Delayed Continuous Outcomes at Scheduled Assessments

Why Scheduled Continuous Assessments Need Special Handling

In a response-adaptive trial, a participant may be randomized today but have the relevant continuous outcome measured only at a later scheduled visit. If enrollment is staggered, participants assessed at the same calendar-time update can have very different follow-up: some have completed the endpoint visit, some have completed an earlier visit, and others have not yet reached any relevant assessment. An allocation update must distinguish these states. A value that has not yet been measured is not a zero, and an earlier-visit measurement is not automatically the endpoint.

This tutorial treats a continuous endpoint measured at a prespecified visit, such as change in symptom score at day 28. The focus is how to use that endpoint for allocation while assessments are pending. It is not a time-to-event analysis: the endpoint is defined at a scheduled visit, not by the time an event occurs.

Definition: A scheduled endpoint is matured for a participant when the prespecified assessment has been completed and its endpoint value is available for the allocation update. A scheduled endpoint is pending when that assessment has not yet been completed or its result is not yet available. A measurement at an earlier visit is interim information, not a matured endpoint, unless the trial’s endpoint definition explicitly says otherwise.

The central bookkeeping unit is the participant, not the calendar week or enrollment cohort. For each participant, record the randomization arm, enrollment date, scheduled assessment date, whether the assessment has occurred, and whether the measurement is available. This makes it possible to distinguish a participant who has not yet reached the visit from one who reached it but has an unavailable measurement. Those situations may call for different operational responses; neither justifies silently inserting an endpoint value.

Three Ways to Handle Pending Assessments

At a scheduled allocation update, a trial can use only matured endpoints, wait for additional assessments, or use an explicit model to predict pending endpoints. These are different policies, not interchangeable calculations. Matured-only updating is transparent but may use a small and temporally unrepresentative subset. Waiting can improve the amount of observed information but delays the allocation decision. Prediction can use interim measurements, but it depends on a model whose assumptions must be checked.

For illustration, suppose there are two arms, A and B, and a smaller endpoint value is better. One possible allocation rule uses the posterior probability that each arm has the smaller mean. Let \(\mu_A\) and \(\mu_B\) be the arm-specific mean scheduled endpoint values. With a minimum allocation probability \(\epsilon\), where \(0<\epsilon<0.5\), define

q_B=\epsilon+(1-2\epsilon)P(\mu_B<\mu_A\mid \text{data}),\qquad q_A=1-q_B.

Here \(q_A\) and \(q_B\) are the probabilities of assigning the next participant to each arm. The floor prevents either probability from reaching zero or one. This rule is an example, not a universal prescription; a trial must specify its allocation rule in advance. In a matured-only update, “data” in the probability calculation means endpoint values that have actually matured.

Model assumptions for the posterior example: Within each arm, matured endpoint values are independent observations from a normal distribution with arm-specific mean and a known common variance. The arm means have independent normal prior distributions. The scheduled assessment and measurement process must be defined consistently, and the model must be appropriate for the endpoint scale. These assumptions are used for the calculation below; they are not implied merely by the fact that an outcome is continuous.

Worked Example: Updating With Four Matured Assessments

In a hypothetical trial, lower day-28 symptom scores are better. At an allocation update, four endpoint measurements have matured: arm A has scores 9 and 11, and arm B has scores 8 and 9. Three later enrollment cohorts contained eight participants each, or \(3\times 8=24\) participants altogether. Four participants in the earliest of these cohorts have now completed their scheduled assessments; the other \(24-4=20\) endpoints remain pending. Use the four matured values for the current update, not the pending participants.

Assume \(Y\mid\mu_a\sim N(\mu_a,4)\) in each arm and independent priors \(\mu_a\sim N(10,4)\), where the second parameter is a variance. For arm A, \(n_A=2\) and \(\bar{x}_A=(9+11)/2=10\). For arm B, \(n_B=2\) and \(\bar{x}_B=(8+9)/2=8.5\). The posterior mean and variance for an arm with \(n\) matured observations and sample mean \(\bar{x}\) are

V_n=\left(\frac{1}{4}+\frac{n}{4}\right)^{-1},\qquad m_n=V_n\left(\frac{10}{4}+\frac{n\bar{x}}{4}\right).

