Why Specify a Time-Specific Effect?
In a platform trial, treatments may enter at different calendar times, eligibility rules may change, and the standard of care may evolve. A single phrase such as “the effect of treatment” can therefore conceal several distinct questions. It matters whether the question concerns patients eligible during a particular enrollment period, all eligible patients across several periods, or a subgroup defined by the protocol.
A time-specific estimand makes the question explicit: it names the treatment comparison, the outcome and its measurement time, the calendar period, and the eligible population to which the effect refers. Calendar period here means when participants enter or are assigned in the trial. It is not the duration of follow-up after assignment.
The notation emphasizes two choices that must be stated, not left implicit: the calendar period \(t\) and the target population \(G_t\). It also assumes the outcome is defined in a consistent way, including its unit and follow-up time. For a binary outcome, one possible estimand is the risk difference \(P\{Y(1;t)=1\mid G_t\}-P\{Y(0;t)=1\mid G_t\}\). Other scales, such as a risk ratio, answer different questions and should be specified rather than substituted after seeing the data.
Define the Period, Population, and Outcome
A usable calendar period has operational boundaries, such as assignments from 1 January through 30 June of a stated year. Those boundaries should match the scientific question and be set without using observed outcomes to choose a favorable interval. If the trial's eligibility criteria change, the target population must also say which criteria apply. For instance, “all eligible patients entering between January and June” and “eligible patients with biomarker B present in that period” are different populations.
The outcome definition needs equal care. “Clinical response” is incomplete unless the response rule and assessment time are identified. A mean difference in a symptom score at 12 weeks is not the same estimand as a risk difference in response by 12 weeks. If lower scores indicate improvement, a negative mean difference may favor treatment; the direction must be interpreted using the scale's meaning.
Intercurrent events—events after assignment that affect interpretation or measurement, such as discontinuing assigned treatment—also need a stated handling strategy. For example, an outcome measured regardless of discontinuation answers a treatment-assignment question, whereas an outcome defined only while participants remain on treatment asks a different question and may require additional assumptions. The estimand cannot be fully understood from the treatment labels alone.
Worked Example: Mean Effect During a Defined Enrollment Period
Suppose, hypothetically, that a platform trial assigns eligible participants entering from 1 January through 30 June to a new treatment or concurrent control. The outcome is a symptom score at 12 weeks, in points, with lower scores indicating fewer symptoms. Among 50 participants assigned to each group, the mean scores are 18.4 points for treatment and 21.1 points for control. Assume assignment was randomized within this period, the score was measured under the same rule in both groups, and all 100 participants have the defined outcome.
The target is the eligible trial participants entering during this six-month period, and the estimand is the mean treatment-minus-control difference in 12-week score:
The estimated treatment effect is a 2.7-point lower mean score at 12 weeks for assignment to treatment than for assignment to control in this period's target population. Since lower scores represent fewer symptoms, the estimate favors treatment. This calculation does not establish that the treatment caused improvement in every individual, nor does it by itself quantify sampling uncertainty.
Why the Effect Can Change With Calendar Time
The potential outcome under a treatment strategy may depend on when the participant is treated. Background care, diagnostic practice, supportive treatment, and the mix of eligible patients can change over calendar time. Thus \(Y_i(a;t_1)\) need not equal \(Y_i(a;t_2)\), even for the same person under a hypothetical assignment. This is why a treatment effect estimated in one period is not automatically the effect in another.
For each period, define the contrast using that period's target population. The difference between \(\theta(t_1,G_{t_1})\) and \(\theta(t_2,G_{t_2})\) can reflect a genuine change in treatment effect, a change in the population, or both. The estimands alone do not separate these explanations. A comparison that aims to isolate effect modification by time would need to make the target populations and other aspects of the comparison explicit.
An overall effect across periods is also a distinct target. If the intended target is a mixture of period-specific populations, the mixture weights must be defined. For a mean-difference estimand, a mixture with weights \(w_t\), where \(w_t\geq 0\) and \(\sum_t w_t=1\), has effect \(\sum_t w_t\theta(t,G_t)\). Different weights represent different target mixtures and can produce different overall effects, even if each period-specific effect remains unchanged.
