Why the Treatment Plan Must Include Time
With a single treatment decision, it may be enough to compare an outcome under treatment with an outcome without treatment. In longitudinal settings, treatment is assigned repeatedly, and decisions may respond to information that changes over time. “Treat” is therefore incomplete unless we specify when treatment is given and, for an adaptive plan, what information guides each decision.
A regime gives that specification. Its counterfactual outcome is the outcome an individual would have under that plan, including the treatment decisions the plan would generate as the individual’s history evolves. This is a potential outcome indexed by a plan, not simply the observed outcome grouped by the treatment actually received.
Notation for Treatment Sequences and Histories
Suppose treatment is assigned at times \(t=0,1,\ldots,K\), and the outcome \(Y\) is measured after the final treatment decision. Let \(A_t\) denote treatment at time \(t\), and let \(L_t\) denote covariate information available before that decision. For example, \(L_t\) might include a current biomarker or disease-severity measure.
A bar denotes a history or sequence. Thus \(\bar A_t=(A_0,\ldots,A_t)\) is the treatment history through time \(t\), while \(\bar L_t=(L_0,\ldots,L_t)\) is the covariate history through time \(t\). A fixed treatment sequence through the final decision is written \(\bar a=(a_0,\ldots,a_K)\).
A fixed sequence specifies the same planned treatment at each decision time for everyone assigned to that plan. For instance, \(\bar a=(1,1,1)\) represents treatment at all three decisions when \(A_t\) is binary and 1 denotes treatment. The sequence \((0,0,0)\) represents no treatment at any of those decisions. Such a plan is often called a sustained or static regime. Its treatment assignments do not change in response to a person’s evolving covariates.
A dynamic regime instead specifies a decision rule at each time. The rule may use the covariate and treatment history available before the decision. Write \(d_t(\bar L_t,\bar A_{t-1})\) for the treatment assigned at time \(t\) by that rule. At \(t=0\), there is no prior treatment history, so the rule uses baseline information. A dynamic regime is the collection of these rules, \(d=(d_0,\ldots,d_K)\).
The final clause matters. A dynamic rule does not generally name one treatment sequence in advance for every individual. The covariates used at a later decision may themselves differ depending on earlier treatment. Therefore, the rule must be applied in sequence to the counterfactual history produced under that rule.
How a Dynamic Regime Generates a Counterfactual Path
To make the notation explicit, let \(L_t^d\) be the covariate value that would be present at time \(t\) under regime \(d\), and let \(A_t^d\) be the treatment that regime would assign then. The regime generates its treatment path recursively:
For a fixed sequence, by contrast, \(A_t^{\bar a}=a_t\) at each decision. The counterfactual covariate history and final outcome depend on the treatment path. One way to express the connection is to let \(Y^{\bar a}\) denote the outcome under each possible fixed sequence. Once the dynamic rule generates its own sequence for an individual, the regime-specific outcome is the outcome under that generated sequence:
This notation is a compact statement of the recursive idea, not a way to calculate the outcome from the rule alone. To know the counterfactual path, we also need to specify how the covariates and outcome would behave under the treatments. The rule determines which treatment to assign given a history; it does not, by itself, determine what that history or the final outcome will be.
Use the individual’s baseline covariates as the history available for the first decision.
Set \(A_0^d=d_0(L_0^d)\), then consider the covariate history that would follow under that treatment.
At each later decision, evaluate the rule using the counterfactual history accumulated under the regime so far.
After the final treatment, \(Y^d\) denotes the outcome under the treatment path generated by the rule.
Worked Examples
Worked Example: Comparing Two Sustained Plans for One Individual
Suppose three binary treatment decisions are made, and \(Y\) is the number of symptom days in the following month. For a hypothetical individual, suppose the potential outcomes under two sustained plans are \(Y^{(0,0,0)}=12\) days and \(Y^{(1,1,1)}=7\) days. The observed treatment sequence is \((0,0,0)\), and the observed outcome is 12 days.
Under the no-treatment plan, the relevant potential outcome is \(Y^{(0,0,0)}\), which is 12 symptom days for this individual. Under the sustained-treatment plan, the relevant potential outcome is \(Y^{(1,1,1)}\), which is 7 symptom days. The individual-level arithmetic difference between these two stipulated counterfactual values is \(7-12=-5\) days.
The calculation describes the difference between the two potential outcomes in this hypothetical illustration. It does not mean that the individual’s outcome under both plans was observed: only the outcome under the actual sequence was observed. The other value is counterfactual.
