Protocolizing the BLRM Escalation Algorithm

From a Statistical Model to an Executable Rule

A Bayesian dose-toxicity model does not, by itself, specify what a trial team should do next. A protocol must also define when the data are analyzed, which doses may be assigned, how posterior evidence is converted into a decision, and what action follows if no dose is sufficiently safe. Leaving any of these details implicit can produce inconsistent decisions even when everyone agrees on the model.

This tutorial sets out one explicit protocol structure for the Bayesian logistic regression model (BLRM). The numerical choices below are illustrative design choices, not universal standards. A real protocol needs to justify its target toxicity rate, overdose cutoff, prior distribution, cohort size, dose range, and stopping rules before enrollment.

Model, Data, and Overdose Admissibility

Definition: Let $d_i$ be the $i$th positive dose level and $p(d_i)$ the probability that a patient assigned that dose experiences a dose-limiting toxicity (DLT) within the protocol-defined assessment window. A BLRM specifies $$ \operatorname{logit}\{p(d)\}=\log\alpha+\beta\log(d/d_{\mathrm{ref}}), $$ where $d_{\mathrm{ref}}$ is a positive reference dose, $\alpha>0$, and $\beta>0$. The positivity constraint on $\beta$ makes the modeled DLT probability increase with dose. For dose $d_i$, let $y_i$ of $n_i$ patients have an observed DLT. Conditional on the model parameters and assigned doses, the usual likelihood assumes independent Bernoulli outcomes: $$ L(\alpha,\beta)=\prod_i p(d_i)^{y_i}\{1-p(d_i)\}^{n_i-y_i}. $$

The prior distribution for the model parameters must be specified in the protocol. For example, a design may use a prespecified bivariate normal prior for $(\log\alpha,\beta)$, restricted to $\beta>0$. After observing the DLT data, the posterior distribution is proportional to the likelihood multiplied by that prior. The posterior probability of overdose at dose $d_i$ is the posterior probability that its DLT probability exceeds the chosen target boundary $\phi_T$.

Definition: Given an overdose cutoff $\gamma$, dose $d_i$ is admissible when $$ P\{p(d_i)>\phi_T\mid \text{DLT data}\}\leq\gamma. $$ The set of admissible doses is the set of dose levels satisfying this inequality. Equality is admissible under this convention; the protocol must not leave the boundary case unspecified.

This constraint is about posterior probability of exceeding the target boundary. It is not a rule based on posterior mean toxicity. The earlier material on posterior mean toxicity and overdose rules distinguishes these quantities. The protocol should state the target boundary and cutoff numerically and in context; for instance, a hypothetical design might set $\phi_T=0.30$ and $\gamma=0.25$. Neither value should be treated as appropriate for every disease or treatment.

What the Protocol Must Fix in Advance

A usable escalation algorithm is a sequence of decisions, not just a formula. At minimum, prespecify the following items.

Outcome definition and timing: the DLT criteria, assessment window, and the point at which a patient’s outcome is considered available for a dose update.
Cohort rules: the number of patients assigned together and whether a new cohort can start before the preceding cohort’s DLT outcomes are available. The example protocol below uses cohorts of three and waits for the cohort’s required outcomes before updating.
Dose range and starting dose: the ordered dose levels, the starting level, and any dose that cannot be assigned because of formulation, manufacturing, or clinical constraints.
Model and analysis: the BLRM parameterization, priors, target $\phi_T$, cutoff $\gamma$, posterior computation method, and numerical precision used for decisions.
Permitted moves and stopping actions: whether upward dose skipping is permitted, what happens if no dose is admissible, and when enrollment stops for reasons such as reaching a maximum sample size.

For a concrete rule, suppose a hypothetical protocol has ordered dose levels $d_1<d_2<\cdots<d_K$, cohorts of three, a prespecified starting dose $d_1$, and no overlapping cohorts. After each evaluable cohort, the model is updated using all available DLT outcomes. Upward escalation is limited to one dose level at a time. A downward move may select any lower admissible dose; this permits a safety response if the current dose is no longer admissible. These are protocol choices, not mathematical necessities.

Wait for the scheduled analysis.
Use the DLT outcomes available under the prespecified assessment rule. Do not silently treat pending outcomes as non-DLTs.

