Mixed Strategies for Multiple Events

Why Combine Strategies?

A trial may involve several intercurrent events, and the scientific question need not treat them all alike. For example, a question might ask about outcomes regardless of treatment discontinuation, but ask what the outcome would have been had rescue medication not been used. Those choices address different clinical questions: one retains the consequences of discontinuation, while the other defines an outcome under a hypothetical absence of rescue medication.

The aim is not to select one strategy that applies to every event. It is to specify a strategy for each relevant event and then make clear how those choices jointly define the outcome being compared. A list of strategy labels alone is not enough: the event definitions, outcome, treatment contrast, and any important interactions between events must also be stated.

Definition: A mixed-strategy estimand is an estimand for which two or more intercurrent events are handled using potentially different strategies. The strategy assignment is event-specific. Together, the assignments define the outcome of interest under each treatment condition; the estimand then summarizes or contrasts those outcomes in a specified population.

From Event-Specific Choices to One Estimand

Let \(A\) denote randomized treatment, with \(A=1\) for one treatment and \(A=0\) for its comparator. Let the relevant intercurrent events be \(E_1,\ldots,E_K\), such as treatment discontinuation and use of rescue medication. For each event, specify an event strategy \(g_k\). The complete set of choices is the strategy vector \(g=(g_1,\ldots,g_K)\).

For example, \(g_1\) could be treatment policy for discontinuation, while \(g_2\) could be hypothetical for rescue medication. Write \(Y^{a,g}\) for the outcome defined under treatment \(a\) and the full strategy vector \(g\). This notation refers to the outcome targeted by the estimand, not automatically to an observed measurement or to the output of a particular statistical model.

\theta(g)=E\!\left(Y^{1,g}\right)-E\!\left(Y^{0,g}\right).

This mean difference is only one possible population-level summary; another summary could be appropriate for the endpoint and clinical question. Regardless of the summary, the target population, treatment conditions, endpoint, event definitions, event-specific strategies, and population summary together make the estimand interpretable. The earlier material on the five attributes of a trial estimand provides the general structure; the extra task here is to make the several event strategies fit together.

The event strategies must be defined precisely enough to determine what outcome is targeted. Under treatment policy, outcomes are included regardless of the event and its consequences, as specified by the question. Under a hypothetical strategy, the outcome is defined for a scenario in which the event does not occur. A composite strategy incorporates the event into the outcome definition. A while-on-treatment strategy targets outcome up to the event, and a principal-stratum strategy targets a subgroup defined by potential event status. These are different questions, not interchangeable ways to handle a single missing value.

Conditions for a Coherent Mixed Strategy

A mixed strategy is coherent when its components can be read together as one clinical question. The following checks are useful before writing the estimand formally.

Define each event.
Specify what counts as each event, including its timing or operational rule when that matters. “Rescue” must refer to a defined event, not an informal category that could be interpreted in several ways.

Assign a strategy to each event.
State the strategy separately for each event. Do not leave it to the reader to infer that one strategy applies to every event.

Resolve overlap and ordering.
Say what the outcome rule is if events occur in either order or both occur. If one strategy changes whether another event can happen, make the intended joint scenario clear.

State the target outcome and summary.
Name the endpoint, its measurement time or period, the population, and the treatment contrast or other summary. Check that the endpoint is defined under every strategy being invoked.

These checks concern the definition of the estimand. They do not establish that it can be identified from the collected data or reliably estimated. In particular, a hypothetical strategy requires a clear description of the scenario in which the event is absent. Whether the data support estimation of that outcome is a separate question; defining an estimand does not itself supply the assumptions or evidence needed for that task.

A short justification for using a strategy vector is that different event-specific choices define different parts of the outcome question. If discontinuation is handled by treatment policy and rescue by a hypothetical strategy, replacing both with a single “hypothetical” label would remove the treatment-policy choice for discontinuation. Conversely, calling both treatment policy would no longer ask for the outcome in the no-rescue scenario. The pair of choices therefore defines a target that cannot in general be recovered from either label alone.

