From a CRM Recommendation to a Cohort Assignment
A CRM produces a dose recommendation from the toxicity outcomes observed so far. A trial still needs an assignment rule: which dose will the next participants actually receive, and when may that decision be made? In a cohort-based implementation, the answer is specified at the level of a group. Participants in a cohort are assigned together, and the recommendation is reconsidered between cohorts according to a prespecified schedule.
This distinction matters because a dose recommendation is not permission to escalate immediately. If participants at the current dose have not completed the toxicity assessment window, their outcomes are not yet known. Treating “no DLT observed so far” as “no DLT in the assessment window” can make the CRM act on incomplete evidence.
Model, Outcomes, and Timing
Suppose the trial has ordered dose levels \(1,\ldots,J\), and each participant has a binary dose-limiting toxicity (DLT) outcome within a fixed assessment window. Let \(q_j\) be the CRM skeleton probability at dose \(j\), with \(0<q_1<\cdots<q_J<1\). For a one-parameter power CRM, the model is
The model assumes that participants’ binary outcomes are conditionally independent given \(a\) and their assigned doses, that the DLT definition and assessment window are applied consistently, and that the ordered skeleton represents increasing toxicity across doses. The parameter \(a\) is updated from the outcomes; it is not a participant-specific dose assignment.
A participant is evaluable for the binary-window outcome when the protocol has established whether a DLT occurred during the full window. A DLT during the window can establish the outcome before the window ends. In the absence of a DLT, the outcome is not known to be DLT-free until the window is complete. A missed assessment or incomplete follow-up is not automatically a no-DLT outcome.
For a completed cohort \(c\), let \(d_c\) be its assigned dose, \(n_c\) its number of evaluable participants, and \(y_c\) the number who experienced a DLT within the window. Its likelihood contribution is
With prior density \(\pi(a)\), the posterior after completed cohorts is proportional to the prior multiplied by their likelihood contributions:
where \(D\) denotes the completed cohort data and \(\mathcal{C}_{\mathrm{complete}}\) is the set of cohorts whose required outcomes are known. The expression makes the timing rule explicit: a pending outcome must not be entered as a completed no-DLT observation. The introductory material on the posterior target-dose rule defines a common recommendation as the dose whose posterior mean toxicity probability is closest to the target \(\phi\). Whatever prespecified recommendation rule is used, it is applied to the posterior based on the outcomes available under the protocol.
A Cohort-by-Cohort Assignment Rule
A straightforward conservative implementation begins with a protocol-specified starting dose. All participants in a cohort receive the same assigned dose. After the required outcomes are observed, the CRM is updated and produces a recommendation for the next cohort. The recommendation can be the same dose, a lower dose, or a higher dose, subject to the protocol’s assignment restrictions.
Give every participant in the cohort the same dose specified for that cohort. Do not move an individual participant to a different dose because another cohort member has a different outcome.
Follow each participant through the DLT assessment window, or until a DLT within the window establishes the outcome. Keep unresolved participants and cohorts visibly marked as pending.
Use the completed dose and DLT counts to update the posterior and apply the protocol’s CRM recommendation rule.
If a required outcome remains pending, do not assign a higher dose. Under a fully non-overlapping rule, hold the next cohort assignment until the current cohort is complete; a protocol may separately specify whether enrollment can continue at the same dose while waiting.
This is a rule for translating model output into trial conduct, not a new toxicity model. It deliberately separates the data-update schedule from the rule used to choose a dose. A protocol must state both. For example, “use the CRM recommendation” does not say whether the recommendation is recalculated after each participant, after a full cohort, or only after all participants complete follow-up.
Why the Waiting Rule Follows From the Outcome Model
The binary-window likelihood treats each participant as having an outcome for the entire specified window. If a participant has no observed DLT at day 10 of a 28-day window, the participant may still have a DLT on day 20. Entering that participant as a no-DLT outcome would use the factor \(1-p_{d_c}(a)\) before the outcome is established.
The problem is not merely that the estimate might be imprecise. The likelihood contribution itself would be based on a false classification. A time-to-event model could use partial follow-up under its own assumptions, but that is a different model and is not obtained by labeling every participant still event-free as a completed no-DLT case.
Under a fixed-window model and a non-overlapping cohort rule, the logic is therefore direct: determine the cohort’s required outcomes; form the likelihood from those outcomes; update the posterior; then assign the next cohort. If any outcome required by the rule is unresolved, do not use an upward recommendation as grounds to escalate. This protects the interpretation of the data and prevents a new cohort from receiving a higher dose while toxicity information from the current dose is still pending.
Worked Example: Updating After a Complete Cohort
Hypothetical trial. A trial uses a 28-day DLT window and cohorts of three. The protocol starts at dose 1 and waits for all three outcomes before assigning the next cohort. All three participants complete the window without a DLT, so \(n_1=3\) and \(y_1=0\) at dose 1.
For the power CRM, the cohort likelihood contribution is
The exponent on the no-DLT term is \(3-0=3\), because all three participants have completed follow-up without a DLT. The posterior is updated using this contribution and the prior. Suppose the prespecified target-dose rule then recommends dose 2. That recommendation is based on the completed cohort data; it does not alter the doses already received by the first three participants.
The next cohort of three is assigned dose 2. There is no within-cohort escalation: all three receive dose 2, and another recommendation is not made until the required outcomes at dose 2 are available. The likelihood and assignment arithmetic are consistent with the record: three evaluable participants at dose 1, zero DLTs, followed by three participants assigned together at dose 2.
Worked Example: One Pending Outcome Blocks Escalation
Hypothetical trial. Three participants receive dose 2. At day 28, two have completed the window without a DLT. The third participant is at day 10, has had no DLT so far, and has not completed follow-up. Thus two outcomes are established and one is pending.
