Worked Escalation Decisions Under Overdose Control

Why Work Through Decisions Cohort by Cohort?

A dose-finding rule is useful only when it can be applied consistently after each cohort’s outcomes arrive. Under posterior overdose control, the decision has two distinct parts: first determine which doses satisfy a prespecified safety constraint, then select a dose from that eligible set according to the protocol’s target-based rule. Keeping these parts separate prevents a dose that appears close to the target from being selected when its posterior safety risk is too high.

This tutorial follows a hypothetical trial with completed toxicity windows and a Bayesian model for dose-specific toxicity. It focuses on the arithmetic and logic of repeated decisions. It does not address partially observed toxicity windows.

Model, Target, and Decision Rule

Definition: Let $p_j(\theta)$ be the probability of dose-limiting toxicity (DLT) at dose $j$ under model parameter $\theta$. Let $\phi$ be the target toxicity rate. A dose is an overdose under parameter value $\theta$ if $p_j(\theta)>\phi$. After observing data $D$, its posterior overdose probability is $q_j=P\{p_j(\theta)>\phi\mid D\}$. Given a prespecified limit $\xi$, dose $j$ passes overdose control if $q_j\leq \xi$.

The target $\phi$ is a toxicity-rate threshold, whereas $\xi$ is a probability threshold for uncertainty about whether that toxicity rate is exceeded. They are different quantities. Passing the safety rule does not mean that the dose is known to be safe; it means that its posterior probability of exceeding $\phi$ is no greater than $\xi$, under the specified model and data.

For the worked example, the model has three possible toxicity curves, indexed by $\theta$. Conditional on a curve, patient DLT outcomes are independent Bernoulli observations at the administered dose. The curve probabilities are known within each state, and the prior assigns probabilities to the three states. All toxicity windows are complete, and the recorded outcome is DLT or no DLT. These assumptions allow a standard binomial likelihood update.

Model state	Prior probability	Dose 1 toxicity	Dose 2 toxicity	Dose 3 toxicity
A	0.50	0.05	0.10	0.20
B	0.30	0.10	0.20	0.40
C	0.20	0.20	0.40	0.65

The curves are monotone across the three ordered doses. Set the target rate to $\phi=0.25$ and the maximum acceptable posterior overdose probability to $\xi=0.25$. The protocol starts at dose 1. After each completed cohort, it selects, among eligible doses, the dose with posterior mean toxicity closest to $\phi$, subject to increasing by at most one dose level from the current dose. A lower eligible dose can be selected if the current dose fails safety control. If no dose is eligible, treatment assignment stops. This selection convention is one explicit way to complete the safety rule; other protocols may specify a different target-based rule.

Posterior Updating and Overdose Calculation

If $n_j$ patients have been treated at dose $j$, of whom $y_j$ experienced DLT, the dose-specific binomial likelihood under state $\theta$ is proportional to $p_j(\theta)^{y_j}\{1-p_j(\theta)\}^{n_j-y_j}$. With independent patient outcomes conditional on the model state, likelihood contributions across doses multiply. Bayes’ rule therefore updates each state’s weight by multiplying its current posterior probability by the cohort likelihood and normalizing the resulting weights to sum to one.

w_\theta^{\,\mathrm{new}} = \frac{w_\theta^{\,\mathrm{old}} \prod_j p_j(\theta)^{y_j}\{1-p_j(\theta)\}^{n_j-y_j}} {\sum_{\theta'} w_{\theta'}^{\,\mathrm{old}} \prod_j p_j(\theta')^{y_j}\{1-p_j(\theta')\}^{n_j-y_j}}

Once the weights are updated, calculate the posterior overdose probability by summing the weights of states whose toxicity rate exceeds the target at that dose. For example, in this model, at dose 2 only state C exceeds 0.25; at dose 3, states B and C exceed 0.25. The posterior mean toxicity is a separate summary, calculated by weighting the state-specific toxicity rates. It guides selection among eligible doses but does not replace the safety calculation.

Justification: The posterior event $\{p_j(\theta)>\phi\}$ is the union of the model states that exceed the target at dose $j$. Since the states are mutually exclusive, its posterior probability is the sum of their posterior probabilities: $$ q_j=\sum_{\theta:\,p_j(\theta)>\phi}P(\theta\mid D). $$ Thus the safety check requires identifying the relevant states and adding their updated weights, not comparing the posterior mean alone with $\phi$.

Worked Example: Starting Dose and First Escalation

Suppose the first cohort of three patients is treated at dose 1 and none experiences DLT. Calculate the updated state probabilities and decide whether escalation to dose 2 is allowed.

