From Escalation Evidence to a Final Recommendation
At the end of dose escalation, the question changes. During escalation, a dose decision governs what participants may receive next; at the end, the trial must apply its prespecified rule to the accumulated evidence and state whether a dose is recommended. These decisions use related posterior information, but they need not use the same rule. A dose that was permissible for escalation is not automatically the final recommendation.
The final decision commonly combines two distinct ideas. First, a dose must satisfy the protocol’s overdose-control condition. Second, among doses that satisfy that condition, the protocol uses a rule involving the posterior probability that toxicity lies in a clinically defined target interval. The ordering and any additional thresholds or tie-breakers must be specified by the protocol; posterior evidence alone does not dictate a unique final-dose rule.
Assume here that all dose-limiting toxicity outcomes included in the analysis are available and that the Bayesian logistic regression model (BLRM) has been fitted to the trial data. For each candidate dose \(d\), the model gives a posterior distribution for its DLT probability \(p(d)\). The examples use hypothetical completed data and posterior summaries. They do not imply that raw cohort proportions are themselves posterior probabilities.
Make the Protocol Rule Explicit
A final selection rule needs more than the phrase “choose the dose with the best posterior evidence.” It must say which doses are eligible, what is optimized or thresholded among them, and what happens if none qualifies. For example, one protocol might select the admissible dose with the largest target-interval probability. Another might recommend the highest admissible dose whose target-interval probability exceeds a minimum. Those rules can select different doses from the same posterior distribution.
Write the admissible set as
If the protocol says to maximize target-interval probability among admissible doses, its rule is
This expression is meaningful only if \(\mathcal{A}\) is nonempty and the protocol specifies how to handle ties. If the protocol instead requires a minimum target-interval probability \(q_{\min}\) and selects the highest qualifying dose, the eligible set is
when \(\mathcal{E}\) is nonempty. In that rule, “highest” refers to the dose level, not to the target-interval probability. If no dose qualifies, the protocol’s no-recommendation outcome applies; one should not silently replace the rule with a different ranking.
Estimating the Posterior Probabilities
In practice, posterior event probabilities can be computed from posterior draws. Suppose \(p^{(m)}(d)\) is the toxicity probability at dose \(d\) in draw \(m\), for \(m=1,\ldots,M\). The target-interval probability is estimated by counting draws inside the interval, and the overdose probability by counting draws above the overdose boundary:
Here \(I\{\cdot\}\) equals 1 when the condition is true and 0 otherwise. These are Monte Carlo estimates of posterior probabilities, not proportions of trial participants who experienced DLTs. If the posterior is represented by direct calculation rather than simulation, the same selection logic applies to the calculated probabilities.
The rule can be justified as a constrained decision: first exclude doses that violate the protocol’s overdose condition, then apply the protocol’s stated objective or threshold within the remaining set. This ordering prevents a high target-interval probability from overriding an overdose restriction. It also makes clear that the final recommendation is conditional on the chosen model, posterior, boundaries, cutoff, and protocol rule.
Worked Example: Highest Target Probability Among Admissible Doses
Suppose a hypothetical trial has complete DLT outcomes at four candidate doses: D1, 0/3; D2, 1/6; D3, 2/6; and D4, 3/3 participants with DLT. After fitting the BLRM, the analysis uses \(M=10{,}000\) posterior draws per dose. The target interval is 0.20 to 0.30, the overdose boundary is \(p_O=0.33\), and the admissibility cutoff is \(\xi=0.25\). The protocol selects the admissible dose with the largest target-interval probability.
| Dose | Draws in target interval | Draws above 0.33 | Estimated target probability | Estimated overdose probability |
|---|---|---|---|---|
| D1 | 1,700 | 100 | 0.1700 | 0.0100 |
| D2 | 3,100 | 400 | 0.3100 | 0.0400 |
| D3 | 3,600 | 2,100 | 0.3600 | 0.2100 |
| D4 | 2,500 | 5,100 | 0.2500 | 0.5100 |
For D3, for example, the target estimate is \(3{,}600/10{,}000=0.3600\), and the overdose estimate is \(2{,}100/10{,}000=0.2100\). Since \(0.2100\leq0.25\), D3 is admissible. D4 is not admissible because \(0.5100>0.25\). D1 and D2 are also admissible, with overdose estimates 0.0100 and 0.0400, respectively.
Among admissible doses D1, D2, and D3, D3 has the largest estimated target-interval probability: \(0.3600>0.3100>0.1700\). The protocol therefore recommends D3. This conclusion follows from the stated rule, not just from D3 having a higher dose level or a higher observed DLT rate. The observed cohort counts describe the data; the two probabilities used in the decision are posterior quantities from the fitted model.
Why the Rule Changes the Answer
A rule that maximizes target-interval probability and a rule that selects the highest qualifying dose express different priorities. The first favors the dose with the strongest posterior support for target-range toxicity. The second may favor a higher dose, provided it passes the overdose condition and a target-probability threshold. Neither rule is universally correct in the abstract: the protocol must state which one governs the recommendation.
Worked Example: Highest Qualifying Dose Is Not the Maximum-Probability Dose
In a second hypothetical trial, the target interval is again 0.20 to 0.30, with overdose boundary \(p_O=0.33\) and cutoff \(\xi=0.25\). The protocol requires an estimated target-interval probability of at least \(q_{\min}=0.20\), then recommends the highest dose satisfying both conditions. Posterior draws give the following summaries:
| Dose | Target-interval draws out of 5,000 | Overdose draws out of 5,000 | Target probability | Overdose probability |
|---|---|---|---|---|
| D1 | 1,250 | 100 | 0.2500 | 0.0200 |
| D2 | 1,750 | 500 | 0.3500 | 0.1000 |
| D3 | 1,400 | 1,000 | 0.2800 | 0.2000 |
| D4 | 1,000 | 2,000 | 0.2000 | 0.4000 |
The estimates are obtained by division by 5,000; for instance, D3 has target probability \(1{,}400/5{,}000=0.2800\) and overdose probability \(1{,}000/5{,}000=0.2000\). D1, D2, and D3 pass the overdose cutoff. D4 does not, because \(0.4000>0.25\).
