End-to-End BLRM Dose-Escalation Workshop

From Protocol Rules to a Complete Study

A dose-escalation design is not a sequence of dose increases triggered mechanically by the observed DLT rate. Each cohort contributes information about a dose-toxicity curve; the model combines that information with a prior, and a prespecified rule turns the posterior into a decision. This workshop follows one hypothetical study from its starting dose to its final recommendation. The calculations are illustrative: posterior summaries are assumed to come from numerical integration under the model stated below, and are rounded.

The study uses a target toxicity interval of 0.20 to 0.30 and an overdose cutoff of 0.25. The cutoff applies to a posterior probability, not to the posterior mean toxicity rate. The distinction is central: a dose can have a mean below the toxicity boundary yet still have too much posterior probability of exceeding it.

Model, Prior, and Decision Rules

Definition: For dose $d$, the two-parameter Bayesian logistic regression model (BLRM) is $$ \operatorname{logit}\{p(d)\}=\alpha+\beta\log(d/4), $$ where $p(d)$ is the probability of a DLT during the protocol-defined observation window at dose $d$, $\alpha$ is the log-odds of a DLT at the reference dose 4 mg, and $\beta$ is the dose-response slope. In this example, $\alpha\sim N(\operatorname{logit}(0.20),0.8^2)$, and $\beta$ has a $N(1.1,0.4^2)$ distribution truncated to $\beta>0$. The parameters are independent before observing trial data.

The positive-slope restriction encodes the assumption that DLT probability increases with dose over the modeled range. For a cohort of $n_j$ patients treated at dose $d_j$, let $y_j$ be the number with a DLT. Conditional on the model parameters, patient outcomes are assumed independent Bernoulli observations with probability $p(d_j)$; cohorts contribute the likelihood

L(\alpha,\beta)=\prod_j {n_j\choose y_j}p(d_j)^{y_j}\{1-p(d_j)\}^{n_j-y_j}.

This likelihood assumes complete DLT ascertainment for the observation window, a consistent DLT definition, and no unmodeled change in risk across patients at the same dose. Patients with pending follow-up should not simply be counted as DLT-free; the earlier material on incomplete DLT follow-up addresses that issue. The posterior is proportional to this likelihood times the specified prior.

For this study, a dose is admissible when its posterior probability of exceeding the upper target boundary is no greater than 0.25. Among admissible doses, the final recommendation will be the dose with the greatest posterior probability that its DLT rate lies in the target interval. The starting dose is selected separately, using nonclinical and clinical safety information and the protocol; it is not selected by pretending that no-data posterior estimates establish clinical safety.

Why the Overdose Probability Is a Model-Based Quantity

Write $\theta=\operatorname{logit}(0.30)$. Since the inverse-logit function is increasing, the overdose event at dose $d$ can be written as a condition on the regression parameters:

p(d)>0.30 \quad\Longleftrightarrow\quad \alpha+\beta\log(d/4)>\theta.

Thus the posterior overdose probability is the posterior mass on one side of a boundary in the $(\alpha,\beta)$ parameter space. It is not obtained by comparing the posterior mean $E\{p(d)\mid\text{data}\}$ with 0.30. The calculation integrates over uncertainty in both parameters.

For example, the posterior probability at 6 mg is

P\{p(6)>0.30\mid\text{data}\} = \int I\{\alpha+\beta\log(6/4)>\operatorname{logit}(0.30)\} \,\pi(\alpha,\beta\mid\text{data})\,d\alpha\,d\beta,

where $I\{\cdot\}$ equals 1 when its condition is true and 0 otherwise, and $\pi(\alpha,\beta\mid\text{data})$ is the posterior density. Numerical integration or posterior draws can estimate this probability. If 10,000 posterior draws are used and 5,400 satisfy the overdose condition, the estimated probability is $5{,}400/10{,}000=0.540$; the dose fails a 0.25 cutoff.