Worked Example: Treatment Policy for Discontinuation, Hypothetical for Rescue

Hypothetical trial. A 24-week trial compares an active treatment with control in adults with a chronic condition. The endpoint is improvement from baseline in a symptom score at week 24, measured in points, with larger values indicating greater improvement. Treatment discontinuation and initiation of rescue medication are defined as two intercurrent events. The clinical question is the mean treatment difference if participants had not initiated rescue medication, while retaining outcomes and subsequent care after treatment discontinuation.

Specify the joint strategy. For discontinuation, use treatment policy: the target includes the consequences of discontinuation and subsequent care. For rescue medication, use a hypothetical strategy: define the endpoint in the scenario where rescue medication is not initiated. The target does not mean “no intercurrent events”; discontinuation remains under the treatment-policy strategy. The hypothetical intervention on rescue must be described consistently with the trial question, including how it applies if discontinuation has already occurred.

Suppose, purely for illustration, that the target population is the randomized population and the hypothetical mean week-24 improvements under this joint strategy are 5.2 points for active treatment and 3.7 points for control. The mean difference, active treatment minus control, is

\theta(g)=5.2-3.7=1.5\text{ points}.

The estimand is a 1.5-point difference in mean improvement under the stated joint strategy. The numbers are illustrative target means, not evidence that a particular analysis can identify or estimate them. In an actual report, the statement would also need to say how the rescue-free scenario is defined and how outcomes are summarized for participants who discontinued treatment.

Worked Specification with an Explicit Outcome Rule

Mixed strategies can become ambiguous when one event changes how another is handled. A useful safeguard is to write down the outcome rule in plain language before presenting a compact estimand statement. The next example uses a composite strategy to illustrate that safeguard; it does not imply that the same rule is suitable for every clinical question.

Worked Example: Treatment Policy for Discontinuation, Composite for Rescue

Hypothetical trial. In a 12-week trial, the endpoint is a symptom-improvement score from 0 to 10, measured in points, with larger values indicating more improvement. The question is the mean difference at week 12 in the randomized population. Outcomes after treatment discontinuation are retained. If a participant initiates rescue medication at any time before week 12, the composite endpoint is defined as 0; otherwise, it is that participant’s week-12 improvement score.

The event strategies are treatment policy for discontinuation and composite for rescue. This defines an outcome for each participant under the question: discontinuation by itself does not replace the week-12 score, while rescue does. If both events occur, the rescue rule sets the endpoint to 0, and the treatment-policy rule still means that discontinuation does not remove the participant from the target.

Suppose a hypothetical group of 20 active-treatment participants includes 8 who use rescue medication; the other 12 have a mean improvement of 5.5 points. The composite mean is

\bar{x}_1=\frac{8(0)+12(5.5)}{20} =\frac{66}{20}=3.30\text{ points}.

Suppose the 20 control participants include 10 who use rescue medication, and the other 10 have a mean improvement of 4.0 points. Their composite mean is

\bar{x}_0=\frac{10(0)+10(4.0)}{20} =\frac{40}{20}=2.00\text{ points}.

The illustrative difference in these group summaries is \(3.30-2.00=1.30\) points in favor of active treatment. This arithmetic shows how the specified composite outcome affects the summary; it does not turn the observed group summaries into a general conclusion about efficacy. The estimand is still defined by the target population, endpoint rule, treatment comparison, and population summary—not by these particular numbers.

Check the Joint Question, Not Just the Labels

A useful review is to read the full estimand statement as if it were a clinical question. It should be possible to answer: who is in the target population; which treatments are compared; what outcome and time point are relevant; what happens to the outcome under each event strategy; and what population summary is being compared?

Pay particular attention to events that can overlap, occur in different orders, or affect the availability or meaning of the endpoint. For instance, a hypothetical strategy for an event that commonly follows another event may be unclear unless the target scenario says whether the second event is still allowed. Likewise, an endpoint that is undefined after an event cannot be made well-defined merely by naming a treatment-policy strategy; the endpoint rule itself must address that situation. Some cases require a different strategy or a more explicit outcome definition.