Each of D1, D2, and D3 also meets the target-probability threshold: \(0.2500\geq0.20\), \(0.3500\geq0.20\), and \(0.2800\geq0.20\). The protocol selects the highest of these qualifying doses, D3. D2 has the largest target-interval probability, but selecting D2 would apply a maximum-probability rule instead of the stated highest-qualifying-dose rule. The posterior summaries are the same; the protocol’s decision rule changes the result.
When No Dose Qualifies
A recommendation rule must also cover an empty eligible set. If every dose exceeds the overdose cutoff, none is admissible, even if one has the largest target-interval probability. Depending on the protocol, the appropriate final outcome may be no recommended dose. That is a decision produced by the prespecified criteria; it is not a statistical failure to choose the apparently “least bad” dose.
Worked Example: No Admissible Dose Means No Recommendation
Suppose a third hypothetical trial uses the target interval 0.20 to 0.30, overdose boundary \(p_O=0.33\), and cutoff \(\xi=0.25\). The protocol recommends the admissible dose with the largest target-interval probability and states that no dose is recommended if none is admissible. Among 10,000 posterior draws, dose A has 2,100 draws in the target interval and 3,100 above the overdose boundary; dose B has 3,200 and 4,800, respectively.
For A, the estimated target probability is \(2{,}100/10{,}000=0.2100\), while the overdose probability is \(3{,}100/10{,}000=0.3100\). For B, the corresponding estimates are \(3{,}200/10{,}000=0.3200\) and \(4{,}800/10{,}000=0.4800\). Both doses fail admissibility because \(0.3100>0.25\) and \(0.4800>0.25\). Thus \(\mathcal{A}\) is empty, and the protocol’s outcome is no recommended dose. B’s larger target-interval probability does not undo its overdose-control failure.
Diagnostics, Alternatives, and Limitations
Posterior probabilities estimated by simulation have Monte Carlo error. For an estimated event probability \(\hat{\pi}\) based on \(M\) effectively independent draws, its approximate Monte Carlo standard error is \(\sqrt{\hat{\pi}(1-\hat{\pi})/M}\). For the overdose estimate 0.2100 in the first example with 10,000 draws, this is approximately \(\sqrt{0.2100(0.7900)/10{,}000}=0.00407\). This is small relative to the distance from 0.25, but an estimate near the cutoff deserves more draws or a more precise calculation before classifying the dose. With correlated simulation draws, the effective number of independent draws can be less than the raw draw count.
Check that the target interval, overdose boundary, cutoff, candidate dose set, and selection rule all match the protocol. Confirm that each probability uses the correct event: target-interval probability concerns \(p(d)\) between the two target limits, whereas overdose probability concerns \(p(d)\) above the overdose boundary. Inspect whether the probabilities are close to a decision threshold, whether posterior computation is stable, and whether the reported recommendation follows the stated rule, including its tie and no-eligible-dose provisions.
Several tempting substitutes answer different questions. Selecting the dose with the highest observed DLT rate or the highest observed response ignores posterior uncertainty and does not implement the overdose constraint. Selecting the dose with the lowest posterior mean toxicity does not establish that the probability of exceeding the overdose boundary is acceptable. Selecting the dose with the largest target-interval probability without first enforcing admissibility can recommend a dose the protocol rules out. A utility-based decision rule is another possible approach, but it requires a prespecified utility function and is not interchangeable with a target-probability rule.
The final recommendation is conditional on the model and the protocol’s definitions. It does not prove that the selected dose is safe for every participant, that its true toxicity probability lies in the target interval, or that it is optimal under every clinical objective. Model fit, dose range, prior assumptions, and the amount of information at each dose affect posterior probabilities. The rule makes the decision reproducible; it cannot remove uncertainty or compensate for a poorly specified model or protocol.
Check Your Understanding
Answer each question using the stated rule. Solutions follow the questions.
- A protocol uses overdose cutoff 0.25. At dose X, the posterior overdose probability is 0.24 and the target-interval probability is 0.18. Is X admissible?
- Three doses are admissible, with target-interval probabilities 0.22, 0.31, and 0.27. Under a rule that maximizes target-interval probability among admissible doses, which dose is selected?
- Under a highest-qualifying-dose rule, doses D1, D2, and D3 have overdose probabilities 0.02, 0.10, and 0.20 and target probabilities 0.25, 0.35, and 0.28. The overdose cutoff is 0.25 and the minimum target probability is 0.20. Which dose is selected?
- With 8,000 posterior draws, 1,600 exceed the overdose boundary at a candidate dose. What is the estimated overdose probability? If the cutoff is 0.25, is the dose admissible?
- Every candidate dose has posterior overdose probability greater than the protocol cutoff. What follows under a rule that only recommends admissible doses?
Solutions. (1) Yes. Since \(0.24\leq0.25\), X is admissible; its target probability is not part of the admissibility test. (2) Select the dose with probability 0.31, since it has the largest target-interval probability in the admissible set. (3) All three doses pass the overdose condition and exceed the minimum target probability, so the highest qualifying dose, D3, is selected. (4) The estimate is \(1{,}600/8{,}000=0.2000\), so the dose is admissible because \(0.2000\leq0.25\). (5) The admissible set is empty, so no dose is recommended under that rule.