For $y_1=0$ of $n_1=3$, the likelihood in each state is $\{1-p_1(\theta)\}^3$. Multiplying by the prior gives unnormalized weights:

\begin{aligned} A &: 0.50(0.95)^3=0.4286875,\\ B &: 0.30(0.90)^3=0.2187000,\\ C &: 0.20(0.80)^3=0.1024000. \end{aligned}

The normalizing total is $0.7497875$. Dividing each unnormalized weight by this total gives posterior weights approximately $(0.5717,\,0.2917,\,0.1366)$ for states A, B, and C. Dose 2 is an overdose only in state C, so $q_2=0.1366$, which is below $\xi=0.25$. Dose 3 is an overdose in states B and C, so $q_3=0.2917+0.1366=0.4283$, which exceeds the limit.

The posterior mean toxicity at dose 2 is $0.5717(0.10)+0.2917(0.20)+0.1366(0.40)=0.1701$. Dose 2 is eligible and is the next dose level; dose 3 is ineligible. Therefore the protocol escalates from dose 1 to dose 2. This decision reflects both the safety screen and the one-level escalation restriction.

Following the Trial Across Further Cohorts

The next update shows why “the dose is below target on average” is not a safety rule. After the first cohort, the dose-3 posterior mean is relatively close to the target, but its overdose probability is still above the specified limit. A target-based selection criterion is applied only after the overdose screen.

Worked Example: No DLT at Dose 2 and a Stay Decision

Treat a second cohort of three at dose 2; assume none experiences DLT. Update the posterior using the full data: three patients with no DLT at dose 1 and three with no DLT at dose 2.

The unnormalized weights can be calculated directly from the prior and both cohort likelihoods:

\begin{aligned} A &: 0.50(0.95)^3(0.90)^3=0.3125131875,\\ B &: 0.30(0.90)^3(0.80)^3=0.1119744,\\ C &: 0.20(0.80)^3(0.60)^3=0.0221184. \end{aligned}

Their sum is $0.4466059875$, giving posterior weights approximately $(0.6998,\,0.2507,\,0.0495)$. At dose 3, the overdose probability is the sum for states B and C: $q_3=0.2507+0.0495=0.3002$, above 0.25. At dose 2, $q_2=0.0495$, so dose 2 remains eligible.

The posterior mean toxicity at dose 2 is approximately $0.6997(0.10)+0.2507(0.20)+0.0495(0.40)=0.1399$. Since dose 3 fails safety control, and dose 2 is eligible, the next cohort stays at dose 2. A zero-DLT cohort has reduced the posterior weight on state C, but it has not reduced the combined posterior probability of states B and C enough to make dose 3 eligible.

Worked Example: One DLT at Dose 2 and Continued Treatment

At the next cohort, treat three more patients at dose 2 and observe one DLT. Starting from the posterior weights after the preceding two cohorts, update using the likelihood $3p(1-p)^2$ in each state. The common factor 3 cancels when the weights are normalized, so the likelihood factors are $p(1-p)^2$.

\begin{aligned} A &: 0.6997(0.10)(0.90)^2=0.05668,\\ B &: 0.2507(0.20)(0.80)^2=0.03209,\\ C &: 0.0495(0.40)(0.60)^2=0.007132. \end{aligned}

Normalizing these values gives posterior weights approximately $(0.5910,\,0.3346,\,0.0744)$. Thus $q_2=0.0744$, because only state C exceeds the target at dose 2, while $q_3=0.3346+0.0744=0.4090$. Dose 2 passes the overdose limit and dose 3 does not.

The posterior mean toxicity at dose 2 is approximately $0.1558$. The available eligible dose closest to the target is still dose 2, so the rule stays at dose 2. The observed DLT has increased the posterior weight on the higher-toxicity states, but it has not made the current dose fail the stated safety criterion.

Worked Example: Multiple DLTs Trigger De-Escalation

Now suppose the following cohort of three at dose 2 has two DLTs. From the previous posterior, the likelihood factors, omitting the common binomial coefficient, are $p^2(1-p)$. Calculate the new posterior and determine the next action.

\begin{aligned} A &: 0.5910(0.10)^2(0.90)=0.005319,\\ B &: 0.3346(0.20)^2(0.80)=0.01071,\\ C &: 0.0744(0.40)^2(0.60)=0.007139. \end{aligned}

The normalized weights are approximately $(0.2296,\,0.4622,\,0.3082)$. At dose 2, only state C is above the target, so $q_2=0.3082>0.25$: dose 2 fails the safety screen. At dose 1, no state exceeds 0.25, so $q_1=0$. The protocol therefore de-escalates to dose 1, the highest available lower dose that passes the constraint.