The mixed strategy can also be clinically awkward even when it is mathematically describable. A hypothetical intervention that prevents one event while leaving all consequences of another event unchanged may not correspond to a plausible care pathway. This does not automatically make the estimand invalid, but it makes the clinical interpretation and any assumptions needed to estimate it especially important. The later material on individual events examines such issues in greater detail.

Worked Example: Changing One Event Strategy Changes the Question

Hypothetical trial. Consider the same 12-week symptom-improvement endpoint, but suppose the target question is the mean difference under a hypothetical strategy for discontinuation and a treatment-policy strategy for rescue. Under this joint strategy, the target outcome is defined in a scenario where treatment is not discontinued, while rescue use and its consequences remain part of the outcome question.

Suppose the illustrative target mean improvements under this strategy are 4.8 points for active treatment and 3.9 points for control. The treatment contrast is

\theta(g)=4.8-3.9=0.90\text{ points}.

This 0.90-point contrast is not an alternative calculation of the 1.30-point contrast in the preceding example. The event strategies, and therefore the target outcomes, differ. One cannot decide which estimand is preferable by comparing these numerical values alone: the choice depends on which clinical question is relevant. Further, the example gives target means for illustration; it does not establish that the no-discontinuation scenario is identified from trial data.

Common Pitfalls and Limitations

Using one strategy label for all events. Saying “hypothetical estimand” may conceal that only one event is hypothetical. Name each event and its assigned strategy.
Leaving the joint scenario implicit. Independent labels may not say what happens when events overlap or when one event affects another. Add an outcome rule or clarify the intended scenario.
Confusing the estimand with the analysis. The estimand states the target; an estimator is a procedure for learning about it, and an estimate is a result from data. A strategy label does not specify an estimator.
Assuming a strategy makes missing outcomes disappear. A defined outcome under a strategy may still be unobserved. Missing-data handling and estimation require additional assumptions and methods.
Treating every combination as clinically meaningful. A combination can be formally stated yet describe an implausible or poorly motivated scenario. Explain why the joint question matters and be transparent about its limits.

A concise final specification can be checked against a simple template: “In [population], compare [treatments] on [endpoint and time period], using [strategy] for [event one] and [strategy] for [event two], with [rule for overlap or ordering], and summarize by [population-level contrast].” If any bracket cannot be filled precisely, the estimand needs more work before an analysis is chosen.

Key takeaway: A mixed-strategy estimand is one coherent clinical question built from explicit, event-specific choices. State the full joint outcome rule, including relevant event interactions, and then state the population summary. Do not confuse that target with the method used to estimate it.

Check Your Understanding

For each question, focus on the target being defined, not on selecting an estimator.

A trial handles discontinuation by treatment policy and rescue medication hypothetically. What does each choice say about the target outcome?
Why may the statement “treatment policy for discontinuation and composite for rescue” need an additional rule for participants who experience both events?
In a hypothetical trial, the two strategy-defined mean outcomes are 6.1 points under treatment and 4.6 points under control. What is the mean difference, treatment minus control, and what does it mean?
Does defining a hypothetical strategy establish that its outcome can be identified from the observed trial data? Explain.
What information should accompany event-specific strategy labels to make a mixed-strategy estimand interpretable?

Solutions. (1) Treatment policy retains outcomes and consequences after discontinuation as specified by the question; the hypothetical rescue strategy targets the outcome in a scenario where rescue is not initiated. (2) Without an overlap rule, it may be unclear whether the composite rule overrides the ordinary endpoint or how the participant’s endpoint is defined. (3) The difference is \(6.1-4.6=1.5\) points: the treatment mean exceeds the control mean by 1.5 points under the stated strategy-defined target. (4) No. Defining the target and identifying it from observed data are separate tasks; identification depends on additional conditions and information. (5) State the target population, treatments, endpoint and time period, precise event definitions, assigned strategy for each event, relevant overlap or ordering rules, and the population summary.