Worked Example: Starting at 2 mg

Suppose preclinical evidence and the protocol support beginning at 2 mg, the lowest dose in the planned grid of 2, 4, 6, and 8 mg. The study enrolls a cohort of three at 2 mg, all with completed DLT follow-up and no observed DLTs.

At this dose, the cohort likelihood contribution is

{3\choose 0}p(2)^0\{1-p(2)\}^3=\{1-p(2)\}^3.

For illustration, assume the specified numerical integration gives posterior overdose probabilities of 0.020 at 2 mg and 0.100 at 4 mg. Both are below the 0.25 cutoff. The protocol permits escalation by one dose level, so the next cohort is assigned to 4 mg. The calculation supports that decision under the model and rule; it does not mean the probability that the 4 mg dose is “safe” is 0.900. The calculated quantity is specifically the posterior probability that its DLT rate exceeds 0.30.

Following the Cohorts

The table records the hypothetical study path. Posterior overdose probabilities are rounded outputs from the model and prior above, conditional on all listed cohorts up to that point. “Posterior mean $p(d)$” summarizes the estimated DLT rate; the overdose decision uses the separate probability in the final column. Each follow-up is complete before the next decision.

Decision point	Cumulative cohort data	Dose considered	Posterior mean $p(d)$	$P\{p(d)>0.30\mid\text{data}\}$	Decision
After 2 mg	2 mg: 0/3	4 mg	0.15	0.100	Escalate one level
After 4 mg	2 mg: 0/3; 4 mg: 0/3	6 mg	0.16	0.100	Escalate one level
After 6 mg	2 mg: 0/3; 4 mg: 0/3; 6 mg: 2/3	6 mg	0.26	0.420	Do not continue at 6 mg; de-escalate
After return to 4 mg	4 mg total: 1/6; 6 mg: 2/3; 2 mg: 0/3	4 mg	0.23	0.200	4 mg remains admissible
Final review	Same completed data	2 mg	0.09	0.010	Admissible, but less target evidence

After the first cohort at 4 mg, the model gives a posterior overdose probability of 0.100 at 6 mg. Since this is below 0.25 and 6 mg is the next planned level, escalation is allowed. The next cohort has two DLTs among three patients at 6 mg. Its likelihood contribution is ${3\choose2}p(6)^2\{1-p(6)\}$. This substantial evidence of toxicity raises the posterior overdose probability at 6 mg to 0.420, above the cutoff. The protocol therefore does not continue at that dose and de-escalates to an admissible lower dose.

At the final review, the dose-specific posterior probabilities of being inside the target interval are 0.100 at 2 mg and 0.490 at 4 mg. The posterior overdose probability at 4 mg is 0.200, so 4 mg passes the stated admissibility rule. The posterior overdose probabilities at 6 mg and 8 mg are 0.540 and 0.700, respectively, so neither is admissible. These estimates support recommending 4 mg under the prespecified selection rule. They do not establish that every patient will tolerate 4 mg or that 4 mg is optimal for efficacy; the endpoint being modeled here is DLT probability.

Worked Example: Updating After 2 of 3 DLTs

At 6 mg, two of three patients experience a DLT. The binomial coefficient is ${3\choose2}=3$, so the likelihood for this cohort is $3p(6)^2\{1-p(6)\}$. Combining it with the likelihood contributions from the previous cohorts gives the posterior used for the next decision.

Suppose the numerical integration estimates $P\{p(6)>0.30\mid\text{data}\}=0.420$. The cutoff is 0.25, and $0.420>0.25$, so 6 mg is not admissible. A posterior mean of 0.26 at 6 mg is not the reason for this decision: the rule compares a posterior probability with the cutoff. The protocol next considers de-escalation to 4 mg, whose posterior overdose probability is 0.140 at this decision point. Since $0.140\leq0.25$, 4 mg is admissible.