Under the stated non-overlapping cohort rule, the cohort is not complete. The data record at this point is not three evaluable participants with zero DLTs. The two completed outcomes contribute \(\{1-p_2(a)\}^2\); the pending participant does not yet contribute a completed no-DLT factor. The protocol holds the next cohort assignment, so it cannot escalate to dose 3 on the basis of an assumed outcome.
Suppose the third participant experiences a DLT on day 18, within the 28-day window. The completed cohort then has \(n_2=3\) and \(y_2=1\), with likelihood contribution
Now the CRM can update using all three outcomes and produce the next recommendation. Whether that recommendation is dose 1, 2, or another permitted dose depends on the prior, skeleton, target, accumulated data, and recommendation rule; the assignment rule does not assume the result. The key decision is that escalation was not allowed while one required outcome was unresolved.
Worked Example: A Cohort Does Not Change Dose Midway
Hypothetical trial. A cohort contains four participants assigned to dose 3. The first participant has a DLT on day 6, the second completes the 28-day window without a DLT, and the other two are still being followed. The protocol specifies a cohort size of four, requires all four outcomes before the next assignment, and assigns one dose to the cohort.
The participant with a DLT has an established positive outcome, and the participant completing the window has an established negative outcome. The other two outcomes remain pending. At this point, the completed-outcome count is two, with one DLT; the cohort count for a final CRM update is not yet \(n_c=4\). The two pending participants cannot be counted as no-DLT outcomes, and the next cohort cannot be escalated.
If the remaining participants later complete the window without DLTs, the full cohort has \(n_c=4\) and \(y_c=1\), giving \(L_c(a)=p_3(a)\{1-p_3(a)\}^3\). The CRM update is then based on that completed cohort. Neither the early DLT nor a later recommendation changes the doses of the four people already assigned to dose 3.
Operational Checks and Alternatives
A sound protocol makes the assignment process auditable. At every decision point, the trial team should be able to state the dose and cohort size, each participant’s window status, the number of established DLT outcomes, whether an update is permitted, and the resulting next-cohort assignment. Ambiguous phrases such as “once toxicity is known” are inadequate if they do not say how pending follow-up, missed visits, or early DLT events are handled.
| Approach | Assignment timing | Main operational distinction |
|---|---|---|
| Non-overlapping cohorts | Wait for the required outcomes from one cohort before assigning the next. | Simple timing and clear cohort likelihood contributions, but accrual may be slower. |
| Individual sequential assignment | Reconsider assignment after each participant’s outcome is available. | Can use smaller batches, but decisions depend on individual outcome timing and may be difficult to coordinate. |
| Rolling or overlapping enrollment | New participants may enter while earlier outcomes are pending, under explicit allocation constraints. | May reduce waiting, but requires rules for pending data and escalation; it is not equivalent to treating pending outcomes as no DLTs. |
Cohort sizes and waiting rules involve trade-offs. Larger cohorts reduce the frequency of decision meetings but expose more participants to a dose before the next model-based update. Smaller cohorts permit more frequent reassessment but can increase operational burden. Overlapping enrollment may improve accrual, but a protocol must specify how recommendations are formed when outcomes are incomplete and must prevent escalation based on unobserved outcomes.
These rules do not replace other trial safeguards. A CRM recommendation and a no-escalation timing rule do not, by themselves, define dose exclusions, stopping criteria, or responses to every safety concern. Those must be specified separately. Nor does waiting guarantee that the model is correct: the assignment process still depends on the DLT definition, assessment quality, skeleton, prior, and the assumptions linking participants’ outcomes to dose.
Common Pitfall
Correction: In a fixed 28-day binary-window model, absence of a DLT by day 10 does not establish absence of a DLT during the full window. Keep the outcome pending and follow the protocol’s assignment rule. Do not escalate until the required outcomes are observed. Using partial follow-up requires a model and a rule designed for partial follow-up, not an undocumented substitution.
Another mistake is to confuse a recommendation with an assignment. The CRM may recommend a different dose after an update, but participants already assigned to a cohort remain at their assigned dose. The protocol controls when that recommendation becomes the next cohort’s assignment.
Check Your Understanding
Answer each question using the fixed-window, non-overlapping cohort rule described above. Solutions follow the questions.
- Three participants complete a 28-day window at dose 1 with zero DLTs. What are \(n_c\) and \(y_c\), and what is the cohort likelihood contribution?
- Three participants are assigned dose 2. Two complete follow-up without DLTs; the third is only 12 days into the window with no DLT so far. Can the next cohort be escalated under the stated rule? Explain.
- A cohort of four at dose 3 has one DLT and three participants who complete the window without DLTs. Give the likelihood contribution in terms of \(p_3(a)\).
- Why must a protocol distinguish a CRM recommendation from the dose assigned to participants already enrolled in a cohort?
Solutions. (1) All three outcomes are evaluable, so \(n_c=3\) and \(y_c=0\). The contribution is \(p_1(a)^0\{1-p_1(a)\}^3=\{1-p_1(a)\}^3\). (2) No. One outcome is pending because the full window is incomplete. Under the non-overlapping rule, the cohort is not complete and the next assignment is held; the pending participant must not be counted as a no-DLT outcome. (3) Here \(n_c=4\) and \(y_c=1\), so the contribution is \(p_3(a)^1\{1-p_3(a)\}^3=p_3(a)\{1-p_3(a)\}^3\). (4) The recommendation is a model-based proposal for a subsequent assignment. The dose already given to a cohort is fixed for those participants; separating the two prevents an update from being mistaken for permission to change their treatment or to enroll the next cohort before required outcomes are known.