From Cohort Outcomes to the Next EWOC Assignment
EWOC dose assignment is sequential: after a cohort’s toxicity outcomes are observed, the posterior distribution of the maximum tolerated dose (MTD) is updated, and candidate doses are checked against the feasibility bound. The calculation must keep two ideas distinct. The likelihood describes how compatible the observed outcomes are with each possible MTD; the feasibility rule then uses the updated posterior to limit the chance of assigning above the MTD.
The examples use a hypothetical, fixed grid of four doses, \(d_1<d_2<d_3<d_4\), and a target dose-limiting toxicity (DLT) probability of 0.25. Let \(M\) denote the MTD. The posterior support is the grid itself: \(M=d_k\) means dose \(d_k\) is the highest dose whose modeled DLT probability is at or below the target. The response model below supplies a toxicity probability at every administered dose under each MTD scenario.
| MTD scenario | At \(d_1\) | At \(d_2\) | At \(d_3\) | At \(d_4\) |
|---|---|---|---|---|
| \(M=d_1\) | 0.25 | 0.50 | 0.70 | 0.85 |
| \(M=d_2\) | 0.10 | 0.25 | 0.50 | 0.70 |
| \(M=d_3\) | 0.05 | 0.15 | 0.25 | 0.50 |
| \(M=d_4\) | 0.02 | 0.08 | 0.15 | 0.25 |
Write \(p_{kj}\) for the modeled DLT probability at dose \(d_j\) under scenario \(M=d_k\). The table is a hypothetical model, not observed trial data. Each row is nondecreasing across doses, and its probability at the scenario’s MTD equals the target. Let \(w_k=P(M=d_k\mid\text{data})\) be the current posterior scenario probability.
Updating the Posterior After a Completed Cohort
Suppose a cohort of \(n\) participants receives dose \(d_j\), and \(y\) experience a DLT in the specified observation window. Under scenario \(M=d_k\), assume each participant’s DLT outcome is Bernoulli with probability \(p_{kj}\), and outcomes are conditionally independent given the scenario. The cohort likelihood is binomial:
Bayes’ rule multiplies the previous scenario probability by this likelihood and normalizes over scenarios. The binomial coefficient is the same under every scenario for a fixed cohort, so it cancels from the normalized update:
The same result can be obtained by computing unnormalized weights \(u_k=w_k^{\mathrm{old}}L_k\), summing them, and dividing each by that sum. This form is convenient for hand calculation. With multiple completed cohorts, multiply each scenario’s current weight by the likelihood for the new cohort; previous outcomes are already represented in the current posterior.
Worked Cohort Updates
Worked Example: One DLT Among Three at \(d_2\)
Suppose the current posterior is \((0.10,0.20,0.40,0.30)\) for \(M=(d_1,d_2,d_3,d_4)\). A hypothetical cohort of three receives \(d_2\), and one has a DLT. Calculate the updated posterior, then check the candidates using \(\alpha=0.25\).
For \(y=1,n=3\), the likelihood under a scenario is \(3p_{k2}(1-p_{k2})^2\). The four probabilities at \(d_2\) are \(0.50,0.25,0.15,0.08\). The likelihoods and unnormalized weights are:
The weights sum to \(0.3128658\). Normalizing gives approximately \((0.1199,0.2697,0.4157,0.1948)\). The feasibility probabilities are \(F_1=0\), \(F_2=0.1199\), \(F_3=0.1199+0.2697=0.3896\), and \(F_4=0.8052\). Thus \(d_1\) and \(d_2\) are feasible, while \(d_3\) and \(d_4\) are not; the highest feasible candidate is \(d_2\).
Worked Example: Two Successive Cohorts Change the Feasibility Check
Continue with the same hypothetical prior and model. First, three participants receive \(d_2\) and none has a DLT. Then, after updating and applying the rule, a cohort receives the selected dose \(d_3\); one of three has a DLT. Follow the posterior and determine the next assignment using \(\alpha=0.25\).