Worked Example: Different Effects in Two Calendar Periods
Consider hypothetical randomized comparisons using the same 12-week symptom score and the same eligibility rules. In an early period, the treatment mean is 17 points and the control mean is 20 points, giving an estimated difference of \(17-20=-3\) points. In a later period, the corresponding means are 19 and 20 points, giving \(19-20=-1\) point. Assume randomization and outcome measurement were carried out comparably within each period.
The two period-specific estimates are therefore \(-3\) and \(-1\) points. They are estimates for different calendar-period questions, not two estimates that must be averaged without qualification. Suppose the desired target mixture assigns weight 0.60 to the early-period population and 0.40 to the later-period population. Its weighted mean-difference estimate is
The \(-2.2\)-point result describes this particular 60:40 target mixture. An equal-weight mixture would instead give \(0.50(-3)+0.50(-1)=-2.0\) points. The choice of weights is part of the estimand; it should reflect the target population rather than be treated as a purely computational convenience.
Changing the Eligible Population Changes the Question
Eligibility defines who is included in the target population, not merely who happens to have complete data. A protocol may restrict entry by disease subtype, biomarker status, prior therapy, or other criteria. A treatment effect among people meeting one set of criteria need not equal the effect among people meeting a broader or revised set. This remains true even when the outcome, follow-up time, and calendar period are held fixed.
The distinction is especially important when criteria change during a platform trial. “Eligible under the rules in force at the time” defines a time-varying target that may differ from “eligible under the revised rules throughout the period.” Either can be a legitimate question, but they are not interchangeable. The population definition should make clear whether membership is determined by the rules in force at entry or by one fixed set of criteria applied retrospectively.
Worked Example: Restricting the Target to a Biomarker Group
Suppose, hypothetically, that during one fixed calendar period, eligible participants are classified at baseline as biomarker-positive or biomarker-negative. The outcome is response by 12 weeks. Among biomarker-positive participants, 30 of 60 assigned to treatment respond, compared with 20 of 60 assigned to control. Among biomarker-negative participants, 18 of 40 assigned to treatment respond, compared with 16 of 40 assigned to control. Assume randomization within these groups, the same response definition and follow-up in all groups, and complete outcome ascertainment.
For the biomarker-positive population, the treatment and control response risks are \(30/60=0.50\) and \(20/60=0.3333\). The risk-difference estimate is
or 16.67 percentage points higher response risk under treatment in this subgroup. For the biomarker-negative population, the risks are \(18/40=0.45\) and \(16/40=0.40\), so the corresponding estimate is \(0.45-0.40=0.05\), or 5 percentage points.
If the target is the entire eligible population represented by these groups, there are 100 participants per assignment group. The overall treatment risk is \((30+18)/100=0.48\), and the overall control risk is \((20+16)/100=0.36\). The overall risk difference is \(0.48-0.36=0.12\), or 12 percentage points. The subgroup and overall estimates answer different population questions; the overall value is not a replacement for the biomarker-positive estimand.
Identification: What a Randomized Contrast Supports
A causal interpretation requires more than subtracting two sample summaries. For the basic within-period randomized comparison, the treatment strategies must be sufficiently well defined, each participant's observed outcome under the assigned strategy must correspond to the relevant potential outcome (consistency), and one participant's assignment must not alter another participant's outcome (no interference). In addition, random assignment within the period must make assignment independent of the potential outcomes in the randomized target population, and both strategies must have a positive probability of assignment.
Under these conditions, the population contrast in observed mean outcomes by assignment equals the average potential-outcome contrast for the randomized population. The justification is short. Let \(A\) denote randomized assignment in period \(t\), with \(A\) independent of \(Y(1;t)\) and \(Y(0;t)\), and let \(Y=Y(A;t)\) by consistency. Then
The middle equality follows from randomization, and the final equality follows from linearity of expectation. This argument identifies the effect for the randomized target population under the stated assumptions. Extending the result to all eligible people in a wider setting, including people who were not enrolled, requires an additional basis for generalizing beyond trial participants; randomization alone does not provide it.
