From Cohort Counts to Dose-Specific Boundaries
In dose escalation, each cohort contributes information about a dose-response curve, but the observed DLT rate at a dose is not itself the probability that the dose is overly toxic. The relevant quantity for an overdose boundary is a posterior probability: among the toxicity curves still plausible after the observed cohort counts, how much posterior probability lies above the protocol’s toxicity limit at this particular dose?
The calculation is dose-specific. A curve can put a low dose below the limit and a higher dose above it; uncertainty about the curve determines how often each event occurs in the posterior. This tutorial works through that connection on a small grid and shows how cohort counts change the resulting boundary.
The two thresholds have different roles. The toxicity limit \(\phi\) defines which toxicity curves count as exceeding the acceptable level. The cutoff \(\delta\) controls how much posterior probability of exceeding that limit the decision rule will tolerate. For example, \(\phi=0.30\) and \(\delta=0.25\) mean that the event of interest is toxicity above 30%, and the rule allows a dose only when the posterior probability of that event is at most 25%.
How Cohort Counts Enter the BLRM
For dose levels \(d_1,\ldots,d_J\), let cohort \(j\) have \(n_j\) evaluable patients and \(y_j\) DLTs. A standard binomial-response assumption is that, conditional on the model parameters, outcomes within a cohort are independent Bernoulli observations with common DLT probability \(p(d_j)\). Cohorts are conditionally independent given the parameters. The likelihood, omitting binomial coefficients that do not depend on the parameters, is
Here \(\theta\) denotes the dose-response parameters. In a two-parameter Bayesian logistic regression model (BLRM), a common specification is
The reference quantities \(d_{\mathrm{ref}}\) and \(p_{\mathrm{ref}}\) set the model’s dose scale and reference toxicity; \(\alpha\) and \(\beta\) describe the curve’s level and dose trend. A prior distribution on the parameters, updated by the likelihood, gives a posterior distribution for the whole curve. The posterior overdose probability is then the posterior mass of the parameter values for which \(p(d;\theta)>\phi\).
This expression makes the boundary’s meaning explicit: it is a probability over uncertain model parameters, not the toxicity probability \(p(d;\theta)\) at a single parameter value. In practice the integral is commonly approximated using posterior draws. If \(M\) draws are available, the estimate is the fraction of draws whose toxicity at the target dose exceeds \(\phi\).
Worked Example: A Small Grid and Posterior Weights
Worked Example: A Small Grid and Posterior Weights
Suppose a hypothetical trial has dose levels 10, 20, and 40 mg. After one patient at 20 mg experiences a DLT, consider four representative BLRM curve configurations, A–D. Their prior weights are unnormalized; only their relative values matter. The likelihood for this one-DLT cohort is \(p(20)\). The table gives each configuration’s toxicity probabilities on the grid, its prior weight, and its likelihood.
| Configuration | \(p(10)\) | \(p(20)\) | \(p(40)\) | Prior weight | Likelihood \(p(20)\) |
|---|---|---|---|---|---|
| A | 0.18 | 0.516 | 0.70 | 0.0678 | 0.516 |
| B | 0.32 | 0.632 | 0.80 | 0.0215 | 0.632 |
| C | 0.10 | 0.2695 | 0.45 | 0.0935 | 0.2695 |
| D | 0.25 | 0.391 | 0.60 | 0.0541 | 0.391 |
Each configuration is increasing across doses, as required for this illustrative monotone grid. For a single observed DLT, the Bernoulli likelihood is \(p(20)^1\{1-p(20)\}^0=p(20)\). Multiplying prior weight by likelihood gives the unnormalized posterior weights:
Their sum is \(0.09492415\). Dividing each weight by this sum gives posterior weights A \(0.3686\), B \(0.1431\), C \(0.2655\), and D \(0.2228\), to four significant figures. The rounded weights sum to 1.0000.
Set the toxicity limit to \(\phi=0.30\). At 10 mg, only configuration B exceeds the limit, so the overdose probability is \(0.1431\). At 20 mg, A, B, and D exceed it, so the probability is \(0.3686+0.1431+0.2228=0.7345\). At 40 mg, all four configurations exceed it, giving probability 1.000. With \(\delta=0.25\), the 10 mg dose meets the overdose-probability rule, while 20 and 40 mg do not.
The count was only one DLT in one patient at 20 mg, yet the posterior overdose probability at that dose is about 73.45%. That result reflects both the prior information and uncertainty about the response curve; it is not the observed rate of 100% treated as a precise estimate.
Why the Boundary Is a Posterior Event
The event \(p(d)>\phi\) is evaluated separately for every posterior parameter value. In a simulation-based calculation, the direct estimate is
where \(\theta^{(1)},\ldots,\theta^{(M)}\) are posterior draws. If 7,346 of 10,000 draws exceed the limit at a dose, the estimated posterior overdose probability there is \(7,346/10,000=0.7346\). Monte Carlo error matters near the decision cutoff: a small number of draws can make a probability estimate unstable, so use enough draws and check that the estimate is stable across simulation batches.
