Aligning Historical Endpoint Definitions

Why endpoint alignment comes before borrowing

A historical control group can be relevant to a current trial and still measure a different outcome. One trial may define response as an improvement of at least 4 points; another may use at least 5 points. A study may count events through day 35 while the current trial assesses them through day 28. A historical composite may include hospitalization or death, whereas the current primary endpoint counts death alone. These are not small bookkeeping differences: they can change what a participant’s outcome means and the likelihood contributed to the analysis.

Power-prior borrowing discounts a historical likelihood; it does not turn one endpoint into another. If historical data describe an outcome different from the current outcome, raising that likelihood to a discount power does not make it a likelihood for the current endpoint. The first task is therefore to define a common analysis outcome and determine whether the records contain enough information to construct it.

Definition: A common analysis outcome is an outcome that can be defined and measured consistently for the current and historical control data, including its clinical meaning, population, assessment time or window, event components, and rules for handling relevant events. Endpoint alignment is the process of determining whether the available records can be expressed in terms of that outcome without changing its intended meaning.

The endpoint should be specified before inspecting which definition produces the most favorable historical result. In clinical-trial terms, state what outcome is measured, for whom, under what treatment condition, at what time, and how events that affect interpretation are handled. A shared endpoint name is not enough. Conversely, different labels do not always prevent alignment if the underlying records contain the information needed to apply one prespecified definition to both studies.

Conditions for constructing a common outcome

A defensible construction requires both a scientific target and usable data. The target is the endpoint definition to be used in the current analysis. The historical records must then support applying that definition at the participant level, or through sufficient summaries when the chosen likelihood permits it. The construction should not depend on unverified assumptions about unrecorded outcomes.

For a binary endpoint, for example, let \(Y_i^*=1\) indicate that participant \(i\) meets the common event definition and \(Y_i^*=0\) otherwise. Under a binomial model with a common event probability \(\theta\), and independent participant outcomes, historical data with \(y_0\) events among \(n_0\) participants have likelihood

L_0(\theta\mid D_0^*) \propto \theta^{y_0}(1-\theta)^{n_0-y_0}.

Here \(D_0^*\) denotes the historical data after applying the common endpoint definition. With a fixed power-prior discount \(a\), where \(0\leq a\leq 1\), the historical contribution is proportional to \(L_0(\theta\mid D_0^*)^a\). This operation discounts the contribution of the aligned historical data; it does not repair a mismatch in the definition of \(Y_i^*\).

Alignment checks: Before constructing \(D_0^*\), verify that (1) the same clinical event or measurement is targeted, (2) assessment times and windows can be made consistent, (3) event components and thresholds can be applied from the records, (4) missing or unobserved outcomes are not silently classified as non-events, and (5) the resulting outcome has a defensible interpretation for both control groups.

The model also needs to be appropriate for the constructed outcome. For the binomial likelihood above, outcomes must be binary under the common rule, and participants must contribute independent observations under the assumed model. If the data are clustered, repeated, or otherwise dependent, an appropriate likelihood must account for that structure. Alignment alone does not establish independence or make historical and current control event probabilities exchangeable; those are separate modeling judgments.

Reconstructing a common endpoint from records

Worked Example: Different Response Thresholds

Suppose a current trial defines response as an improvement of at least 5 points on a symptom scale. A hypothetical historical control study used a threshold of at least 4 points and reports 27 responders among 60 participants. Its participant-level records also retain each participant’s baseline and follow-up scores. In those records, 18 participants improved by at least 5 points. Can the historical records support the current endpoint?

Construction. The common endpoint is response at the current trial’s threshold: improvement of at least 5 points. Because the historical records contain the scores needed to apply that rule, recode each historical participant using the common threshold. This gives \(y_0=18\) events among \(n_0=60\), rather than using the reported count of 27 at the historical threshold.

For illustration, suppose the current trial has 24 responders among 80 control participants under the same threshold. The respective observed proportions are

\hat{p}_0=\frac{18}{60}=0.300,\qquad \hat{p}_1=\frac{24}{80}=0.300.

At a fixed discount \(a=0.5\), the aligned historical likelihood contributes

L_0(\theta\mid D_0^*)^{0.5} \propto \left\{\theta^{18}(1-\theta)^{42}\right\}^{0.5} = \theta^9(1-\theta)^{21}.

