they may be used in an analysis, not the basic indexing convention.

Worked Example: Exam Scores Under Tutoring and Control

Consider one hypothetical student. Let \(a=1\) mean access to a defined tutoring program and \(a=0\) mean control. The outcome is the student’s score, in exam points, on the same end-of-term exam. Suppose the two potential outcomes for this student are \(Y_i(0)=72\) points and \(Y_i(1)=81\) points. Use the selector notation to identify the outcome indexed by each treatment state.

For control, substitute \(a=0\) into the formula:

Y_i(0)=(1-0)(72)+0(81)=1(72)+0(81)=72\text{ points}.

For treatment, substitute \(a=1\):

Y_i(1)=(1-1)(72)+1(81)=0(72)+1(81)=81\text{ points}.

The subscripts and state labels make clear that both numbers concern the same student and the same exam outcome, with the treatment state differing. The values are hypothetical inputs for this example; the calculation only shows how the notation selects a state-specific value.

What the Notation Does—and Does Not—Say

Potential outcomes are indexed by both a unit and a treatment state. The notation therefore keeps two questions separate: which unit? and under which state? A group mean, such as the average score among students assigned to tutoring, is a summary across units; it is not the definition of an individual student’s \(Y_i(1)\). Likewise, \(Y_i(0)\) is not a generic average for all students in control.

The two values for a unit must use a common outcome definition if they are to be compared meaningfully. If \(Y_i(0)\) were a score at the end of one term and \(Y_i(1)\) a score on a different exam, the symbols would not describe the same outcome target. Similarly, changing the treatment definition while keeping the same label \(1\) can change what the notation means. State the treatment and outcome in words before interpreting the indexed values.

This notation is also distinct from a regression prediction. A fitted model may produce a predicted outcome for a unit under a specified covariate pattern and treatment setting. A potential outcome, by definition, is the unit-level outcome indexed by a treatment state. The model and its assumptions may be used in an analysis involving such quantities, but a prediction from a model and the definition of \(Y_i(a)\) are not interchangeable.

Worked Example: Counting Symptom-Free Days

In a hypothetical example, the treatment is use of a specified symptom-management app during a seven-day period, coded \(a=1\); control means no app, coded \(a=0\). The outcome is the number of symptom-free days in that period. For one unit, suppose the daily indicators of a symptom-free day would be \(0,1,0,0,1,0,0\) under control and \(1,1,0,1,1,0,1\) under treatment. Define the outcome as the sum of the seven daily indicators.

Under control, the outcome is

Y_i(0)=0+1+0+0+1+0+0=2\text{ symptom-free days}.

Under treatment, it is

Y_i(1)=1+1+0+1+1+0+1=5\text{ symptom-free days}.

For the treatment-state index \(a=1\), the selector gives

Y_i(1)=(1-1)(2)+1(5)=0+5=5\text{ symptom-free days}.

The daily indicators are just a way to construct the stated outcome scale; the potential outcome is the resulting seven-day count. This example also shows why the measurement period belongs in the definition: a count over seven days is not automatically the same outcome as a count over a different period.

Conditions for a Clear Definition

The notation is simple, but its interpretation depends on a few basic specifications. These are conditions for saying what the quantities mean, rather than statistical assumptions that make a particular estimator valid.

Two treatment states are identified. The coding must say which state is control (\(0\)) and which is treatment (\(1\)). If the treatment variable has more than two meaningful states, two-state notation does not describe all of them without further definition.
The unit is identified. The index \(i\) must refer to the same unit under both states. Changing from a person to a clinic, for example, changes the unit being described.
The outcome target is shared. Both indexed outcomes must refer to the same type of outcome, scale, and measurement target. The treatment state changes; the meaning of the outcome should not silently change with it.
The treatment states are described precisely enough to interpret. A label such as “program” is not informative unless the intended state is clear enough to distinguish it from control in the study context.

