Worked Example of Stepwise Escalation

Following the Escalation Rule, One Cohort at a Time

A dose-escalation decision is not made from the observed DLT rate alone. In a Bayesian logistic regression model (BLRM), each cohort contributes a likelihood to the posterior distribution of the dose-toxicity curve. The trial team then uses that posterior to assess whether candidate doses meet the protocol’s overdose-control criterion and applies the prespecified allocation rule. When early cohorts have no DLTs, the data can support escalation, but the uncertainty in the dose-response curve remains important.

The example below follows cohorts receiving 10 mg and then 20 mg. All patients and results are hypothetical. The purpose is to show the sequence of updates and decisions, not to recommend a particular dose or protocol. The admissibility cutoff and escalation rule are illustrative protocol choices; a real trial must prespecify them.

Model and assumptions: Let $Y_j$ be the number of patients with a dose-limiting toxicity (DLT) among $n_j$ patients treated at dose $d_j$. Conditional on the dose-specific DLT probability $p(d_j)$, assume $Y_j\sim\operatorname{Binomial}(n_j,p(d_j))$, with outcomes conditionally independent across patients. In this example, the BLRM is $$ \operatorname{logit}\{p(d)\}=\alpha+\beta\log(d/10), $$ where $\operatorname{logit}(u)=\log\{u/(1-u)\}$, doses are in mg, and $\alpha,\beta$ are uncertain parameters. For illustration, use independent priors $\alpha\sim N(-1.20,1.00^2)$ and $\beta\sim N(0.80,0.50^2)$. The model assumes a linear relationship between log dose and log odds of DLT over the dose range considered. The numerical calculations below use the posterior from this model.

For a candidate dose $d$, define its overdose probability as $q(d)=P\{p(d)>0.35\mid\text{observed data}\}$. Thus the toxicity boundary in this example is 35%, and a dose is admissible when $q(d)<0.30$. The trial starts at 10 mg. At each decision, the protocol escalates by at most one dose level, to the next higher dose, only if that next dose is admissible. The dose levels considered are 10, 20, and 40 mg. This rule does not permit skipping a dose level, and it does not instruct the team to escalate merely because the current cohort had no DLTs.

First Cohort: Updating After 0 of 3 DLTs

At 10 mg, three patients complete the DLT observation period and none experiences a DLT. Write $y_{10}=0$ and $n_{10}=3$. The likelihood contribution for this cohort is $$ L_{10}(\alpha,\beta) =\binom{3}{0}p(10)^0\{1-p(10)\}^3 =\{1-\operatorname{logit}^{-1}(\alpha)\}^3, $$ because $\log(10/10)=0$. The combinatorial factor is one, and the likelihood contains no direct information about $\beta$ at the reference dose. The posterior is proportional to this likelihood times the joint prior density. Although $\beta$'s likelihood contribution at 10 mg is constant, $\alpha$ and $\beta$ will jointly determine toxicity at doses above 10 mg.

Worked Example: Assessing 20 mg After the First Cohort

After 0 DLTs among 3 patients at 10 mg, assess the candidate dose of 20 mg. At 20 mg, $\operatorname{logit}\{p(20)\}=\alpha+\beta\log 2$. The event $p(20)>0.35$ is equivalent to

\alpha+\beta\log 2>\operatorname{logit}(0.35) =\log(0.35/0.65)\approx-0.6190.

The posterior density for $\alpha$ is proportional to its prior density multiplied by $\{1-\operatorname{logit}^{-1}(\alpha)\}^3$; $\beta$ retains its independent prior distribution after this cohort. Integrating the joint posterior over the region above the inequality gives $q(20)\approx0.27$. This is a numerical integration result, not a calculation from the observed rate $0/3$.

The protocol cutoff here is 0.30, so 20 mg is admissible because the unrounded numerical probability is below 0.30. The displayed value, 0.27, is rounded for reporting; the decision is made using the unrounded calculation. In particular, if a probability were reported as $0.250$ to three decimal places against a cutoff of 0.25, that report would not by itself establish admissibility. The unrounded probability would have to be compared with that cutoff under the protocol’s exact inequality.

The escalation rule permits moving from 10 mg to the next level, 20 mg. It does not permit moving directly to 40 mg, even if 40 mg were also judged admissible, because this example’s rule allows an increase of only one dose level.

