From Overdose Probabilities to Admissibility
A posterior overdose probability answers a dose-specific question: given the data and model, how much posterior probability is assigned to the dose having a toxicity probability above the overdose boundary? Constructing the admissible dose set applies a prespecified rule to those probabilities. The result is a classification of candidate doses for consideration, not a ranking of doses and not, by itself, a recommendation of which dose to give.
Let \(d_1<d_2<\cdots<d_K\) be the candidate doses, let \(D\) denote the observed dose-limiting toxicity data, and let \(p(d_j;\theta)\) be the modelled probability of a dose-limiting toxicity at dose \(d_j\), given model parameters \(\theta\). Write \(\phi\) for the upper toxicity-probability boundary that defines overdose, and \(\gamma\) for the prespecified posterior probability cutoff. Both boundaries must be specified for the rule to be applied.
The probability in this rule is conditional on the observed data and the specified model and prior. It concerns uncertainty about the dose-specific toxicity probability, not the probability that an individual patient will experience a toxicity. The cutoff \(\gamma\) is a decision-rule threshold; it is not a confidence level, a target toxicity rate, or a probability that the model is correct.
Applying the Rule to Posterior Draws
For the Bayesian logistic regression model (BLRM), posterior draws of the parameters induce posterior draws of the toxicity probability at each dose. For posterior draw \(m\), calculate \(p(d_j;\theta^{(m)})\) and record whether it exceeds \(\phi\). The fraction of draws exceeding the boundary estimates \(q_j\).
Here \(M\) is the number of posterior draws used, and \(I\{\cdot\}\) is 1 when its condition holds and 0 otherwise. The posterior probability itself is the quantity of interest; \(\widehat q_j\) is a Monte Carlo estimate of it. In practical computation, classifications based on estimates close to \(\gamma\) deserve more scrutiny because finite simulation can move an estimate slightly above or below the cutoff.
The rule should be applied to all candidate doses using the same \(\phi\), \(\gamma\), posterior model, and dataset. These choices should be prespecified as part of the design rather than adjusted after seeing which doses pass. Otherwise the resulting classification no longer has the interpretation of applying the stated decision rule.
Worked Example: Classifying a Dose Grid
Suppose a hypothetical BLRM analysis uses overdose boundary \(\phi=0.30\) and admissibility cutoff \(\gamma=0.25\). For each of five candidate doses, 1000 posterior draws are evaluated. The numbers of draws with toxicity probability above 0.30 are 42, 131, 219, 278, and 463, respectively.
For the first dose, \(\widehat q_1=42/1000=0.042\). Since \(0.042\leq0.25\), it is admissible. For the second, \(\widehat q_2=131/1000=0.131\), also at or below the cutoff. For the third, \(\widehat q_3=219/1000=0.219\), so it too is admissible.
For the fourth dose, \(\widehat q_4=278/1000=0.278\), which is greater than 0.25; it is inadmissible. The fifth dose has \(\widehat q_5=463/1000=0.463\), also greater than the cutoff. Thus the estimated admissible set on this grid is the first three candidate doses. This conclusion applies to the classification rule only; it does not identify which of those three doses should be selected.
Why a Monotone BLRM Gives an Ordered Set
Under the positive-slope BLRM, the modelled toxicity probability is nondecreasing with dose for every fixed parameter value. This feature gives a useful mathematical check on the classification. If \(d_j<d_k\), then \(p(d_j;\theta)\leq p(d_k;\theta)\) for every \(\theta\). Therefore, whenever \(p(d_j;\theta)>\phi\), it must also be true that \(p(d_k;\theta)>\phi\). The overdose event at the lower dose is contained in the overdose event at the higher dose.
The probability inequality follows because the probability of a subset cannot exceed the probability of the containing event, under the same posterior distribution. Consequently, if the model is monotone and all doses are evaluated consistently, the theoretical overdose probabilities are nondecreasing across the ordered grid. Applying a common cutoff then yields an initial segment of admissible doses: once a dose is inadmissible, higher candidate doses will also be inadmissible.
This is a structural implication of the model and rule, not a claim that observed DLT proportions must increase at every dose. Random cohort outcomes can look nonmonotone even when the fitted dose-toxicity model is constrained to be monotone. The classification uses posterior probabilities under the model, not a direct ordering of observed proportions.
Specify the overdose boundary \(\phi\), the cutoff \(\gamma\), the candidate-dose grid, and the model and posterior to be used.
At each candidate dose, estimate \(P\{p(d_j;\theta)>\phi\mid D\}\) from posterior draws or an equivalent posterior calculation.
Mark a dose admissible when its posterior overdose probability is at most \(\gamma\); otherwise mark it inadmissible.
Confirm the calculation and, under the monotone BLRM, check that the probabilities and classifications are ordered across doses.
Worked Example: Handling a Probability at the Cutoff
In a hypothetical analysis, let \(\gamma=0.25\), and suppose 200 posterior draws are used at each of two successively higher candidate doses. At the lower dose, 49 draws exceed the overdose boundary; at the higher dose, 50 do.
The estimated probabilities are \(49/200=0.245\) and \(50/200=0.250\). With the rule defined as \(q_j\leq\gamma\), the lower dose is admissible because \(0.245\leq0.25\). The higher dose is also admissible because \(0.250=0.25\). Equality belongs to the admissible category under this definition. A rule using a strict inequality would classify the boundary case differently, which is why the inequality convention must be stated rather than left implicit.
These are estimated probabilities based on finite posterior simulation. If the underlying posterior probabilities were near the cutoff, a larger number of draws or a more precise calculation would help determine whether the apparent equality is numerical or substantive. Rounding a probability before comparing it with \(\gamma\) can also create a false tie, so comparisons should use adequate numerical precision.