Compare the Questions Before Comparing the Numbers
| Target question | What is held fixed? | What changes? |
|---|---|---|
| Effect in a specified period | Population definition, outcome, and follow-up | Calendar period of assignment |
| Effect in a specified subgroup | Calendar period, treatment strategies, and outcome | Eligibility or subgroup definition |
| Effect across several periods | Outcome and treatment strategies | Target mixture and its period weights |
| Effect on another scale | Period and target population | Summary measure, such as mean difference versus risk difference |
These alternatives are not competing estimators of one universally defined quantity. They are different estimands. The choice depends on the decision the result is intended to inform. A current-period decision may call for the current eligible population; a policy intended for a broader mix of patients may call for a prespecified mixture.
Diagnostics, Pitfalls, and Limitations
Before interpreting an estimate, check whether the trial records support the intended estimand. Confirm that the calendar boundaries are recorded and applied consistently, that eligibility is determined using the stated rules, and that the outcome and follow-up time are defined in the same way across groups. Check that both treatment strategies were available for randomized assignment in the period. If a treatment was not yet open, there is no randomized within-period contrast for that treatment at that time.
A marked change in control outcomes across periods can flag that period-specific effects should not be casually pooled. It does not by itself show that the treatment effect changed: the underlying patient mix, care, measurement, or other conditions may also have changed. Conversely, similar control outcomes do not prove that the populations or treatment effects are equivalent. An estimand specifies the question; it does not resolve these empirical uncertainties.
- Mistake: treating “the trial effect” as a fixed quantity. A pooled contrast may refer to a particular collection of periods and participants. State which target it represents and how periods are combined.
- Mistake: confusing follow-up time with calendar time. “Response at 12 weeks” is an outcome measurement time; “participants enrolled from January to June” defines a calendar period. Specify both.
- Mistake: interpreting a subgroup estimate as the overall effect. Restriction to a biomarker group changes the target population. Report the group to which the estimate refers.
- Mistake: assuming randomization answers a broader population question automatically. Randomization supports a causal comparison in the randomized population under its assumptions; extending beyond that population requires further justification.
The main limitation is that a carefully stated estimand does not guarantee precise or unbiased estimation. Small period-specific samples can yield imprecise estimates, and missing outcomes, inconsistent outcome measurement, protocol changes, or departures from the randomized comparison can undermine interpretation. The value of specifying the estimand is that it makes the intended question visible, so such limitations can be assessed against the right target.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A trial measures a fatigue score 8 weeks after assignment for participants enrolled from April through September. Which phrase describes follow-up time, and which describes calendar period?
- In one period, mean outcome scores are 14.2 in treatment and 16.0 in control, where lower is better. Calculate and interpret the treatment-minus-control estimate.
- Why is an effect among biomarker-positive participants not automatically the effect in all eligible participants?
- Two period-specific mean differences are \(-2\) and \(-4\) points. Calculate the overall effect for a target mixture with weights 0.25 and 0.75, respectively.
- Name two assumptions that support a causal interpretation of a randomized within-period contrast, and explain why randomization does not automatically extend that interpretation to people outside the trial.
Solutions. (1) Eight weeks after assignment is the follow-up time; April through September is the enrollment calendar period. (2) \(14.2-16.0=-1.8\) points. The estimated mean score is 1.8 points lower under treatment in that period's target population, which favors treatment because lower is better. (3) The subgroup and full population have different membership and may have different distributions of treatment response; they are distinct targets. (4) \(0.25(-2)+0.75(-4)=-0.5-3=-3.5\) points for the stated target mixture. (5) Examples include consistency, no interference, and independence of randomized assignment from potential outcomes; positivity is also required. Randomization identifies the effect for the randomized population, while generalizing to nonparticipants requires assumptions or evidence about how their potential outcomes relate to those of participants.