This is not a conclusion that dose 2 is certainly too toxic. It is a rule-based decision: the posterior probability that its toxicity exceeds the target is larger than the prespecified limit. The decision follows from the model, prior, accumulated outcomes, target, and threshold together.

Decision Workflow, Diagnostics, and Alternatives

Update the model.
For every state, multiply its current weight by the likelihood of the completed cohort outcomes, then normalize across states.

Apply the safety screen.
At each dose, add posterior weights for states whose toxicity rate exceeds $\phi$; retain doses with $q_j\leq\xi$.

Select within the eligible set.
Use the protocol’s target-based selection rule and any escalation restrictions. Do not let target proximity override failed overdose control.

Check operational consistency.
Verify the cohort is fully observed, the correct dose and outcomes entered, the weights sum to one, and the assignment follows the protocol’s stopping and dose-level rules.

Several diagnostics help catch errors. Posterior weights must be nonnegative and sum to one, apart from rounding. Each $q_j$ must lie between zero and one. Under the monotone curves in this example, overdose probabilities should not decrease as dose increases; if they do, recheck which states exceed the target and how their weights were updated. When a cohort has $y$ DLTs among $n$, the likelihood must include $p^y(1-p)^{n-y}$; using only $p^y$ discards the information in patients without DLT. A common binomial coefficient may be omitted for posterior weights when it is identical across states.

This rule differs from unconstrained target-probability CRM selection, which may choose the dose with estimated toxicity closest to $\phi$ without imposing this particular posterior event-probability limit. It also differs from EWOC, which is commonly expressed through a posterior constraint on the maximum tolerated dose relative to the assigned dose. These approaches encode safety differently, so identical outcomes need not produce identical recommendations. A dose-skipping restriction is an additional operational constraint: it limits the next assignment, but it does not change the posterior overdose probability itself.

The example’s finite set of three curves makes the calculations transparent, but it also limits what the model can represent. Posterior risk is conditional on the prior and the assumed dose-toxicity curves; omitted curve shapes or a poorly chosen prior can affect the decision. A permissive $\xi$ allows more posterior uncertainty about exceeding the target, while a stricter value can slow escalation or leave no dose eligible. The target, threshold, starting dose, dose-selection rule, and action when no dose is eligible should therefore be specified before trial conduct. The introductory material on posterior overdose control establishes the basic constraint; the key practical lesson here is to recompute it after every completed cohort rather than treating an earlier eligibility decision as permanent.

Common Mistakes

Using the posterior mean as the overdose probability. The mean toxicity rate is not $P\{p_j(\theta)>\phi\mid D\}$. Identify and sum the posterior weights of states above the target.
Confusing $\phi$ and $\xi$. The target $\phi$ is a toxicity rate; $\xi$ is the tolerated posterior probability of exceeding that rate.
Ignoring the no-DLT contribution. For a cohort with one DLT among three, the likelihood includes both $p$ and $(1-p)^2$, not just $p$.
Escalating because the current cohort had few DLTs. Decisions use the posterior from all observed cohorts and the stated rule, not only the most recent cohort.
Treating eligibility as certainty. Passing means $q_j\leq\xi$, not that the dose cannot exceed the target or that its toxicity rate is known exactly.

Key takeaway: Make cohort-by-cohort decisions in a fixed order: update the posterior, calculate overdose probabilities, exclude doses that fail the prespecified limit, and only then apply the target-based selection and dose-level restrictions.

Check Your Understanding

Answer each question using the definitions and rules above. Solutions follow the questions.

At a dose, states A, B, and C have posterior weights 0.60, 0.25, and 0.15. If only state C has toxicity above $\phi$, what is the posterior overdose probability?
With $\xi=0.25$, is a dose eligible when its posterior overdose probability is 0.25? Explain.
Under state-specific toxicity $p=0.20$, what likelihood factor applies to a cohort of three with one DLT, omitting the common binomial coefficient?
Why can a dose with posterior mean toxicity below the target still fail overdose control?

Solutions. (1) $q_j=0.15$, the posterior weight on the only overdose state. (2) Yes. The stated rule is $q_j\leq\xi$, so equality passes. (3) $p(1-p)^2=(0.20)(0.80)^2=0.128$. (4) A mean below target does not determine the probability that the rate exceeds target: enough posterior weight may lie on states above $\phi$ for $q_j$ to exceed $\xi$.