For A, \(V_A=(0.25+0.50)^{-1}=1.333\) and \(m_A=1.333(2.5+5)=10\). For B, \(V_B=1.333\) and \(m_B=1.333(2.5+4.25)=9\). Thus \(\mu_A-\mu_B\) has posterior mean \(10-9=1\) and variance \(1.333+1.333=2.667\), so its standard deviation is \(\sqrt{2.667}=1.633\). Therefore,

P(\mu_B<\mu_A\mid\text{matured data}) =P(\mu_A-\mu_B>0) =\Phi\left(\frac{1}{1.633}\right) \approx \Phi(0.612) \approx 0.730.

With \(\epsilon=0.10\), the next participant is assigned to B with probability \(q_B=0.10+0.80(0.730)=0.684\), and to A with probability \(q_A=1-0.684=0.316\). The 20 pending participants are not counted as observed outcomes in this update. The result is conditional on the stated normal model and prior; it does not imply that B is certainly better.

Using an Earlier Visit to Forecast the Endpoint

When a participant has an interim measurement, a prespecified longitudinal model can map that value to a predictive distribution for the scheduled endpoint. For example, suppose the endpoint is the day-28 score and the day-14 score is available. A conditional model might be

Y_{28}\mid Y_{14}=y \sim N(\alpha+\beta y,\sigma^2_{\mathrm{pred}}).

The coefficients and predictive variance must come from a justified model—for example, a model estimated using suitable prior data or a prespecified trial model. A point forecast \(\alpha+\beta y\) is not an observed day-28 score. If forecasts enter an allocation score, the algorithm should explicitly say how they are used and whether their uncertainty is incorporated. The model should also be checked for calibration at the scheduled visit, including whether forecast errors vary across the range of interim values or across arms.

Worked Example: A Plug-In Predictive Allocation Update

Consider another hypothetical update with day-28 scores, where lower is better. Six matured participants in arm A have a mean score of 10, and six in arm B have a mean of 9. Two additional participants in each arm have day-14 scores but have not yet completed day 28. For illustration, suppose a prespecified conditional model is \(Y_{28}\mid Y_{14}=y\sim N(2+0.8y,9)\). The day-14 scores are 8 and 10 in A, and 6 and 8 in B.

The forecasts are \(2+0.8(8)=8.4\) and \(2+0.8(10)=10.0\) in A; they are \(2+0.8(6)=6.8\) and \(2+0.8(8)=8.4\) in B. A plug-in expected-completion mean for A is \((6(10)+8.4+10.0)/8=78.4/8=9.80\). For B it is \((6(9)+6.8+8.4)/8=69.2/8=8.65\). These are forecast-augmented summaries, not means of eight observed day-28 scores in either arm.

For a transparent illustrative rule, let \(S_a=\exp(-0.2\tilde{x}_a)\), where \(\tilde{x}_a\) is the forecast-augmented mean, and allocate in proportion to the scores subject to a 0.10 floor. The unadjusted share for B is

\frac{\exp(-0.2(8.65))} {\exp(-0.2(9.80))+\exp(-0.2(8.65))} = \frac{\exp(-1.73)}{\exp(-1.96)+\exp(-1.73)} \approx 0.557.

Applying the floor gives \(q_B=0.10+0.80(0.557)=0.546\) and \(q_A=0.454\). This plug-in rule uses forecast means but does not propagate the predictive variance of 9 into the allocation score. That is a limitation of this particular illustration, not a general property of predictive updating. A trial choosing this approach should assess whether forecast uncertainty materially changes its allocation decisions and should not describe forecasts as observed endpoint data.

Why the Posterior Calculation Has This Form

The normal update in the first example follows by combining the normal likelihood with the normal prior. If the prior density for \(\mu\) is proportional to \(\exp[-(\mu-10)^2/(2\cdot4)]\), and the likelihood from \(n\) observations with mean \(\bar{x}\) is proportional to \(\exp[-n(\bar{x}-\mu)^2/(2\cdot4)]\), the terms involving \(\mu\) in the combined exponent are

-\frac{1}{2}\left\{\frac{(\mu-10)^2}{4} +\frac{n(\bar{x}-\mu)^2}{4}\right\}.

Collecting the coefficients of \(\mu^2\) gives posterior precision \(1/4+n/4\). Collecting the linear terms gives posterior mean \((10/4+n\bar{x}/4)/(1/4+n/4)\). Completing the square therefore yields the \(V_n\) and \(m_n\) used above. The calculation applies to endpoint values in the likelihood. A pending participant with no endpoint measurement does not contribute a likelihood term for that endpoint under a matured-only analysis.

