Why Incomplete Follow-Up Changes an Interim Decision
A BLRM dose decision concerns the probability of a DLT within a specified assessment window, not simply whether a DLT has been reported so far. If a patient has not completed that window and has not had a DLT, the patient's endpoint is unresolved. Calling that patient DLT-free would turn “not observed yet” into “observed not to occur,” which is not justified.
This distinction matters during escalation. A cohort may appear to have few DLTs simply because some patients have not yet had enough time for their outcomes to be known. The posterior from the observed outcomes can still be calculated, but the decision rule should account for the unresolved follow-up separately. In particular, a low posterior overdose probability based on mature outcomes is not, by itself, a reason to escalate while the most recent cohort remains under observation.
The relevant endpoint is the binary outcome by the end of \(\tau\). An adverse event counts as a DLT only if it meets the protocol’s DLT definition and classification rules. A pending patient who experiences a confirmed DLT during the window can contribute that event immediately; the patient need not wait until \(\tau\) ends for an event that has already occurred. By contrast, absence of a DLT so far does not establish a completed DLT-free outcome.
BLRM Evidence and the Pending-Outcome Distinction
For dose \(d_j\), let \(p_j(\theta)\) denote the modelled probability of a DLT by \(\tau\), conditional on BLRM parameters \(\theta\). The BLRM uses the dose–toxicity model to obtain a posterior distribution for this probability from the available outcomes. Write \(D_{\mathrm{obs}}\) for the outcomes that are known at an interim review. For a protocol-defined overdose boundary \(\phi_O\), the posterior overdose probability at dose \(d_j\) is
A common overdose-control rule calls \(d_j\) admissible when \(q_j < \xi\), where \(\xi\) is the protocol-specified cutoff. The values of \(\phi_O\) and \(\xi\) are design choices; they must be fixed in the protocol rather than chosen after seeing the interim results. This probability summarizes posterior evidence about toxicity at the dose. It does not measure whether all patients currently receiving that dose have completed follow-up.
If a dose has \(x_j\) confirmed DLTs and \(m_j\) completed DLT-free outcomes, their binary-outcome likelihood contribution is
A pending patient with no observed DLT contributes neither factor. The patient's endpoint is not yet known. A pending patient with an already confirmed DLT contributes as a DLT; the fact that the rest of the window is still open does not erase an event already observed within it. Thus a convenient interim data summary keeps three counts distinct: confirmed DLTs, completed DLT-free outcomes, and pending outcomes.
A Conservative Interim Rule
A practical conservative rule separates the evidence calculation from the action safeguard. The following formulation is one transparent option; the protocol must specify which patients’ follow-up blocks escalation and what actions are allowed when a safety concern arises.
For every treated patient, record whether a DLT is confirmed, the \(\tau\)-window is complete without a DLT, or the outcome is pending. Do not count a pending no-DLT-to-date patient as completed DLT-free.
Use confirmed DLTs and completed DLT-free outcomes in the binary BLRM likelihood. A pending patient without a confirmed DLT contributes no binary outcome yet.
For each candidate dose, calculate \(q_j\) from the current posterior and compare it with the protocol cutoff \(\xi\). A dose failing \(q_j < \xi\) is not admissible under this rule.
Do not escalate above the dose of a cohort whose required DLT outcomes remain pending. Hold at the current dose, defer the decision, or take a safety action such as de-escalation or stopping as permitted by the protocol.
The pending-follow-up gate is a decision safeguard, not a claim that the posterior probability \(q_j\) has been calculated incorrectly. The posterior may be valid for the observed endpoints while still being too immature to support escalation. A protocol can also require that no relevant patients from earlier cohorts remain pending, or specify a narrower gate; that choice should be explicit and clinically justified.
Why a Pending Outcome Does Not Count as a DLT-Free Outcome
The likelihood makes the distinction precise. If a pending patient were incorrectly counted as DLT-free, the analysis would add a factor \(1-p_j(\theta)\). But the patient’s binary endpoint is unknown. For an unobserved binary endpoint \(Y\), summing over its possible values gives
Thus an unobserved binary endpoint supplies no binary-outcome likelihood factor. Inserting \(1-p_j(\theta)\) would act as if \(Y=0\) had actually been observed, shifting the posterior toward lower toxicity without evidence that the assessment window was completed.