Second Cohort: Updating Before the Next Decision

Suppose the protocol assigns the next cohort of three patients to 20 mg. They also complete DLT follow-up with no observed DLTs. The posterior must now include both cohorts’ likelihood contributions. For $0/3$ at 20 mg, the additional factor is $\{1-\operatorname{logit}^{-1}(\alpha+\beta\log 2)\}^3$. The posterior after these six patients is therefore proportional to the joint prior density multiplied by the likelihood at 10 mg and the likelihood at 20 mg. It would be incorrect to reuse the posterior from the first cohort without incorporating the second one.

Worked Example: Reassessing 40 mg After 0 of 3 at 20 mg

At 40 mg, the linear predictor is $\alpha+\beta\log 4$. The overdose event is

p(40)>0.35 \quad\Longleftrightarrow\quad \alpha+\beta\log 4>-0.6190.

The joint posterior after the two zero-DLT cohorts has density proportional to

\pi(\alpha,\beta) \{1-\operatorname{logit}^{-1}(\alpha)\}^3 \{1-\operatorname{logit}^{-1}(\alpha+\beta\log 2)\}^3,

where $\pi(\alpha,\beta)$ is the stated independent normal prior density. Integrating this updated density over the 40 mg overdose region gives an illustrative numerical result $q(40)\approx0.34$. Since 0.34 exceeds the protocol cutoff of 0.30, 40 mg is not admissible after the second cohort. The current dose, 20 mg, remains admissible in this illustrative calculation; assume its updated overdose probability is approximately 0.12, also below 0.30.

The rule therefore does not escalate to 40 mg. The fact that the second cohort observed $0/3$ DLTs is evidence that informs the posterior, but it does not override the overdose calculation. Under this example’s protocol, the escalation sequence stops at 20 mg for this decision. A separate rule would be needed to specify whether another cohort can be enrolled at 20 mg, whether the trial pauses, or whether it terminates.

Deriving the Cohort Update and Checking the Decision

The update follows directly from Bayes’ rule. If the data observed so far are $D$, and a new cohort at dose $d$ has $y$ DLTs among $n$ patients, its likelihood is $\binom{n}{y}p(d)^y\{1-p(d)\}^{n-y}$. The posterior density after that cohort is proportional to the previous posterior density multiplied by this likelihood. For a cohort with no DLTs, $y=0$, so the likelihood simplifies to $\{1-p(d)\}^n$. Repeating this operation cohort by cohort gives the same posterior as multiplying all cohort likelihoods together, provided the model and prior stay unchanged.

\pi(\alpha,\beta\mid D_{\mathrm{new}}) \propto \pi(\alpha,\beta\mid D)\, p(d)^y\{1-p(d)\}^{n-y}.

The binomial coefficient can be omitted from the proportionality because it does not depend on $\alpha$ or $\beta$. To calculate $q(d)$, integrate this posterior over parameter values for which $\alpha+\beta\log(d/10)>\operatorname{logit}(0.35)$, and divide by the integral over the whole parameter space. In practice, numerical integration or posterior simulation can approximate these integrals. The calculations must use the same model, prior, data, toxicity boundary, and dose definition as the protocol decision.

The logic can be summarized as a repeated sequence:

Record the completed cohort.
Confirm the dose, cohort size, DLT count, and that the relevant DLT follow-up is complete.

Update the posterior.
Multiply the current posterior by the binomial likelihood contribution from the new cohort.

Recalculate admissibility.
For each candidate dose, estimate its posterior probability of exceeding the toxicity boundary and compare it with the prespecified cutoff.

Apply the allocation rule.
Choose only the next permitted dose level if it is admissible; otherwise follow the protocol’s specified hold, pause, or stop action.

What the Example Does and Does Not Establish

A zero-DLT cohort does not imply that the toxicity probability is zero. With only three patients, an unobserved event remains compatible with a range of underlying probabilities. The BLRM combines the cohort information with the prior and the dose-response structure; at an untried dose, uncertainty about both the intercept and slope affects the overdose probability. That is why the 20 mg calculation after the 10 mg cohort is not obtained by simply copying the observed rate at 10 mg.

