From admissibility to dose choice
An admissible dose has passed a protocol’s specified posterior overdose-probability rule. That establishes eligibility; it does not, by itself, establish that the dose is the most attractive choice among eligible candidates. To choose within the admissible set, a protocol needs a further decision rule. One natural rule is to favor the dose with the greatest posterior probability that its toxicity probability lies in the target interval.
This distinction matters because dose escalation is constrained by safety but also aims to identify a dose with a suitable toxicity profile. A dose may be admissible while having a relatively low chance of being in the target interval. Conversely, a dose with the greatest target-interval probability cannot be selected under a rule that excludes it for excessive overdose probability. Eligibility and preference answer different questions.
The target event concerns uncertainty about the dose’s underlying toxicity probability, represented by the posterior distribution after observing the data. It is not the probability that a particular patient will experience a DLT, nor the probability that the true toxicity probability is fixed at a particular value. The model and posterior incorporate uncertainty in the toxicity curve parameters.
Choosing by posterior target probability
Suppose the protocol’s objective, after applying the admissibility rule, is to choose a dose with the greatest posterior chance of lying in the target interval. The selection is then $$ d^*=\arg\max_{d_j\in A}\pi_j, $$ where \(A\) is the admissible set. This rule is conditional on the stated objective and on the admissibility rules; it is not a universal clinical decision rule.
The justification follows from a simple utility interpretation. Assign utility 1 if a selected dose’s toxicity probability is in the target interval and utility 0 otherwise. The posterior expected utility of selecting \(d_j\) is $$ E\{I(\phi_L\leq p_j\leq\phi_U)\mid D\} =P(\phi_L\leq p_j\leq\phi_U\mid D)=\pi_j, $$ where \(I(\cdot)\) is 1 when its condition holds and 0 otherwise. Maximizing posterior expected utility over eligible doses therefore means maximizing \(\pi_j\) over \(A\). A different utility, such as one that also values efficacy or penalizes particular toxicities differently, could produce a different ranking.
This rule requires a specified target interval, a posterior distribution from the chosen model and data, a clearly defined admissible set, and a prespecified tie-breaking rule if multiple doses have equal target probabilities. The BLRM framework used for dose escalation models DLT probability across doses; its positive-slope assumption makes toxicity probability increase with dose for each parameter draw. Even then, the target-interval probabilities need not increase with dose: they can rise as doses approach the target and fall once the posterior mass shifts beyond it.
Worked Example: Selecting the Dose With the Highest Target Probability
Suppose a hypothetical trial uses target interval \([0.20,0.30]\), overdose cutoff \(\eta=0.25\), and the rule \(q_j\leq\eta\). Posterior summaries from the fitted model are shown below. Assume the listed doses are the full candidate set and there are no additional protocol restrictions.
| Dose (mg) | Posterior target probability \(\pi_j\) | Posterior overdose probability \(q_j\) |
|---|---|---|
| 5 | 0.12 | 0.02 |
| 10 | 0.42 | 0.08 |
| 20 | 0.48 | 0.22 |
| 30 | 0.24 | 0.41 |
Apply the eligibility rule to each dose: \(0.02\leq0.25\), \(0.08\leq0.25\), and \(0.22\leq0.25\), so 5, 10, and 20 mg are admissible. The 30 mg dose is not admissible because \(0.41>0.25\). Among eligible doses, compare target probabilities: \(0.12\), \(0.42\), and \(0.48\). Thus $$ \max(0.12,0.42,0.48)=0.48, $$ which occurs at 20 mg. Under the stated target-probability rule, select 20 mg. This conclusion does not mean that 20 mg is certainly in the target interval; it has a posterior probability of 0.48 of being in that interval.
When a protocol may select the highest admissible dose
Some protocols specify a different selection rule: among doses satisfying the overdose constraint, choose the highest dose. Such a rule can be appropriate when the protocol explicitly defines the decision that way—for example, when it seeks the greatest dose that meets a specified safety screen. In that case, the rule is $$ d^*=\max\{d_j:d_j\in A\}. $$ The posterior target probabilities may still be reported to describe how well each dose aligns with the target, but they do not determine the choice under a highest-admissible-dose rule.
The highest-admissible-dose rule is not equivalent to maximizing posterior target probability. It selects by dose ordering after applying the safety screen, whereas the target-probability rule selects by the posterior probability of a toxicity event. The two rules agree only when the highest admissible dose also has the largest \(\pi_j\) among admissible doses, or when the protocol explicitly prioritizes the highest eligible dose regardless of that ranking.
Worked Example: A Protocol Rule Takes Priority Over the Ranking
Consider a second hypothetical trial with target interval \([0.20,0.30]\), overdose cutoff \(\eta=0.20\), and the protocol rule “select the highest admissible dose.” The fitted-model summaries are:
| Dose (mg) | \(\pi_j\) | \(q_j\) |
|---|---|---|
| 5 | 0.10 | 0.01 |
| 10 | 0.38 | 0.06 |
| 20 | 0.34 | 0.18 |
| 30 | 0.16 | 0.36 |
The admissibility comparisons are \(0.01\leq0.20\), \(0.06\leq0.20\), and \(0.18\leq0.20\); hence \(A=\{5,10,20\}\) mg. Since \(0.36>0.20\), 30 mg is excluded. The highest member of \(A\) is 20 mg, so the stated protocol selects 20 mg. However, the largest target probability among admissible doses is \(0.38\), at 10 mg, compared with \(0.34\) at 20 mg. Therefore, a target-probability-maximizing rule would select 10 mg instead. The protocol must say which objective governs; these summaries alone do not make the rules interchangeable.
