Composite-Strategy Estimands

Why Build the Event Into the Outcome?

An intercurrent event can change what a treatment contrast means. For example, a treatment may affect both a symptom score and the chance that participants need rescue medication. A composite strategy defines a new outcome that incorporates the event itself, rather than treating the event as something to disregard or imagining it did not happen. The resulting contrast answers a specific clinical question: how do the treatment groups compare on this combined outcome?

The event and the outcome must be combined using a rule that has a defensible clinical interpretation. One common rule assigns a prespecified value to anyone who experiences the event and uses the measured outcome for those who do not. Another rule defines success as achieving a favorable clinical outcome without experiencing the event. These are different composite outcomes, and they need not produce the same treatment contrast.

Definition: A composite-strategy estimand incorporates an intercurrent event into the outcome variable through a prespecified rule. Let $A\in\{0,1\}$ denote treatment assignment, $E(a)\in\{0,1\}$ indicate whether the event would occur under assignment $a$, and let $Y(a)$ be the clinical outcome when the event does not occur. A composite outcome $Q(a)$ is defined by a rule that combines $E(a)$ and $Y(a)$. A common form assigns a fixed value $c$ when the event occurs: $$ Q_c(a)= \begin{cases} Y(a), & E(a)=0,\\ c, & E(a)=1. \end{cases} $$ For a higher-is-better scale, $c$ may represent a poor outcome, if that coding is clinically justified. The estimand may be the mean contrast $\theta_c=E\{Q_c(1)\}-E\{Q_c(0)\}$; a positive value favors treatment 1 on that scale.

The event definition, observation period, outcome scale, and composite rule are all part of the estimand. The event must be specified clearly enough that it can be determined under either treatment assignment. The rule should also make clinical sense: an assigned value is not merely a convenient way to fill in a number, but a statement about how event occurrence ranks or combines with the outcome.

Mean Contrasts and the Role of Event Occurrence

For a fixed-value composite, let $p_a=P\{E(a)=1\}$. When $p_a<1$, define $\mu_{a,0}=E\{Y(a)\mid E(a)=0\}$, the mean outcome among those who would not experience the event under assignment $a$. Splitting the composite mean by event status gives:

E\{Q_c(a)\}=(1-p_a)\mu_{a,0}+p_a c,\qquad p_a<1.

This conditional-mean expression is valid only when $p_a<1$, because otherwise the event-free group has probability zero and $\mu_{a,0}$ is undefined. The definition of the composite mean itself remains valid in all cases. In particular, if $p_a=1$, everyone receives the composite value $c$, so $E\{Q_c(a)\}=c$. More generally, define $Z(a)$ as $Y(a)$ for event-free individuals and zero for individuals with the event. Then:

E\{Q_c(a)\}=E\{Z(a)\}+p_a c.

This form does not require an event-free conditional mean and also covers $p_a=1$. When both arms have $p_a<1$, subtracting the two arm means yields:

\theta_c =(1-p_1)\mu_{1,0}-(1-p_0)\mu_{0,0}+c(p_1-p_0).

The contrast therefore reflects both the event rates and the outcomes among people who do not experience the event. The conditional means refer to event-free groups defined under different treatment assignments. In general, those groups are not the same people, so their difference is not automatically a causal effect among a common “event-free” subgroup.

Worked Examples

Worked Example: A Continuous Outcome With an Assigned Event Value

Suppose a hypothetical randomized trial measures symptom improvement on a 0-to-10 scale, where higher scores are better. Rescue medication during follow-up is the intercurrent event. The prespecified composite assigns $c=0$ to anyone who uses rescue medication and otherwise uses the symptom score. In hypothetical summaries, 20% of treatment-group participants experience the event and the remaining participants have a mean score of 6.5. In the comparison group, the event rate is 10% and the event-free mean score is 6.0. Assume event status and the required event-free scores are observed for everyone.