A time-to-event model can use the length of event-free follow-up as information, but that requires a model and likelihood for event times or censoring. A fixed-window binary BLRM does not acquire that information merely because a patient has remained event-free for part of \(\tau\). Unless the protocol specifies and justifies a time-to-event approach, partial follow-up should not be converted into a completed binary outcome.
Worked Example: The Cohort Looks Safe, but Follow-Up Is Pending
Suppose a hypothetical cohort of three patients receives dose \(d_2\). Two patients complete a 28-day DLT window without a DLT. The third has no DLT recorded by day 12, but the window is still open. Assume the protocol uses \(\phi_O=0.30\) and \(\xi=0.25\), and the BLRM fit to known outcomes gives \(q_3=0.18\) for the next higher dose \(d_3\).
The observed counts at \(d_2\) are \(x_2=0\) confirmed DLTs, \(m_2=2\) completed DLT-free outcomes, and \(r_2=1\) pending outcome. The known-outcome likelihood contribution at \(d_2\) is
The pending patient does not add a third factor of \(1-p_2(\theta)\). Since \(q_3=0.18<0.25=\xi\), dose \(d_3\) satisfies the posterior overdose cutoff. However, under the stated conservative gate, the cohort at \(d_2\) has an unresolved DLT outcome, so escalation to \(d_3\) is deferred. The decision is to hold or wait, not to escalate on the basis of the posterior cutoff alone.
Interim Events and Reassessment
The rule is not “wait no matter what.” A confirmed DLT during an open window is new safety information and should be classified and acted on under the protocol. The BLRM likelihood then includes that confirmed event, and the posterior overdose probability may change. De-escalation, suspension, or stopping can be allowed immediately when the observed safety evidence warrants it; waiting for every other patient’s window to close is not a reason to ignore a DLT that has already occurred.
When the remaining windows close, outcomes are reclassified. A patient who completes \(\tau\) without a DLT now contributes a completed DLT-free outcome. The BLRM is updated with the newly known endpoints, posterior overdose probabilities are recalculated, and the pending-follow-up gate can be removed if its requirements are satisfied. A dose can remain inadmissible after follow-up completes; maturity of the data does not guarantee that escalation is allowed.
Worked Example: A Confirmed DLT During an Open Window
In a hypothetical cohort at \(d_2\), one patient has completed 28 days without a DLT. A second patient develops a protocol-defined DLT on day 9. A third patient is event-free on day 14 but has not completed the window. Let \(\phi_O=0.30\), \(\xi=0.25\), and suppose the BLRM posterior using known endpoints gives \(q_2=0.31\) at \(d_2\) and \(q_1=0.12\) at the lower dose \(d_1\).
The counts are \(x_2=1\), \(m_2=1\), and \(r_2=1\). The known-outcome likelihood is
The event-free patient at day 14 is not counted as a second DLT-free outcome. Since \(q_2=0.31\) is not less than \(0.25\), \(d_2\) fails the stated overdose cutoff. Since \(q_1=0.12<0.25\), \(d_1\) passes it. A conservative action is to de-escalate to \(d_1\) or otherwise follow the protocol’s safety action, while the remaining outcome is monitored. The confirmed DLT is usable now; the event-free partial follow-up is not a completed negative endpoint.
When Follow-Up Completes
Worked Example: Reopening the Escalation Decision
Consider a hypothetical cohort of four patients treated at \(d_2\). One had a confirmed DLT, two completed the 28-day window without DLT, and one was pending at the first interim review. At that review, suppose the posterior overdose probability for \(d_3\) was \(q_3=0.22\), with cutoff \(\xi=0.25\). The conservative rule deferred escalation because one patient was pending.