Final Recommendation and Its Limits

Check the data state.
Confirm the DLT window is complete for every cohort used in the update and that dose, follow-up, and DLT coding agree with the protocol.

Apply overdose control.
At each dose, compare $P\{p(d)>0.30\mid\text{data}\}$ with 0.25. Exclude doses above the cutoff from escalation or final selection as specified.

Use the final selection criterion.
Among admissible doses, compare posterior probabilities of lying in the target interval. Here 4 mg has probability 0.490, exceeding 0.100 at 2 mg.

Report the scope of the conclusion.
Recommend 4 mg under this model, endpoint, prior, target interval, and overdose rule. Do not present it as proof of efficacy or as a guarantee of individual safety.

This approach is more explicit than escalation based only on the observed proportion of patients with DLTs: the BLRM borrows information across doses through its dose-response structure and accounts for parameter uncertainty. A simple rule based on cohort counts can be easier to implement and less dependent on a parametric curve, but it does not estimate the same posterior quantities. Conversely, a model-based rule can be sensitive to the prior and to misspecification. A positive-slope logistic curve may be inadequate if toxicity is nonmonotone, if important patient heterogeneity is omitted, or if the dose grid requires extrapolation beyond the data. Diagnostics include checking posterior predictive fit at observed doses, inspecting posterior uncertainty and sensitivity to reasonable prior alternatives, and reviewing whether observed cohort patterns conflict with the assumed increasing curve. These checks inform interpretation; they do not authorize changing the protocol rule after seeing the data without justification.

Common Mistakes and Diagnostics

Using the posterior mean as the overdose rule. A mean below 0.30 does not imply that $P\{p(d)>0.30\mid\text{data}\}$ is below 0.25. Compute the posterior tail probability directly.
Treating an observed rate as the true toxicity probability. Two DLTs among three patients gives an observed rate of $2/3$, not a known value of $p(6)$. The likelihood updates uncertainty; it does not remove it.
Escalating before follow-up is complete. A patient without a recorded DLT but with an incomplete observation window is not necessarily a non-DLT. Follow the protocol’s pending-outcome method.
Trusting the curve without checking it. A strong nonmonotone pattern or posterior predictions that poorly describe observed cohorts can signal model limitations. Reconcile the data and model assumptions before relying on an extrapolated dose decision.
Calling the recommendation a proven maximum tolerated dose. The example selects a dose under a specified posterior target and overdose rule. That is not a guarantee of safety, an efficacy conclusion, or a universal dose recommendation.

Key takeaway: A defensible BLRM recommendation links four things explicitly: completed cohort data, a stated dose-toxicity model and prior, posterior overdose probabilities, and a prespecified rule for escalation and final selection. The posterior mean alone cannot replace the overdose control calculation.

Check Your Understanding

Answer each question using the model and hypothetical study above. Solutions follow the questions.

Why does 0 DLTs among 3 patients at a dose not imply that the DLT probability at that dose is zero?
At a dose, the posterior overdose probability is 0.270. Is it admissible under the study rule? Explain.
At final review, why is 4 mg recommended over 2 mg even though 2 mg has a lower posterior mean toxicity estimate?
What does the positive-slope restriction imply about the modeled dose-toxicity curve, and name one pattern that would prompt a model-fit review.

Solutions. (1) The cohort likelihood is $\{1-p(d)\}^3$, which is positive for many values of $p(d)$; observing no DLTs updates uncertainty but does not set the unknown probability to zero. (2) No. The rule requires the posterior overdose probability to be no greater than 0.25, and $0.270>0.25$. (3) Both doses are admissible, but the prespecified final criterion selects the admissible dose with the greatest posterior probability of being in the 0.20 to 0.30 target interval; that probability is 0.490 at 4 mg versus 0.100 at 2 mg. (4) It implies that modeled DLT probability increases with dose. A marked nonmonotone pattern in observed cohort outcomes would prompt checking the data and assessing whether the model adequately describes the trial.