Scheduling, Diagnostics, and Alternatives

A scheduled-assessment update should use the actual assessment status at its cutoff time. A simple operational procedure is to classify every randomized participant, calculate the chosen allocation rule from the eligible information, and retain the same rules at later updates. This prevents calendar-time convenience from changing which measurements count as endpoints.

Set the cutoff and endpoint.
Specify the scheduled visit, assessment window, and data cutoff for each allocation update.

Classify participant records.
Separate matured endpoint values, interim measurements, assessments not yet due, and due assessments whose results are unavailable.

Apply the prespecified information rule.
Use matured values only, wait for a scheduled batch, or generate model-based predictions using the stated forecast model.

Check the update before randomizing.
Verify arm-specific counts, visit timing, model inputs, allocation floors, and that no pending endpoint has been entered as observed.

Useful diagnostics include the number of matured and pending assessments by arm and enrollment cohort; the distribution of time from enrollment to measurement; forecast errors once endpoints mature; and whether scheduled visits are completed within the prespecified windows. An unexpected arm difference in maturity counts, an accumulation of unavailable results after assessments are due, or systematic forecast errors signals that the update may not be operating as intended. Those checks do not establish that missingness is harmless; they identify conditions needing investigation.

Approach	What enters the update	Main strength	Main concern
Matured-only	Observed endpoint values only	Does not require predicting an unobserved endpoint	May be slow and may reflect only participants with earlier assessment dates
Wait for a scheduled batch	Observed endpoints after the chosen wait	Can increase the observed information before updating	Delays adaptation and may require a prespecified waiting rule
Model-based prediction	Matured values and explicitly modeled interim information	Can use information before the endpoint visit	Allocation can depend on model misspecification and forecast uncertainty

These choices do not have a single universally best answer. Matured-only updating is easiest to audit, but can be unstable when few endpoints have matured. Waiting is attractive when a short delay yields a substantial scheduled batch. Predictive updating can be useful when interim measures are informative about the endpoint, but its benefit depends on the predictive model being suitable for the trial population and visit schedule. Allocation probabilities should also retain any prespecified minimum and maximum constraints, rather than allowing noisy early estimates to produce near-deterministic assignment.

Common Mistakes and Limitations

Treating pending values as zeros. Zero is a real score on some scales, not a placeholder for “not measured.” Entering it changes arm means and can create an artificial treatment difference.
Calling an interim value the endpoint. A day-14 score does not equal a day-28 score unless that equality is part of the endpoint definition. Use a justified conditional model if the interim value is to inform a day-28 forecast.
Ignoring staggered assessment dates. A calendar-time snapshot is not a common follow-up duration. Track each participant’s own scheduled visit and maturity status.
Presenting forecasts as observations. A predicted score is uncertain. Label it as a forecast and describe how predictive uncertainty enters—or does not enter—the allocation rule.
Assuming that a fixed visit schedule solves every delay problem. The schedule defines when an endpoint is due; it does not guarantee that every due measurement is obtained or available. Investigate delayed or unavailable measurements rather than silently excluding them.

Scheduled continuous endpoints also have practical limitations: visit windows can widen the time at which a nominally fixed endpoint is measured, and a model trained in a different population may forecast poorly. If assessment timing or availability differs systematically across arms or patient types, a simple matured-only comparison or a plug-in prediction may not adequately represent the endpoint process. The allocation rule and its assumptions should therefore be evaluated as part of the trial design, not chosen informally after outcomes begin to arrive.

Check Your Understanding

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

Three later cohorts contain eight participants each. Four endpoints have matured. How many remain pending, and why should they not be coded as zero?
In a two-arm trial, the posterior probability that B has a lower mean endpoint than A is 0.65. With \(\epsilon=0.10\), what is the allocation probability for B under the stated rule?
A day-14 measurement is available but the endpoint is scheduled at day 28. What additional ingredient is needed to use the earlier value as a day-28 forecast?
Name one diagnostic for a predictive update and one reason that a forecast-augmented mean is not an observed endpoint mean.

Solutions. (1) \(3(8)=24\) participants total, so \(24-4=20\) remain pending. They have not supplied endpoint values, and zero would be an invented measurement rather than a missing-status label. (2) \(q_B=0.10+0.80(0.65)=0.620\), so \(q_A=0.380\). (3) A prespecified and justified conditional model linking the day-14 measurement to the day-28 endpoint is needed; the forecast also has uncertainty. (4) Check forecast errors once the day-28 endpoints mature. A forecast-augmented mean includes predicted values, which are not measurements of the participants’ completed day-28 endpoints.