The posterior distribution is obtained under assumptions. Cohort outcomes must be correctly classified as DLT or no DLT for the analysis; the binomial model treats patients within a cohort as having the same dose-specific risk and conditionally independent outcomes. The dose-response specification and prior must also be adequate for the dose range being evaluated. Strong disagreement between observed counts and the fitted dose-response pattern, or large changes under reasonable prior or model checks, signals that the boundary may be model-sensitive. The earlier material on updating the BLRM after each cohort covers the sequential timing of these updates; here the emphasis is the dose-by-dose posterior event calculation.
Worked Example: An Exact Single-Dose Calculation
Worked Example: An Exact Single-Dose Calculation
For a transparent calculation, consider a separate hypothetical single-dose analysis. Assume a common DLT probability \(p\), independent Bernoulli outcomes, and a uniform prior \(p\sim\operatorname{Beta}(1,1)\). If 2 of 5 patients have DLTs, conjugate updating gives \(p\mid y\sim\operatorname{Beta}(3,4)\). Find the posterior probability that \(p\) exceeds \(\phi=0.30\).
For positive integer beta parameters, the survival probability can be written as a binomial sum. Here \(a=3\), \(b=4\), and \(a+b-1=6\), so
Thus the posterior probability of toxicity above 30% is 0.7443. Under a cutoff \(\delta=0.25\), this dose would not meet the rule. This conjugate model is not a replacement for the BLRM’s dose-response structure; it is an exact single-dose calculation that illustrates how counts and prior information produce a probability for an exceedance event.
Worked Example: Updating After an Additional Outcome
Worked Example: Updating After an Additional Outcome
Return to configurations A–D and the one-DLT observation at 20 mg in the grid example. Suppose a second patient at 20 mg has no DLT. The combined data are one DLT among two patients, so the likelihood for a configuration is \(p(20)\{1-p(20)\}\). The new unnormalized posterior weights are:
The total is \(0.0532225\) (using the unrounded terms, \(0.0532226\)). At 20 mg, configurations A, B, and D still exceed 0.30. Their weights sum to \(0.0348153\), so the updated overdose probability is
The additional no-DLT outcome lowers the posterior overdose probability from 0.7345 to about 0.6541, but does not bring it below the 0.25 cutoff. The outcome updates every curve’s weight according to its likelihood; it does not simply replace the model-based calculation with the observed proportion \(1/2\).
Derivation, Diagnostics, and Comparisons
The discrete-weight calculation above is a direct version of the continuous posterior integral. If \(\theta_k\) are candidate parameter values with prior weights \(w_k\), Bayes’ rule assigns posterior weight proportional to prior weight times likelihood:
At a target dose, sum these normalized weights only over configurations satisfying \(p(d;\theta_k)>\phi\). This gives the posterior event probability because the indicator is 1 for those configurations and 0 for the rest. For a continuous posterior, the corresponding sum becomes the integral shown earlier; posterior simulation estimates the same quantity by counting draws.
A practical diagnostic is to tabulate the posterior overdose probability at every dose, alongside the observed cohort counts and the model-predicted dose-response behavior. Check that each probability is computed against the same \(\phi\), that a posterior draw is evaluated at the correct dose, and that the boundary comparison uses the intended \(\delta\). If the modeled curve is constrained to be monotone, dose-specific overdose probabilities should also be nondecreasing across ordered doses: for each draw, exceeding \(\phi\) at a lower dose implies exceeding it at a higher dose. A violation in computed estimates can indicate coding, dose-ordering, or simulation error. This monotonicity check need not hold for a model that permits nonmonotone curves.
Observed DLT proportions are useful descriptive summaries but do not account for parameter uncertainty or borrowing of information across doses in a dose-response model. A posterior interval for \(p(d)\) describes uncertainty about the toxicity probability; the overdose probability instead reports the posterior mass on one side of the specified limit. These quantities answer different questions, so an interval endpoint should not be substituted for the posterior exceedance calculation. The posterior mean is also a summary of the distribution, not the exceedance event itself.
The boundary depends on the chosen model, prior, toxicity limit, and posterior computation. Sparse counts, extreme outcomes, or extrapolation beyond well-supported doses can leave substantial uncertainty. A probability near \(\delta\) deserves particular care: Monte Carlo variability or modest model changes may alter admissibility. The rule is a decision aid under stated assumptions, not proof that a dose is safe or a guarantee about future patients.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- In the definition of an overdose boundary, what do \(\phi\) and \(\delta\) each control?
- At a dose, 3,200 of 8,000 posterior draws have toxicity probability above \(\phi\). What is the estimated posterior overdose probability?
- In the grid example, which configurations contribute to the overdose probability at 20 mg when \(\phi=0.30\)?
- For one DLT in one patient, what is the parameter-dependent part of the binomial likelihood?
- Why is an observed DLT proportion not, by itself, the posterior overdose probability?
Solutions. (1) \(\phi\) defines the toxicity event \(p(d)>\phi\); \(\delta\) is the largest posterior probability of that event allowed by the boundary rule. (2) \(3,200/8,000=0.4000\), or 40.00%. (3) A, B, and D exceed 0.30 at 20 mg, so their normalized posterior weights are summed. (4) It is \(p(d)\), because \(p(d)^1\{1-p(d)\}^0=p(d)\). (5) The observed proportion is a data summary; the posterior overdose probability is the model’s posterior mass on curves exceeding the limit, incorporating prior information and uncertainty.