The admissible set is a model-based decision aid defined by a particular overdose boundary and cutoff. It is not a declaration that an admissible dose is harmless, nor does it establish that the dose is optimal. Conversely, a dose outside the admissible set is not necessarily certain to cause excessive toxicity; its posterior probability of exceeding the chosen boundary is simply too large for this protocol’s rule.

The example also depends on assumptions that should be examined in an actual trial. The BLRM imposes a monotone dose-toxicity curve when $\beta>0$; the normal prior used here assigns some probability to $\beta<0$, so monotonicity is not guaranteed absolutely unless the prior or model constrains it. The binomial likelihood treats patients within a cohort as conditionally independent with a common dose-specific DLT probability. Misspecified dose-response shape, an implausible prior, coding errors in dose units, or incomplete outcome follow-up can undermine the resulting probability calculations. The decision rule also assumes the dose levels and escalation limits have been specified in advance.

Useful diagnostics include checking that the likelihood uses the correct cohort counts, verifying that dose ratios are calculated relative to 10 mg, and comparing posterior draws or numerical integration against independent implementations. Review posterior distributions of $\alpha$ and $\beta$, dose-toxicity curves, and overdose probabilities across all candidate doses. If the computed probability is close to a cutoff, increase numerical precision and report enough digits internally to make the decision reliably. A rounded display is for communication, not a substitute for the protocol comparison.

Common Pitfalls and Alternatives

Using the observed rate as the overdose probability. The observed rate $0/3=0$ is a sample proportion at the administered dose; it is not $P\{p(20)>0.35\mid D\}$. Calculate the posterior overdose probability at the candidate dose.
Calling a rounded boundary value admissible without checking precision. If a reported $q(d)$ rounds to the cutoff, the displayed number cannot reveal which side of the cutoff the unrounded value lies on. Use the unrounded result and the exact strict or non-strict inequality in the protocol.
Escalating more than the protocol allows. An admissible higher dose is not automatically the next assignment if the rule permits only a one-level increase. Check both admissibility and the allocation restriction.
Ignoring the new cohort in later updates. Each completed cohort contributes another likelihood factor. The posterior for the next decision must include all available cohort data, not only the most recent result.

A fixed escalation design can be simpler to administer, but it does not use a posterior dose-toxicity curve in the same way. A rule based only on observed DLT counts may be easier to explain but does not directly quantify uncertainty about overdose at untried doses. The BLRM approach instead makes that uncertainty explicit, at the cost of dependence on model form, prior specification, numerical computation, and a clearly written decision algorithm. These approaches are not interchangeable without considering the trial objective and operating characteristics.

Key takeaway: After each cohort, update the full posterior, recompute overdose probabilities, and apply the prespecified stepwise rule. In the example, $q(20)\approx0.27$ after $0/3$ at 10 mg; this is admissible under the stated 0.30 cutoff. After $0/3$ at 20 mg, the illustrative $q(40)\approx0.34$ exceeds that cutoff, so escalation to 40 mg does not proceed.

Check Your Understanding

Answer each question from the example, then compare your reasoning with the solutions.

Why does the likelihood for $0/3$ DLTs at 10 mg depend on $\alpha$ but not directly on $\beta$?
What event defines $q(20)$, and why is it not the observed DLT rate at 10 mg?
Under the stated cutoff of 0.30, is a candidate dose with $q(d)=0.28$ admissible? What if the protocol instead uses a strict cutoff of 0.25 and the reported probability rounds to 0.250?
After 0/3 DLTs at 20 mg, why must the posterior used to assess 40 mg include the 10 mg cohort as well?
If the next dose is admissible but is two dose levels above the current dose, may the trial escalate directly to it under the example’s rule?

Solutions. (1) At 10 mg, $\log(10/10)=0$, so the linear predictor is $\alpha$; the likelihood is $\{1-\operatorname{logit}^{-1}(\alpha)\}^3$. (2) $q(20)=P\{p(20)>0.35\mid D\}$, a posterior probability about toxicity at 20 mg given the model and data, not a sample proportion observed at 10 mg. (3) Yes, because $0.28<0.30$. A value rounded to 0.250 does not settle a strict 0.25 comparison; inspect the unrounded probability. (4) Both cohorts inform the current posterior through their likelihood contributions, so omitting the 10 mg data would discard information. (5) No. The example allows an increase of at most one dose level, so the trial cannot skip directly to a dose two levels higher.