This is a likelihood contribution for the common 5-point response definition. Using 27 historical responders would instead contribute exponents based on the 4-point definition and would answer a different endpoint question. The construction assumes that the scales and score direction are meaningfully comparable and that the recorded scores were collected and interpreted consistently enough to apply the same threshold. If those conditions fail, recoding the numbers does not establish clinical equivalence.

The example also illustrates an important distinction: a reported aggregate count is not always sufficient to harmonize an endpoint. The historical count of 27 at the 4-point threshold does not reveal how many of those participants meet the 5-point threshold. Participant-level scores make that count available; without them, one cannot calculate the common endpoint count from the published summary alone.

Worked Example: Different Assessment Windows

Suppose a current trial defines an event as at least one qualifying event by day 28. A hypothetical historical study reports events through day 35. Its records include event dates and confirm that every participant’s event status was ascertained through day 28. Among 100 historical control participants, 20 had an event by day 35: 15 by day 28 and 5 after day 28. The current endpoint is therefore reconstructible.

Construction. Count only events occurring by day 28. The common historical endpoint has \(y_0=15\) events and \(n_0=100\), so the estimated historical event proportion for this endpoint is

\hat{p}_0=\frac{15}{100}=0.150.

If the hypothetical current control group has 12 events by day 28 among 90 participants, then \(\hat{p}_1=12/90=0.1333\). With a fixed historical discount \(a=0.4\), the historical binary likelihood contribution is

\left\{\theta^{15}(1-\theta)^{85}\right\}^{0.4} = \theta^6(1-\theta)^{34}.

The arithmetic uses the day-28 event definition in both groups. The construction relies on complete day-28 ascertainment for the historical participants, accurate event dates, and the same qualifying-event rule. If the historical records contained only the total of 20 events by day 35, the day-28 count could not be recovered. If some participants’ day-28 status were unknown, treating them as non-events would be unjustified. For time-to-event data with censoring, the survival likelihood must represent the observed event or censoring times appropriately; a simple day-28 binomial count is justified here only because day-28 status is known for everyone.

When components allow a narrower common endpoint

A composite endpoint is a rule that classifies a participant as having an event if one or more specified components occur. If studies use different composites, a common component may sometimes be extracted from the historical records. This can align the endpoint mathematically, but it does not by itself show that the narrower endpoint is the clinically appropriate target or that the studies are otherwise comparable.

Worked Example: Rebuilding a Death-Only Endpoint

A hypothetical historical study defines its primary endpoint as hospitalization or death by day 30. Among 50 control participants, 14 met that composite definition, and the records separately identify 6 deaths. A current trial’s endpoint is death by day 30; 8 of 70 current control participants died by day 30. The historical data can support the death-only endpoint because they retain the death component and its timing.

Construction. Apply the current definition to the historical records: death by day 30. The historical outcome is \(y_0=6\) deaths among \(n_0=50\), not 14 composite events. Its observed proportion is \(6/50=0.120\), while the current control proportion is \(8/70=0.1143\). At \(a=0.5\), the aligned historical likelihood contribution is

\left\{\theta^6(1-\theta)^{44}\right\}^{0.5} = \theta^3(1-\theta)^{22}.

This construction assumes that death is ascertained consistently through day 30 in both studies, that the historical death component is defined comparably, and that death-only is a scientifically defensible outcome for the intended analysis. The 14-person composite count cannot be substituted for the death count: it includes hospitalizations without death. In the reverse direction, if the historical study recorded death only and did not record hospitalization, it could not reconstruct a hospitalization-or-death composite. A missing component cannot be recovered by relabeling the available endpoint.

Why recoding can be valid—and what it cannot establish

The justification for recoding is direct. Suppose each historical participant has a record \(R_i\) containing the information needed to apply a prespecified function \(g\), and define the common outcome as \(Y_i^*=g(R_i)\). If the chosen model specifies the distribution of \(Y_i^*\) under parameter \(\theta\), and its assumptions hold, the likelihood is built from the recoded outcomes. For independent binary outcomes with event probability \(\theta\), each event contributes a factor \(\theta\), and each non-event contributes \(1-\theta\). Multiplying those contributions gives the binomial likelihood used above.