Update and check the BLRM.
Fit the prespecified model and calculate the posterior overdose probability at every assignable dose level.

Form the admissible set.
Include each dose whose unrounded posterior overdose probability is at most $\gamma$.

Choose the next action.
If the admissible set is empty, stop dose escalation and follow the protocol’s no-admissible-dose action. Otherwise, choose the highest admissible dose no more than one level above the current dose.

Apply terminal rules.
Stop if the maximum enrollment or another prespecified stopping condition has been reached; otherwise assign the next cohort at the selected dose.

In Step 4, “no more than one level above” limits upward movement; it does not require the selected dose to be the current dose or the immediately adjacent dose. If the current level is inadmissible, the highest admissible level below it can be selected. If the current dose is the highest level and remains admissible, the rule can select that level again. The protocol should also state whether the trial may re-open a previously closed dose; the example rule treats the admissible set as controlling at each update but does not override any separate clinical or operational closure.

Worked Decisions

Worked Example: Escalating One Level After a Cohort

A hypothetical trial has dose levels 1, 2, 4, and 8 mg, a cohort size of three, target $\phi_T=0.30$, and cutoff $\gamma=0.25$. The current cohort was treated at 2 mg, and its DLT outcomes are complete. Assume the prespecified BLRM and posterior computation yield overdose probabilities of 0.03, 0.08, 0.21, and 0.46 at 1, 2, 4, and 8 mg, respectively. These are illustrative posterior outputs, not values claimed to follow from a particular prior or data set.

For 4 mg, the admissibility comparison is $$ P\{p(4\text{ mg})>0.30\mid \text{data}\}=0.21\leq0.25. $$ Thus 4 mg is admissible. The 8 mg dose is not, since $0.46>0.25$. The admissible set is $\{1,2,4\}$ mg. The one-level upward limit from the current 2 mg dose permits a move at most to 4 mg, so the highest permissible admissible dose is 4 mg. The next cohort of three is assigned to 4 mg, provided no terminal stopping condition has been met.

Worked Example: De-Escalating After the Current Dose Becomes Inadmissible

In another hypothetical trial using the same four dose levels and $\gamma=0.25$, the current cohort is at 8 mg and has completed follow-up. Suppose the posterior overdose probabilities at 1, 2, 4, and 8 mg are 0.02, 0.09, 0.24, and 0.63. The dose-specific comparisons are $0.02\leq0.25$, $0.09\leq0.25$, $0.24\leq0.25$, and $0.63>0.25$. The admissible set is therefore $\{1,2,4\}$ mg.

Because the current 8 mg dose is not admissible, the protocol does not assign the next cohort there. Under the specified downward-move rule, the highest admissible dose below 8 mg is 4 mg. The next cohort is assigned to 4 mg. Notice that 4 mg is admissible at the boundary margin: its posterior overdose probability is 0.24, which is below the cutoff by $0.25-0.24=0.01$. A protocol using a strict “less than” inequality would produce a different boundary convention and must state that choice explicitly.

Worked Example: No Admissible Dose and a Boundary Case

At an interim analysis, suppose the posterior overdose probabilities for 1, 2, 4, and 8 mg are 0.31, 0.48, 0.77, and 0.94, with $\gamma=0.25$. The smallest probability is still above the cutoff: $$ \min_i P\{p(d_i)>\phi_T\mid\text{data}\}=0.31>0.25. $$ No dose is admissible. Under the stated rule, the trial stops dose escalation and does not enroll another cohort at any dose. The protocol should specify whether this means permanent trial termination, a pause for safety review, or another action; continuing at the lowest dose despite its failing the constraint would contradict this algorithm.

For contrast, if the probabilities were 0.02, 0.25, 0.44, and 0.81, the 2 mg dose would be admissible because the rule uses $\leq0.25$. If the current dose were 1 mg, the one-level upward limit would allow selection of 2 mg. Decisions use unrounded probabilities: a displayed value of 0.25 that was rounded from 0.254 is not admissible. If numerical integration error is large enough that the probability could lie on either side of the cutoff, the analysis should be refined before assigning the next cohort rather than resolving the uncertainty by casual rounding.

Why the Decision Rule Is Coherent

The admissibility calculation gives a dose-specific safety screen. The move rule then applies operational restrictions to the screened set. These are separate operations: a dose can satisfy the overdose constraint but remain unavailable for the next cohort because it is more than one level above the current dose. This distinction is useful in implementation and auditing.