Diagnostics, Alternatives, and Limitations
Several checks help distinguish a genuine classification from an implementation problem. First, verify that the overdose boundary is the intended upper toxicity boundary and that the same boundary was used at every dose. Second, verify that the posterior draws come from the intended model and posterior, and that the toxicity probability is evaluated on the correct dose scale. Third, inspect the estimated probabilities in dose order. A clear decrease across doses is inconsistent with the monotonicity implication if the same positive-slope BLRM posterior and calculation were used. It can indicate a coding error, mismatched posterior draws or dose labels, or that a different model or rule was applied.
Monte Carlo uncertainty matters most near the cutoff. If the draw indicators are independent posterior simulations, the approximate Monte Carlo standard error of \(\widehat q_j\) is \(\sqrt{\widehat q_j(1-\widehat q_j)/M}\). For example, with \(\widehat q_j=0.25\) and \(M=1000\), this is approximately \(\sqrt{0.25(0.75)/1000}=0.0137\). This describes simulation error in estimating the posterior probability; it is not uncertainty in the patient-level toxicity rate. A classification close to the cutoff should be checked using more effective posterior draws and appropriate convergence diagnostics. Strong autocorrelation reduces the effective information in a chain, so the raw number of stored draws can overstate precision.
Admissibility is not interchangeable with other summaries. A rule based on posterior mean toxicity compares \(E\{p(d_j;\theta)\mid D\}\) with a toxicity threshold; it does not measure the posterior probability of exceeding that threshold. A rule based on the posterior probability of lying in a target interval asks about a different event. The observed DLT fraction is a data summary and does not account for posterior uncertainty in the dose-toxicity curve. These quantities can lead to different classifications and should not be substituted for one another without explicitly changing the decision rule.
| Quantity or rule | Question it addresses | Why it is not the admissibility rule |
|---|---|---|
| Posterior overdose probability | How much posterior probability is above \(\phi\)? | This is the quantity compared with \(\gamma\). |
| Posterior mean toxicity | What is the posterior average of the dose-specific toxicity probability? | An average does not specify the probability of exceeding \(\phi\). |
| Posterior probability of target-interval toxicity | How much posterior probability lies within a specified interval? | It concerns a different event from exceeding the overdose boundary. |
| Observed DLT proportion | What fraction of observed patients at a dose had DLTs? | It is not a posterior probability about the dose-specific toxicity parameter. |
The resulting set is conditional on the chosen candidate grid, model, prior, data, overdose boundary, and cutoff. It does not establish that every admissible dose is safe in an absolute sense, and it does not account for considerations that are not part of the stated overdose rule. If the set is empty, the rule has classified every candidate dose as inadmissible; that result should be reported as such rather than silently changing the cutoff or treating the least risky dose as automatically admissible.
Worked Example: An Empty Admissible Set
Suppose a hypothetical analysis has three ordered candidate doses and a cutoff \(\gamma=0.25\). Posterior simulation gives overdose probabilities \(0.31\), \(0.44\), and \(0.62\). Each value exceeds 0.25, so none of the three doses is admissible and \(A=\varnothing\).
The result is internally consistent with a monotone BLRM because the probabilities increase with dose. It does not mean that the lowest dose has an overdose probability of zero, nor does the rule relabel it admissible simply because it is the lowest available option. It means only that no candidate dose passes this particular prespecified criterion, conditional on the analysis. What action follows is a separate design decision.
Common Mistakes
- Reversing the inequality. A dose passes when its posterior probability of exceeding \(\phi\) is no greater than \(\gamma\), not when it is greater. A larger overdose probability is more evidence of concern under this rule.
- Confusing the two thresholds. The overdose boundary \(\phi\) is a toxicity-probability threshold inside the model event; \(\gamma\) is the tolerated posterior probability of exceeding that boundary.
- Treating admissibility as selection. Passing the rule means the dose is in the admissible set, not that it is the dose to administer. Selection among admissible doses requires an additional rule.
- Replacing the posterior event with a mean or observed rate. The comparison is based on \(P\{p(d_j;\theta)>\phi\mid D\}\), not on the posterior mean toxicity or the observed DLT proportion.
- Ignoring numerical uncertainty near the cutoff. An estimated probability just on one side of \(\gamma\) may be sensitive to simulation error; check the Monte Carlo precision before treating a borderline result as exact.
Check Your Understanding
For each question, use the stated rule \(q_j\leq\gamma\) for admissibility.
- If \(\phi=0.30\), \(\gamma=0.20\), and a dose has posterior overdose probability \(0.18\), is it admissible? What does the probability refer to?
- With 500 posterior draws, 125 exceed the overdose boundary at a candidate dose. Calculate the estimated overdose probability and classify the dose when \(\gamma=0.25\).
- Under a monotone BLRM, a lower dose has overdose probability 0.12 and a higher dose has probability 0.09. What should be checked?
- Why can a dose with posterior mean toxicity below \(\phi\) still be inadmissible under the overdose-probability rule?
Solutions. (1) Yes: \(0.18\leq0.20\). The value is the posterior probability that the dose-specific toxicity probability exceeds 0.30, conditional on the data and model; it is not an individual patient’s toxicity probability. (2) \(\widehat q=125/500=0.25\). Since equality passes under the stated rule, the dose is admissible. (3) The ordering conflicts with the monotone-model implication \(q_{\text{lower}}\leq q_{\text{higher}}\). Check dose labels, posterior calculations, model specification, and implementation before relying on the classification. (4) The posterior mean and the posterior probability of exceeding \(\phi\) summarize different aspects of the posterior distribution. A distribution can have a mean below \(\phi\) while still assigning more than \(\gamma\) probability to values above \(\phi\); the dose is inadmissible whenever that exceedance probability is greater than \(\gamma\).