A common error is to interpret \(Y_i(1)\) as “the outcome if unit \(i\) has a high value of the treatment.” In this binary notation, \(1\) is the designated treatment state, not a high level on an unspecified continuum. Another error is to treat \(Y_i(0)\) and \(Y_i(1)\) as group-level means. Both errors confuse the unit-level, state-indexed definition with a different quantity.

Worked Example: Checking the Outcome and Treatment Coding

Suppose a hypothetical clinic defines \(a=1\) as receiving a text-message reminder and \(a=0\) as receiving no reminder. The outcome is the number of missed appointments during the next 30 days. For one hypothetical patient, suppose \(Y_i(0)=2\) missed appointments and \(Y_i(1)=1\) missed appointment. Determine what the values mean and evaluate the state-indexed expression for control.

The treatment labels identify the two states, and both outcomes use the same patient, 30-day period, and count scale. The example therefore gives interpretable values under the stated definitions. For control, set \(a=0\):

Y_i(0)=(1-0)(2)+0(1)=2+0=2\text{ missed appointments}.

The result is the control-indexed count for this patient, not the average number of missed appointments among all patients receiving no reminder. If the outcome under treatment had instead been defined as missed appointments over 60 days, the pair would not share the same outcome target, and the two numbers would need clearer definitions before being interpreted together.

Common Pitfalls and Practical Checks

Before using potential-outcome notation, write down the unit, the treatment coding, and the outcome in words. Then check that the subscript and argument match those definitions. These quick checks catch many avoidable interpretation errors.

Reversing the state labels. If the study codes treatment as \(0\) and control as \(1\), do not silently use the convention stated here. Record the study’s coding explicitly.
Confusing a treatment label with an outcome value. In \(Y_i(1)\), the \(1\) labels the treatment state. It does not mean that the outcome equals one.
Changing the measurement target between states. Check that units, time period, and outcome definition match for \(Y_i(0)\) and \(Y_i(1)\).
Replacing an individual quantity with a group summary. Keep \(Y_i(a)\), a unit-level potential outcome, distinct from a mean or other summary calculated across units.
Assuming the notation itself makes an analysis valid. Defining potential outcomes does not, by itself, establish that a comparison or estimate identifies a causal quantity.

The main alternative to this notation is to report observed outcomes by treatment group without indexing outcomes by state. Such summaries can be useful descriptions, but they do not provide the same unit-by-state vocabulary. Potential-outcome notation is useful precisely because it states which outcome is being discussed and under which treatment condition, while leaving questions about data and identification to the relevant analysis.

Key takeaway: \(Y_i(0)\) and \(Y_i(1)\) name the potential outcomes for the same unit under control and treatment. Define the unit, both treatment states, and the shared outcome target before interpreting either value.

Check Your Understanding

Answer each question using the unit, treatment state, and outcome definitions in the prompt.

A study codes a dietary program as \(1\) and usual care as \(0\). What does \(Y_8(0)\) denote if the unit is a participant and the outcome is their cholesterol measurement at six months?
For one hypothetical unit, \(Y_i(0)=10\) days and \(Y_i(1)=7\) days. Evaluate the selector expression for \(a=1\), including units.
Why is \(Y_i(1)\) not the mean outcome among all units receiving treatment?
A researcher defines \(Y_i(0)\) as a 30-day count and \(Y_i(1)\) as a 90-day count. What must be checked before the two values can be treated as potential outcomes for the same outcome target?
In the binary selector formula, what does the argument \(a\) identify, and what values can it take?

Solutions. (1) It denotes participant 8’s potential cholesterol measurement at six months under usual care, the state coded \(0\). (2) \(Y_i(1)=(1-1)(10)+1(7)=0+7=7\) days. (3) The subscript \(i\) specifies one unit, whereas a mean summarizes outcomes across multiple units. (4) The researcher must establish a common outcome target, including a comparable measurement period; as stated, the periods differ. (5) \(a\) identifies the treatment state and can be \(0\) or \(1\) in this binary notation.