For the treatment group, $p_1=0.20$, $\mu_{1,0}=6.5$, and $c=0$. Its composite mean is: $$ (1-0.20)(6.5)+(0.20)(0)=0.80(6.5)=5.20. $$

For the comparison group, $p_0=0.10$, $\mu_{0,0}=6.0$, and $c=0$. Its composite mean is: $$ (1-0.10)(6.0)+(0.10)(0)=0.90(6.0)=5.40. $$

The estimated treatment-minus-comparison contrast is $5.20-5.40=-0.20$ points. Although the event-free mean is higher in the treatment group, its composite mean is lower because that group has the higher event rate. The result is a contrast on the prespecified composite scale; it is not a comparison restricted to participants who remained event-free.

Worked Example: A Binary Composite Success Outcome

In a hypothetical trial, define success as both remaining free of an intercurrent event and achieving a clinical response. Any event occurrence counts as failure, as does lack of response among event-free participants. In a treatment group of 100 participants, 18 experience the event; among the 82 without the event, 49 respond. In a comparison group of 100, 10 experience the event; among the 90 event-free participants, 54 respond. Assume event status and response are recorded for all participants.

The treatment-group success proportion is the number who satisfy both parts of the rule divided by all 100 randomized participants: $$ \hat p_{\text{success},1}=\frac{49}{100}=0.49. $$ The comparison-group success proportion is: $$ \hat p_{\text{success},0}=\frac{54}{100}=0.54. $$

The estimated difference is $0.49-0.54=-0.05$, or 5 percentage points lower in the treatment group. The denominators are all randomized participants, not just those without the event. The composite therefore counts the event as failure rather than excluding participants who experience it.

Worked Example: When Everyone in One Arm Experiences the Event

Consider a hypothetical 0-to-10 outcome with higher scores better and a composite value $c=0$ for event occurrence. Suppose everyone assigned to treatment 1 experiences the event. In the comparison group, 20% experience the event, and the event-free mean outcome is 5.0. Assume the composite is fully observed.

For treatment 1, $p_1=1$, so the composite outcome is always $0$, and $E\{Q_c(1)\}=0$. There is no event-free treatment-1 group, so its conditional mean $\mu_{1,0}$ is undefined and must not be used.

For the comparison group, $p_0=0.20<1$, so: $$ E\{Q_c(0)\}=(1-0.20)(5.0)+(0.20)(0)=4.0. $$ Thus the treatment-minus-comparison contrast is $0-4.0=-4.0$ points. This calculation uses the composite means directly and does not invent an event-free outcome mean for the treatment group.

Assumptions, Estimation, and Diagnostics

The estimand is defined before considering how to estimate it. For a randomized trial, a causal interpretation of the treatment contrast requires that assignment be randomized, that each participant’s observed composite outcome correspond to the outcome under the treatment actually assigned (consistency), and that one participant’s assignment not affect another participant’s outcome. The target population and the treatment conditions must also be specified. These conditions support interpreting the arm-specific means as means under assignment to each treatment.

To estimate the contrast, apply the same prespecified composite rule to each participant and calculate the relevant arm-level summaries. For a fully observed continuous composite, this could be the difference between the two sample means. For a binary composite, it could be the difference between the two sample proportions. The examples assume the event and all outcome information needed by the rule are observed; if a required component is missing, the composite may also be missing. A composite strategy does not, by itself, solve missing-data problems.

Before interpreting the estimate, check that the event definition and ascertainment period were applied consistently in both arms, that the score was coded in the intended direction, and that the event rule was followed for every participant. Report event frequencies alongside the composite contrast: they help readers understand which components may have contributed to the result. Also examine the event-free outcome distributions and the number of observations contributing to them. Large differences in event rates or unusual outcome distributions can make the composite contrast hard to summarize with a single mean.

Specify the clinical question.
Define the population, treatment comparison, event, observation period, and outcome scale.

State the composite rule.
Describe exactly how event occurrence and the event-free outcome determine each participant’s composite value.

Check the observed components.
Verify that event status is known and that outcomes required for event-free participants are available or appropriately handled.

Estimate and interpret the target contrast.
Summarize composite outcomes by treatment and explain what higher or lower values mean clinically.

