Why Treatment History Needs Its Own Notation
When treatment is assigned at several times, “treated” may not describe a unit’s exposure well enough. A patient might receive treatment at the first decision but not the second; another might receive it only at the second. Even if both are called treated at some point, their treatment histories differ, and their outcomes under those histories need not be the same.
Potential-outcomes notation makes this distinction explicit. It asks what outcome a particular unit would have under each possible sequence of treatments, not just under a single label such as “treatment” or “control.” These are conceptual, unit-level outcomes: the notation defines what comparisons mean, but by itself it does not tell us how to identify or estimate them from observed data.
Potential Outcomes for a Complete Sequence
Suppose there are \(K\) treatment decision times, ordered from 1 to \(K\). Let \(A_t\) denote the treatment received by a unit at time \(t\), and let \(a_t\) denote a particular value that treatment could take. The observed treatment history through time \(t\) is \(\bar A_t=(A_1,\ldots,A_t)\); a specified history is \(\bar a_t=(a_1,\ldots,a_t)\). The bar indicates the ordered collection of treatments, not an average.
The order matters. For two decision times, \(Y_i(1,0)\) means the outcome under treatment at time 1 followed by no treatment at time 2. It is generally a different potential outcome from \(Y_i(0,1)\), which reverses that order. If treatments are binary and there are \(K\) decision times, there are \(2^K\) possible complete sequences, and the notation allows a potential outcome for each one.
A potential outcome also needs a well-defined interpretation: the treatment values must describe sufficiently clear interventions. For example, if “treatment” includes versions with meaningfully different effects, the intervention must specify which version is intended or the notation must account for those versions. The unit-level notation \(Y_i(\bar a)\) also treats the outcome as depending on unit \(i\)’s sequence; its usual interpretation presumes that one unit’s assigned treatment does not change another unit’s outcome.
Worked Example: Two Decisions, Four Potential Outcomes
Consider a hypothetical patient with two treatment decisions, each coded 1 for treatment and 0 for no treatment. Let \(Y_i(a_1,a_2)\) be the patient’s symptom score at the end of the second period, with lower scores representing fewer symptoms. Suppose the patient’s potential outcomes are as follows.
| Sequence | Potential symptom score | Interpretation |
|---|---|---|
| \((0,0)\) | 12 | No treatment at either decision |
| \((0,1)\) | 10 | No treatment first, treatment second |
| \((1,0)\) | 8 | Treatment first, no treatment second |
| \((1,1)\) | 6 | Treatment at both decisions |
The sequence-specific contrast between treatment at both decisions and no treatment at either is
For this hypothetical patient, the score under \((1,1)\) is 6 points lower than under \((0,0)\). This is a contrast between two complete sequences. It is not the effect of a single, unspecified treatment, and it does not establish that the same contrast holds for other patients.
How This Differs from a Single-Treatment Potential Outcome
In the familiar single-decision setting, a unit has potential outcomes \(Y_i(1)\) and \(Y_i(0)\): the outcomes under treatment and no treatment, respectively. Each treatment assignment in that setting is enough to specify the intervention being compared. With repeated decisions, a single value such as 1 does not usually specify what happens at the other times. The sequence notation fills in those missing decisions.
For instance, in a two-decision setting, \(Y_i(1,1)\) refers to sustained treatment at both decisions. It is not automatically the same object as \(Y_i(1)\) in a single-decision study. A writer could define shorthand for a particular sequence, but must state which sequence the shorthand represents. Without that definition, “the outcome under treatment” is ambiguous.
Worked Example: The Same Final Treatment, Different Histories
Suppose a hypothetical unit has a binary treatment at each of three decision times. The final outcome is days in hospital, and the unit’s potential outcomes under two sequences are \(Y_i(1,1,0)=4\) and \(Y_i(0,1,0)=7\).
Both sequences specify no treatment at the third decision. They differ at the first decision: the first sequence starts treatment immediately, while the second does not. The sequence contrast is
Under these hypothetical potential outcomes, the first sequence yields 3 fewer hospital days for this unit than the second. Describing both histories only as “no treatment at the final decision” would discard information that distinguishes the interventions being compared. Likewise, calling both “ever treated” would conceal the timing and pattern of treatment.
Outcomes Measured Before the Sequence Is Complete
The outcome’s measurement time determines how much of the treatment sequence is relevant to its index. If an outcome is measured after the first decision but before the second, it is indexed by the first treatment, not by a treatment that has not yet been assigned. Write \(Y_{i,t}(\bar a_t)\) for unit \(i\)’s outcome measured at time \(t\) under the treatment sequence through time \(t\). For example, \(Y_{i,1}(1)\) is an outcome after the first treatment decision under treatment at that decision.