L_0(\theta\mid D_0^*) = \prod_{i=1}^{n_0} \theta^{Y_i^*}(1-\theta)^{1-Y_i^*} = \theta^{\sum_iY_i^*}(1-\theta)^{n_0-\sum_iY_i^*}.

This derivation explains why the common definition must be applied to the records before the likelihood is formed. It also shows the limit of the argument: it establishes the likelihood for the recoded endpoint under the stated model, not that the historical and current control distributions are identical. Differences in patient populations, care, measurement, or other trial features can remain relevant to borrowing.

Endpoint alignment is most credible when the common rule is specified independently of the observed treatment results, the required data are recorded for both groups, and the measurement process has a defensible correspondence. It is less credible when the target is selected after seeing which recoding makes the groups look similar, when assessment practices differ in undocumented ways, or when a clinically important component is unavailable. Sensitivity analyses can show how conclusions vary under plausible definitions, but they cannot identify an unobserved component without additional information.

Diagnostics, alternatives, and limitations

Before using an aligned endpoint for power-prior borrowing, inspect the mapping and the resulting records. Check the number of participants for whom the common outcome is actually determinable, the event counts before and after recoding, the distribution of assessment times, and whether the same threshold or component rule was applied to all relevant participants. Unexpectedly large changes in counts can be legitimate, but they should prompt verification of score direction, units, visit windows, and event coding. Keep a reproducible record of the rule and of any participants whose outcomes could not be classified.

The main alternatives depend on why alignment is uncertain:

Use a narrower common endpoint. This is reasonable when both studies recorded the required component and the narrower outcome remains relevant to the scientific question.
Model endpoint-specific outcomes separately. If the endpoints are related but not interchangeable, a model that represents their distinct definitions may be more defensible than forcing one likelihood onto both. Such a model requires adequate data and assumptions about the relationship between outcomes.
Do not borrow the incompatible endpoint. If the historical outcome cannot be reconstructed and no defensible common model is available, excluding that historical likelihood avoids treating unlike outcomes as though they were the same.
Use a validated measurement bridge. A bridge may be appropriate when endpoint instruments differ but a reliable relationship has been established. The relationship and its uncertainty must be represented; an assumed numerical conversion alone is not validation.

A common pitfall is to reason, “The historical study’s endpoint is close to the current endpoint, and the power-prior discount is small, so the mismatch is harmless.” A discount reduces the historical likelihood’s influence; it does not change what that likelihood measures. The correction is to align the endpoint from the records using a defensible, prespecified rule, or not to use that likelihood for the current outcome. Even after alignment, the resulting borrowing remains conditional on the model and the broader assumptions supporting historical borrowing.

Key takeaway: Borrow only a historical likelihood that represents the intended common outcome. A construction is defensible when the scientific definition is clear, the historical records support applying it, and the resulting data satisfy the assumptions of the analysis model. Similar labels, discounted likelihoods, and convenient summaries do not substitute for those checks.

Check Your Understanding

Answer each question using the endpoint definition and available records, not just the endpoint label.

A historical study reports 31 responders using a 3-point threshold among 100 participants. The current endpoint uses a 5-point threshold. Historical participant-level scores are unavailable. Can the common historical responder count be calculated from the information given?
A historical study has event dates and confirms complete ascertainment through day 30. Its reported day-60 event count is 25, of which 9 events occurred after day 30. How many historical participants meet the day-30 endpoint if 100 participants were enrolled?
A historical composite includes hospitalization or death. The records identify each component separately. What historical event count should be used for a current death-only endpoint if 7 of 60 historical participants died by the common assessment time?
Why does choosing a small power-prior discount not resolve a mismatch between a historical hospitalization-or-death endpoint and a current death-only endpoint?

Solutions. (1) No. The 3-point count does not determine how many participants meet the 5-point threshold; the participant-level scores or a sufficient tabulation at that threshold are needed. (2) \(25-9=16\) participants had an event by day 30, so the common historical event count is 16 of 100, assuming the stated complete ascertainment and a consistent event rule. (3) Use 7 deaths among 60 participants, not the composite count, because death alone is the current endpoint. (4) Discounting reduces the weight of the historical likelihood but leaves its event definition unchanged; it cannot make a composite-event likelihood represent a death-only outcome.