How the Composite Choice Changes Interpretation

The assigned value $c$ is a substantive choice. If both arms have event rates below one, changing $c$ changes the mean contrast by:

\theta_{c_2}-\theta_{c_1}=(p_1-p_0)(c_2-c_1).

This follows because the event-free outcome terms do not depend on $c$, while the event terms in the contrast are $c(p_1-p_0)$. If event rates differ, a change in the assigned value can change the size of the contrast and, in some settings, its direction. If event rates are equal, changing $c$ does not change this mean difference, though the composite still needs a clinically interpretable scale. Therefore, the event value or ranking rule should be justified and prespecified, not selected after seeing which coding favors a preferred conclusion.

A composite strategy differs from a treatment-policy estimand, which includes outcomes after the event as they occur under assigned treatment, and from a hypothetical estimand, which asks about outcomes under a specified scenario in which the event did not occur. A composite instead changes the outcome definition so that the event contributes directly to the endpoint. These strategies answer different questions; none is universally best.

The approach is useful when event occurrence is itself clinically important and a combined endpoint is meaningful. It can avoid treating event occurrence as irrelevant, but it has limitations. A single summary can conceal whether the contrast arose mainly from event frequencies or from event-free outcomes. A fixed score may impose an arbitrary distance between event occurrence and measured outcomes, especially when the score is not naturally on the same scale. Composite outcomes can also combine components with different clinical importance. Presenting component frequencies and explaining the coding rule helps make these limitations visible.

Common Pitfalls

Using a conditional mean when its group does not exist. If $p_a=1$, then $E\{Y(a)\mid E(a)=0\}$ is undefined. Use $E\{Q_c(a)\}=c$ for that arm, or the unconditional representation $E\{Z(a)\}+p_ac$.
Changing the denominator. A binary composite success proportion uses all participants in the treatment arm when event occurrence is coded as failure. Restricting the denominator to event-free participants answers a different question.
Calling the event-free contrast a treatment effect in a common subgroup. The event-free groups under different assignments may consist of different people. Their conditional means do not generally identify a causal effect for a shared subgroup.
Treating the composite value as a neutral imputation. The value or rule expresses how the event ranks or combines with the clinical outcome. It is part of the clinical question, not merely a computational convenience.
Assuming the composite resolves missingness. If event status or an outcome required by the composite is unobserved, the composite may be unobserved too; missing-data assumptions and methods are still needed.

Key takeaway: A composite-strategy estimand compares treatment groups on an outcome that explicitly incorporates the intercurrent event. Its interpretation depends on the event definition and the prespecified combining rule, and its mean contrast can reflect both event rates and event-free outcomes.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

On a higher-is-better scale, what does assigning $c=0$ to participants who experience an event mean for the composite definition?
If an arm has event probability $p_a=1$, what is its fixed-value composite mean, and why is the event-free conditional mean unavailable?
A treatment arm has event rate 0.30 and event-free mean 8; the comparison arm has event rate 0.10 and event-free mean 7. Both use $c=0$. Calculate the two composite means and the treatment-minus-comparison contrast.
For a binary endpoint, 12 of 80 participants experience the event and 40 of the remaining participants respond. If success requires no event and response, what is the success proportion?
Why should a composite contrast not automatically be interpreted as an effect among people who would remain event-free under either treatment?

Solutions: (1) Event occurrence is assigned the composite value zero, which is treated as the specified poor outcome on this scale; this coding requires clinical justification. (2) The composite mean is $c$, because every participant receives that value. The event-free group has probability zero, so its conditional mean is undefined. (3) Treatment mean: $(1-0.30)(8)+(0.30)(0)=5.6$. Comparison mean: $(1-0.10)(7)+(0.10)(0)=6.3$. The contrast is $5.6-6.3=-0.7$ points. (4) There are $80-12=68$ event-free participants, of whom 40 respond; the composite success proportion is $40/80=0.50$, not $40/68$. (5) The event-free groups under the two assignments need not contain the same people, so comparing their conditional means does not generally isolate a causal effect in a common event-free subgroup.

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