Cutoffs, ties, and posterior summaries
Admissibility depends on the precise cutoff convention. If the rule is \(q_j\leq\eta\), a dose with \(q_j=\eta\) is admissible; under a strict rule \(q_j<\eta\), it is not. Protocols should state the inequality, the overdose event, and any other eligibility conditions. Rounding posterior probabilities before applying a cutoff can also change classification near the boundary, so comparisons should use unrounded values where possible.
Posterior target probabilities are usually obtained from posterior draws. For each draw, calculate the dose-specific toxicity probability and record whether it lies in the target interval. If \(M\) draws are used and \(M_j\) fall in the interval at dose \(d_j\), the Monte Carlo estimate is $$ \widehat{\pi}_j=\frac{M_j}{M}. $$ Similarly, if \(R_j\) draws exceed \(\phi_U\), estimate \(q_j\) by \(\widehat{q}_j=R_j/M\). These are numerical approximations to posterior probabilities, so values close to a selection threshold or a near-tie deserve attention to Monte Carlo precision.
Worked Example: Checking a Boundary and a Tie
Suppose a hypothetical protocol uses the inclusive rule \(q_j\leq0.20\). Posterior calculations give \(q=0.20\) at 15 mg and \(q=0.23\) at 25 mg. The 15 mg dose meets the overdose criterion exactly, because \(0.20\leq0.20\); 25 mg does not, because \(0.23>0.20\). If the protocol instead stated the strict rule \(q_j<0.20\), 15 mg would also fail, since \(0.20\not<0.20\).
Now suppose the target probabilities for the admissible doses 5, 10, and 15 mg are \(0.31\), \(0.44\), and \(0.44\). The maximum is \(0.44\), attained at both 10 and 15 mg. A target-probability rule alone therefore identifies a tie, not a unique dose. If the protocol prespecifies choosing the lower dose in a tie, it selects 10 mg; if it specifies the higher dose, it selects 15 mg. The tie-breaker is part of the decision rule, not a conclusion supplied by the equal probabilities.
Diagnostics, alternatives, and limitations
Before applying a selection rule, check that the posterior probabilities were computed for every candidate dose using the same model, data, target interval, and overdose definition. Confirm that the dose set and all protocol restrictions are applied consistently. Posterior draws from a monotone BLRM should respect the model’s dose-ordering assumption within each draw; a violation in the calculation may indicate a coding or indexing error. Also examine the size of differences between \(\pi_j\) values: small differences may not support a practically meaningful ranking, especially when Monte Carlo error is material.
The main alternatives encode different priorities. Maximizing \(\pi_j\) favors the admissible dose with the greatest posterior chance of target-interval membership. Selecting the highest admissible dose favors dose level subject to the safety screen. Selecting the lowest admissible dose is more conservative in dose level but need not maximize target probability. Comparing posterior mean toxicity probabilities with the target midpoint is another possible summary, but it is not the same as comparing posterior probabilities of interval membership: a mean can be near the midpoint even when substantial posterior mass lies outside the interval.
None of these rules removes uncertainty or compensates for a poorly specified model, an unsuitable target interval, sparse data, or candidate doses that poorly cover the relevant range. The target-probability rule also gives equal credit to every toxicity probability inside the interval and none to probabilities outside it; it does not distinguish between values just inside and far inside the interval. A protocol may use other clinical objectives, but those must be stated rather than inferred from the posterior summaries. If no dose satisfies the admissibility criteria, the within-set comparison cannot identify a dose; the protocol must define a separate action.
Check Your Understanding
Use the stated decision rule in each question; distinguish eligibility from preference.
- A protocol admits doses with \(q_j\leq0.15\). Three doses have \((\pi_j,q_j)\) values \((0.25,0.04)\), \((0.50,0.12)\), and \((0.62,0.18)\). Which dose is selected if the rule is to maximize target probability among admissible doses?
- Why can a dose with the greatest posterior probability of being in the target interval still be ineligible?
- With an inclusive overdose cutoff of 0.10, is a dose with \(q_j=0.10\) admissible? How would the answer change under a strict cutoff?
- Two admissible doses have equal target probabilities, each 0.41. What additional information is needed to select one under a target-probability rule?
Solutions. (1) The first two doses are admissible because \(0.04\leq0.15\) and \(0.12\leq0.15\); the third is not because \(0.18>0.15\). Compare the eligible target probabilities, \(0.25\) and \(0.50\), and select the second dose. (2) Its posterior overdose probability may exceed the protocol cutoff, so it fails the eligibility rule even if its target probability is largest. (3) Yes, because \(0.10\leq0.10\). Under a strict rule, \(0.10<0.10\) is false, so it is not admissible. (4) A prespecified tie-breaking rule, such as choosing the lower or higher dose, is needed for a unique selection.