A final outcome measured after \(K\) decisions can be written \(Y_{i,K}(\bar a_K)\), or more simply \(Y_i(\bar a)\) when the measurement time is understood. An intermediate outcome can be indexed by a prefix of a longer sequence. The prefix notation avoids attributing an intermediate outcome to a later treatment decision that has not yet occurred.
Worked Example: An Intermediate Outcome and a Final Outcome
In a hypothetical two-period care plan, let the treatment at each decision be 0 or 1. A symptom score is measured after the first decision, and a final recovery score is measured after the second. Suppose the unit’s relevant potential outcomes are:
- After the first decision: \(Y_{i,1}(0)=9\) and \(Y_{i,1}(1)=7\).
- At the end of the second period: \(Y_{i,2}(1,0)=5\) and \(Y_{i,2}(1,1)=3\).
The first pair compares the symptom score under the two possible first-period treatments. The second pair compares final scores among sequences that both start with treatment but differ in the second-period decision:
Thus, for this hypothetical unit, the final score is 2 points lower under sequence \((1,1)\) than under \((1,0)\). The intermediate score \(Y_{i,1}(1)\) is indexed only by the first treatment, because the second decision comes later. The final contrast concerns the second decision while holding the first treatment fixed in the two specified sequences.
Observed Outcomes and the Potential-Outcome Contrast
For a given unit, only one treatment sequence is actually received. Consistency connects that observed sequence to the corresponding potential outcome: if unit \(i\) receives the sequence \(\bar A_i=\bar a\), then its observed outcome is \(Y_i=Y_i(\bar a)\). This statement identifies the potential outcome matching the observed history; it does not reveal the unit’s outcomes under the other possible histories.
A unit-level causal contrast between two sequences is defined by subtracting their potential outcomes. For example, for sequences \(\bar a\) and \(\bar a'\), the contrast \(Y_i(\bar a)-Y_i(\bar a')\) compares the outcome the same unit would have under each specified sequence. Its direction and units follow the outcome scale. If the outcome is symptom score, the contrast is in score points; if it is time to recovery, it is in units of time.
This is the unit-level version of the fundamental missing-counterfactual problem. A contrast such as \(Y_i(1,1)-Y_i(0,0)\) is well-defined even though only one of its two terms can be observed for unit \(i\). Defining the contrast does not, by itself, provide a way to learn its value from data. In particular, seeing different outcomes among units with different observed sequences does not automatically reveal what would have happened to the same units under the alternative sequence.
Worked Example: Matching the Observed Outcome to One Sequence
Suppose a hypothetical participant receives treatment at the first decision and no treatment at the second, so the observed sequence is \(\bar A_i=(1,0)\). Suppose the participant’s final observed outcome is 8 points on a scale where lower is better.
By consistency, the observed outcome corresponds to the potential outcome for the received sequence:
This information does not give the participant’s \(Y_i(0,0)\), \(Y_i(0,1)\), or \(Y_i(1,1)\). For example, if \(Y_i(0,0)\) is unknown, the unit-level contrast \(Y_i(1,0)-Y_i(0,0)\) cannot be calculated from the observed outcome alone. The observed value is one component of the potential-outcome collection, not a complete record of that collection.
Common Mistakes
- Using one treatment label for several histories. “Treated” may mean treatment at the first decision, at every decision, or at least once. State the sequence being compared.
- Ignoring order. In general, \(Y_i(1,0)\) and \(Y_i(0,1)\) are distinct potential outcomes. Treatment timing is part of the intervention.
- Indexing an outcome by treatment after it was measured. An outcome measured after the first decision is indexed by the treatment history through that measurement, not by a later treatment.
- Treating an observed outcome as if it reveals every potential outcome. Consistency identifies the potential outcome matching the received sequence; outcomes under alternative sequences remain unobserved for that unit.
- Calling a sequence contrast an identified effect. The contrast is a well-defined causal quantity, but estimating or identifying it requires additional assumptions and methods.
Check Your Understanding
Use the notation and distinctions in this tutorial to answer the following.
- With two binary treatment decisions, what intervention does \(Y_i(0,1)\) describe? How does it differ from \(Y_i(1,0)\)?
- For three binary decisions, how many complete treatment sequences are possible?
- A final outcome is measured after the second of two treatment decisions. What does \(Y_i(1,0)\) represent, and what does \(Y_i(1)\) leave unspecified?
- If a unit receives sequence \((0,1)\), which potential outcome is linked to its observed outcome by consistency? What does consistency not reveal?
- Suppose \(Y_i(1,1)=4\) and \(Y_i(1,0)=7\) on a symptom scale. Calculate and interpret the unit-level contrast in context.