Worked Example: Applying a Biomarker-Based Rule
Consider three treatment decisions, \(A_0,A_1,A_2\). A hypothetical adaptive regime assigns treatment when the current biomarker is at least 6 units and assigns no treatment otherwise. For the first decision, use the baseline biomarker \(L_0\); for later decisions, use the biomarker measured before that decision. Suppose that under this regime the individual’s counterfactual biomarker values are \(L_0^d=4\), \(L_1^d=7\), and \(L_2^d=5\), and the outcome under the generated path is 10 symptom days.
At the first decision, \(4<6\), so \(A_0^d=0\). At the second decision, \(7\geq6\), so \(A_1^d=1\). At the third decision, \(5<6\), so \(A_2^d=0\). The rule therefore generates the sequence \(\bar A^d=(0,1,0)\), and this individual’s regime-specific outcome is \(Y^d=Y^{(0,1,0)}=10\) symptom days.
The sequence \((0,1,0)\) is not the definition of the regime itself. It is the sequence this particular history leads the rule to assign. If a different counterfactual biomarker history arose, the same rule could produce a different sequence.
Worked Example: One Rule, Different Individual Treatment Paths
Use the same threshold rule: treat at a decision if the current biomarker is at least 6 units. For two hypothetical individuals, suppose the counterfactual biomarker histories under the rule and their outcomes are as follows.
| Individual | \(L_0^d\) | \(L_1^d\) | \(L_2^d\) | Generated treatment path | \(Y^d\) (symptom days) |
|---|---|---|---|---|---|
| R | 7 | 4 | 8 | (1, 0, 1) | 6 |
| S | 3 | 8 | 7 | (0, 1, 1) | 11 |
For individual R, the rule gives \(A_0^d=1\) because \(7\geq6\), \(A_1^d=0\) because \(4<6\), and \(A_2^d=1\) because \(8\geq6\). Thus \(\bar A^d=(1,0,1)\) and \(Y^d=6\) symptom days. For individual S, the corresponding comparisons are \(3<6\), \(8\geq6\), and \(7\geq6\), giving \(\bar A^d=(0,1,1)\) and \(Y^d=11\) symptom days.
Both individuals are governed by the same decision rule, but their histories generate different treatment paths and their regime-specific outcomes differ. The rule is the intervention specification; the path and outcome are individual-specific counterfactual consequences of following it.
What the Superscript Does—and Does Not—Tell You
The superscript identifies the treatment plan used to define an outcome. \(Y^{(1,1,1)}\) refers to an outcome under a specified fixed sequence; \(Y^d\) refers to an outcome under a specified decision rule. It is important to include the full sequence or rule when the timing of treatment matters. A label such as \(Y^1\) can be ambiguous in a longitudinal setting because it does not say whether treatment is given once, at every time, or only after a particular history.
A regime-specific potential outcome is defined for an individual. Population quantities, such as the mean outcome if everyone followed a particular regime, summarize these individual outcomes across a target population. That summary is distinct from an individual’s \(Y^d\), and defining the notation does not by itself show how to learn its value from observed data.
The distinction between observed and counterfactual outcomes also remains essential. For one individual, only the outcome under the treatment path actually experienced is observed. Outcomes under other sequences or rules are defined as potential outcomes, but are not directly observed for that same individual. A causal analysis must therefore keep separate the plan being considered, the potential outcome indexed by that plan, and the observed data available to study it.
Common Mistakes
- Calling a rule a fixed schedule. A threshold-based rule does not prescribe one sequence for everyone. It assigns treatment as a function of each person’s history.
- Applying a rule to observed history when defining its counterfactual outcome. \(Y^d\) is indexed by the history that would arise under \(d\), not automatically by the history observed under the actual treatment course.
- Leaving treatment timing unspecified. A statement such as “outcome under treatment” is incomplete when treatment can be assigned at several times. State the full sequence or the decision rules.
- Treating a potential outcome as an observed value. In general, an individual’s observed outcome corresponds only to the treatment path actually received; the other regime-specific outcomes remain counterfactual.
- Assuming a regime determines an outcome by itself. A rule specifies assignments given histories. The covariate history and final outcome under that rule are separate counterfactual quantities.
Check Your Understanding
Answer using the notation and distinctions developed above.
- For three treatment decisions, what does \(Y^{(1,0,1)}\) represent, and how does it differ from \(Y^d\) for a threshold-based rule?
- A rule treats when a biomarker is at least 5. An individual’s counterfactual biomarker values at three decisions are 6, 4, and 5. What treatment path does the rule generate?
- Why can two individuals following the same dynamic regime have different treatment sequences?
- If an individual’s observed treatment sequence is \((0,1,0)\), does that mean \(Y^{(1,1,1)}\) is observed for that individual? Explain.
- What information would be missing from the phrase “potential outcome under treatment” in a setting with repeated treatment decisions?