At a later review, the fourth patient completes the window without a DLT. The updated counts at \(d_2\) are now \(x_2=1\), \(m_2=3\), and \(r_2=0\). The likelihood contribution is
Suppose the updated BLRM fit gives \(q_3=0.23\). Then \(0.23<0.25\), so \(d_3\) passes the overdose cutoff, and the follow-up gate is now clear. Under the two stated conditions, escalation to \(d_3\) may be considered. This conclusion is conditional on the hypothetical posterior output and the protocol’s other requirements; completing follow-up does not itself establish that \(d_3\) is safe.
Conservative Sensitivity Checks, Alternatives, and Limitations
A protocol may add a worst-case sensitivity check: temporarily assess the action that would result if all pending patients were treated as DLTs for the purpose of a safety gate. This can be a useful conservative stress test. It must be labelled as a decision overlay, not as the BLRM posterior analysis of observed outcomes. Replacing pending outcomes with assumed DLTs in the likelihood would produce a posterior conditional on fabricated outcomes, not on the actual data.
The main alternatives have different trade-offs:
| Approach | How pending patients are handled | Main implication |
|---|---|---|
| Wait for completion | Defer the escalation decision until the specified outcomes mature. | Preserves the binary endpoint interpretation but can slow accrual or escalation. |
| Known-endpoint BLRM plus an escalation gate | Use completed outcomes and confirmed DLTs in the posterior; prohibit upward movement while specified follow-up is pending. | Separates evidence from a conservative action safeguard. |
| Worst-case safety overlay | Check whether a proposed action remains acceptable under a stated adverse assumption about pending patients. | Can be cautious, but is not a substitute for the posterior based on observed outcomes. |
| Time-to-event model | Uses event timing and partial follow-up under an explicit event-time model. | Can use more information, but requires additional assumptions and is not the fixed-window binary likelihood described above. |
The known-endpoint posterior and a pending-follow-up gate do not eliminate every concern. If the timing of outcome observation is related to toxicity in ways not handled by the analysis, or if DLT classification is delayed or inconsistent, the interim evidence can be misleading. Dose assignment, cohort timing, the DLT window, event adjudication, and the rule for identifying which patients block escalation should therefore be clear. Also, a posterior calculated from few completed outcomes may be highly sensitive to the prior and model assumptions; a pending gate prevents one specific premature action but does not resolve that uncertainty.
A common mistake is to say, “The pending patient has had no DLT, so count that patient as DLT-free; the overdose probability is below the cutoff, so escalate.” The first step is wrong because the endpoint window is incomplete. The correction is to record the patient as pending, omit an unobserved binary endpoint from the likelihood, and apply the protocol’s pending-follow-up safeguard. The posterior cutoff and the follow-up gate answer different questions; both must be satisfied before an escalation permitted by this rule.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A patient has no recorded DLT on day 18 of a 28-day window. Is this a completed DLT-free outcome?
- At a dose, there are two confirmed DLTs, three completed DLT-free outcomes, and one pending patient with no event so far. What is the binary BLRM likelihood contribution?
- A candidate dose has posterior overdose probability \(0.19\), and the cutoff is \(0.20\). The protocol blocks escalation while any patient in the latest cohort is pending. One patient remains pending. Is escalation allowed under this rule?
- A pending patient develops a protocol-defined DLT on day 7. Must the team wait until day 28 before the event can enter the binary BLRM update?
- Why is assigning every pending patient a DLT in the likelihood different from using an all-pending-as-DLT safety stress test?
Solutions. (1) No. The window is incomplete, so the outcome remains pending unless a DLT is confirmed. (2) If the dose-specific DLT probability is \(p_j(\theta)\), the contribution is \([p_j(\theta)]^2[1-p_j(\theta)]^3\); the pending patient adds no binary endpoint factor. (3) No. Although \(0.19<0.20\) passes the posterior cutoff, the separate pending-follow-up gate blocks escalation. (4) No wait is needed for the confirmed event: it is an observed DLT within the window and can enter the update immediately. (5) Inserting DLTs into the likelihood treats assumed events as observed data; a stress test instead evaluates a conservative action under an explicit hypothetical and does not claim those events occurred.