Worked Design Case for a Model-Based Escalation Trial

From a Model to a Trial Decision

A dose-toxicity model is useful only when its implications can be translated into protocol actions. In a model-based escalation trial, the team must update its view of toxicity after each cohort, check whether the next dose satisfies prespecified safety rules, and decide what to do when new observations make the current dose unsuitable. This case follows those steps using a small, hypothetical dose grid and a discrete set of candidate toxicity profiles.

The example uses completed binary dose-limiting toxicity (DLT) outcomes. Every patient has completed the same assessment window before the next cohort decision, so there are no pending outcomes to weight or model. The case therefore isolates the decisions in this trial: updating the dose-toxicity profile, applying a posterior safety constraint, and choosing whether to escalate, hold, or de-escalate.

The Hypothetical Dose Grid and Model

Suppose the trial has four ordered dose levels. Before enrollment, the team specifies four plausible monotone toxicity profiles. Each profile assigns a DLT probability to each dose. The profiles are fixed candidate curves rather than continuously estimated curves; the analysis updates the probability assigned to each candidate.

Assume equal prior probabilities, \(P(M_m)=0.25\), and conditional independence of patients’ DLT outcomes given the profile and the doses they receive. For a cohort of \(n\) patients treated at dose \(j\), with \(y\) DLTs, the likelihood under profile \(M_m\) is proportional to \(p_{jm}^{y}(1-p_{jm})^{n-y}\). The binomial coefficient is common to all profiles and cancels when the posterior probabilities are normalized. Outcomes from different cohorts contribute likelihood factors in the same way.

The protocol defines a target DLT probability of 0.25 and a safety boundary of 0.30. A proposed dose is eligible for escalation only when its posterior probability of exceeding the boundary is at most 0.25. If a proposed escalation fails this check, the next cohort stays at the current dose, provided that dose itself does not trigger the de-escalation rule. De-escalation is required when the posterior probability that the current dose exceeds the boundary is greater than 0.25. These thresholds are hypothetical protocol choices, not universal rules.

Updating After the First Two Cohorts

The first three patients receive dose 1, and none has a DLT. The second cohort of three receives dose 2, also with no DLTs. The data are consistent with lower-risk profiles, but the posterior probability of toxicity at a higher dose must still be checked; observing no DLTs at lower doses does not by itself establish that escalation is safe.

That safety check governs the actual trial path. The cohort at dose 3 in the following examples occurs only after the additional dose-2 cohort provides new evidence and the escalation check is passed. It is not a cohort assigned in violation of the rule.

Rechecking Safety Before Escalation

The next cohort of three receives dose 2, with no DLTs. The likelihood contribution is \((1-p_{2m})^3\) for profile \(M_m\). Multiplying each two-cohort weight by this contribution produces new unnormalized weights of approximately 0.736725, 0.593545, 0.464213, and 0.212218. Their sum is approximately 2.00670.

The calculation illustrates why cohort decisions should be made sequentially. The first two cohorts did not authorize escalation; the protocol held at dose 2. After the additional zero-DLT cohort at dose 2, the posterior probability assigned to the highest-risk profile fell enough for the dose-3 check to pass. The same rule is applied at each decision point rather than assuming that a decision remains appropriate after the data change.

Updating at Dose 3 and De-Escalating

Suppose the dose-3 cohort has one DLT among three patients. The likelihood for a profile with dose-3 probability \(p\) is \({3\choose1}p(1-p)^2\). The coefficient 3 cancels in posterior normalization. After updating the posterior from the previous section, the probabilities for \(M_1\) through \(M_4\) are approximately \(0.2340\), \(0.3143\), \(0.3161\), and \(0.1355\).

At this stage, the probability that dose 3 exceeds 0.30 is about 0.1355, since only \(M_4\) assigns a dose-3 probability above 0.30. The probability that dose 4 exceeds 0.30 is \(0.3161+0.1355=0.4516\): \(M_2\) assigns exactly 0.30 to dose 4, which does not count as exceeding the boundary. Escalation to dose 4 therefore fails the safety rule, and the next cohort stays at dose 3.

Final Dose Decision and Interpretation

For a final toxicity-based recommendation, apply the same boundary to each dose using the latest posterior. At dose 3, the posterior probability of exceeding 0.30 is approximately 0.3674, so dose 3 is ineligible. At dose 4, only \(M_3\) and \(M_4\) exceed 0.30; the probability is approximately \(0.4345+0.3674=0.8019\), so dose 4 is also ineligible. At dose 2, none of the candidate profiles exceeds 0.30, making it eligible under this toxicity constraint.

As a check on the target, the posterior mean DLT probability at dose 2 is approximately \(0.0336(0.04)+0.1645(0.0692)+0.4345(0.0972)+0.3674(0.1948)=0.1265\). Dose 2 is therefore below the 0.25 target on this posterior-mean scale. Under the stated rule, the final toxicity-based recommendation is dose 2: it is the highest dose that meets the posterior safety constraint. This is not evidence that dose 2 is efficacious or optimal for overall benefit; no efficacy outcomes or benefit-risk trade-off were included in this synthetic case.

Diagnostics, Alternatives, and Limitations

Several checks help detect errors in a hand calculation or software implementation. Posterior probabilities must be nonnegative and sum to 1 after every update. The likelihood contribution for a cohort with \(y\) DLTs among \(n\) patients must use \(p^y(1-p)^{n-y}\); reversing \(y\) and \(n-y\) would reverse the evidence. The safety calculation must sum only profiles with \(p_j>0.30\), not those with \(p_j\geq0.30\). In particular, a probability equal to the boundary does not exceed it under this protocol.

A common mistake is to treat a favorable observation at the current dose as automatic permission to escalate. Here, two cohorts without DLTs still left \(P(p_3>0.30\mid D)\approx0.3981\), above the 0.25 limit. The correction is to evaluate the proposed dose’s posterior risk before assigning the next cohort; if the check fails, follow the specified hold action and collect data only at an allowed dose.

This discrete-profile case is transparent and easy to audit, but its conclusions depend on the chosen dose probabilities and prior weights. A continuous dose-toxicity model can represent a broader range of curves, while a rule-based design may use simpler escalation boundaries. Neither alternative removes the need to define safety and stopping rules in advance. This example also assumes fully observed, comparable assessment windows, correct outcome classification, and no material changes in the treatment or patient population across cohorts. Delayed outcomes, model misspecification, or a poorly chosen dose grid require additional modeling and sensitivity checks; they are not resolved by normalizing the posterior weights.