The logic can be written formally. Let $A$ be the set of admissible dose indices, and let $j$ be the current dose index. Under the example policy, the next-dose index is $$ j_{\mathrm{next}}=\max\{i:i\in A,\ i\leq j+1\}, $$ provided the set inside the maximum is nonempty. If $A$ is empty, the protocol stops escalation. If $A$ is nonempty but no admissible dose satisfies $i\leq j+1$, the protocol must define a fallback. In the policy described here, this cannot occur when downward moves to any lower admissible level are allowed and the current index is at least 1: an admissible level below $j$ satisfies the restriction. The formula makes the one-level upward cap and the role of admissibility explicit.

The maximum is unique because the protocol uses strictly ordered dose levels. If two software outputs appear tied only because displayed posterior probabilities have been rounded, the rule still selects by dose index using unrounded probabilities to determine admissibility. If two formulations share a nominal dose level, the protocol needs a separate operational tie-breaker, such as a prespecified formulation preference; the statistical dose ranking cannot resolve that situation.

Diagnostics, Alternatives, and Limitations

A protocolized rule is only as reliable as its inputs and implementation. Before acting on a recommendation, verify that the dose assignments and DLT counts are correct, the assessment window has been applied consistently, and the posterior calculation used the intended prior, target, and cutoff. Check computational diagnostics, including convergence or effective sample size when simulation is used, and verify that posterior overdose probabilities are stable enough for a decision near $\gamma$. Sensitivity to reasonable prior alternatives matters particularly when few patients have been treated or observations are sparse.

The monotone BLRM also imposes a dose-toxicity shape assumption. If posterior summaries or observed outcomes look unexpectedly nonmonotone, do not alter the dose rule ad hoc; check data coding, model fit, prior influence, and whether the model is adequate for the clinical setting. The earlier material on nonmonotone observed DLT rates and all-zero or all-DLT data addresses such patterns. Boundary doses deserve attention as well: posterior estimates near the edge of the modeled range may be sensitive to extrapolation and should not be treated as equally well supported merely because the model returns a probability.

This approach differs from a fixed rule such as the 3+3 design, which bases moves on prespecified observed-count patterns rather than a posterior overdose probability from a dose-toxicity model. It also differs from other model-based dose-finding approaches, which may use a different model or a different allocation objective. A BLRM overdose constraint is transparent about one safety condition, but it does not by itself identify a dose with desirable efficacy, guarantee acceptable risk, or replace clinical review. It is a decision rule for dose assignment, not a complete benefit-risk assessment.

Finally, the exact cohort size, allowable moves, cutoff, and stopping actions are design choices. Their consequences should be examined before the trial through simulation under plausible dose-toxicity scenarios, including scenarios with unusually steep toxicity increases, weak dose-response information, or no dose satisfying the overdose rule. A protocol that is reproducible is not automatically a protocol that performs well; its operating characteristics still require evaluation.

Key takeaway: An executable BLRM algorithm must connect a prespecified posterior overdose calculation to an explicit admissibility boundary, dose-move rule, tie convention, and stopping action. If any of these elements is left to interpretation at the time of a decision, the protocol is incomplete.

Check Your Understanding

Answer each question using the protocol definitions above. Brief solutions are included after the questions.

With $\gamma=0.20$, is a dose admissible if its unrounded posterior overdose probability is 0.20?
The current dose is level 3. The admissible levels are 1, 2, and 4. Under the one-level upward rule, which dose is selected?
Why is a dose with posterior mean DLT probability below $\phi_T$ not automatically admissible?
All assignable doses have posterior overdose probabilities above $\gamma$. What does the example protocol do?

Solutions. (1) Yes: the specified inequality is $P\{p(d)>\phi_T\mid\text{data}\}\leq\gamma$, so equality qualifies. (2) Level 4 is one level above the current level and is admissible, so it is selected as the highest permitted admissible dose. (3) Admissibility is defined by posterior probability that the dose-specific toxicity probability exceeds the target, not by comparing its posterior mean with the target. (4) The admissible set is empty, so dose escalation stops and the protocol’s prespecified no-admissible-dose action is followed; no cohort is assigned contrary to the rule.