For the first cohort, the likelihood under each scenario is \((1-p_{k2})^3\). Multiplying by the prior gives unnormalized weights:
Their sum is \(0.5761314\), so the posterior is approximately \((0.0217,0.1465,0.4264,0.4055)\). The cumulative probabilities below \(d_2,d_3,d_4\) are \(0.0217,0.1682,0.5945\), respectively. Therefore \(d_3\) is the highest feasible dose and may be assigned to the next cohort under this rule.
For the next cohort at \(d_3\), one of three has a DLT. The likelihoods are \(0.189,0.375,0.421875,0.325125\), using the \(d_3\) column of the model table. Retaining full precision from the first cohort, multiply these by its unnormalized weights \((0.01250,0.084375,0.24565,0.2336064)\); the resulting unnormalized weights are \((0.0023625,0.031640625,0.10363359375,0.0759512808)\). Their sum is \(0.21358799955\), giving the updated posterior approximately \((0.0111,0.1481,0.4852,0.3556)\).
Now \(F_2=0.0111\), \(F_3=0.0111+0.1481=0.1592\), and \(F_4=0.6444\). Since \(F_3\leq0.25\) but \(F_4>0.25\), \(d_3\) remains the highest feasible candidate. Each cohort is evaluated using its own administered dose and outcomes; the feasibility check is recalculated from the updated posterior, not copied from the prior decision.
Worked Example: A DLT-Free Cohort Moves Posterior Weight Upward
Return to the prior \((0.10,0.20,0.40,0.30)\), but consider a separate hypothetical cohort of three at \(d_2\) with zero DLTs. Show why the posterior shifts toward higher MTD scenarios and identify the highest feasible dose at \(\alpha=0.25\).
The unnormalized weights are \(0.10(0.50)^3=0.01250\), \(0.20(0.75)^3=0.084375\), \(0.40(0.85)^3=0.24565\), and \(0.30(0.92)^3=0.2336064\). As above, these sum to \(0.5761314\), so the normalized posterior is approximately \((0.0217,0.1465,0.4264,0.4055)\).
The likelihood of seeing no DLTs is larger under scenarios with smaller toxicity probabilities at \(d_2\). In particular, the \(M=d_4\) likelihood is \(0.92^3=0.778688\), compared with \(0.50^3=0.125\) under \(M=d_1\). This supports more posterior weight on higher MTD scenarios, though it does not prove which scenario is true. The feasibility probabilities are \(0\), \(0.0217\), \(0.1682\), and \(0.5945\) at \(d_1\) through \(d_4\). Hence \(d_3\) is the highest feasible candidate.
Why the Update and the Feasibility Check Must Stay Separate
The posterior update summarizes evidence about \(M\), conditional on the model and observed data. The feasibility check applies the chosen risk bound to that summary. A cohort can move posterior mass toward lower MTD scenarios, higher scenarios, or leave the feasibility decision unchanged; the direction depends on the dose administered and the observed outcome. A DLT-free cohort is not automatically a reason to escalate, and a cohort with a DLT is not automatically a reason to de-escalate. The posterior calculation and the bound determine the decision.
A useful computational audit is to verify that every updated weight is nonnegative and that the normalized probabilities sum to one (up to rounding). Then calculate cumulative probabilities from the lowest MTD scenario upward. Since \(F_j\) cannot decrease as \(j\) increases, feasibility on an ordered grid forms a prefix: once a candidate fails the bound, any higher candidate also fails. This monotonicity is a direct consequence of \(F_{j+1}=F_j+w_j\), with \(w_j\geq0\).
A short justification of the sequential calculation follows. Conditional on scenario \(M=d_k\), outcomes from separate completed cohorts are independent under the stated model. Thus the joint likelihood is the product of cohort likelihoods. If the current posterior already includes earlier cohorts, Bayes’ rule has already incorporated their likelihood factors; multiplying by just the new cohort likelihood is equivalent to using the full product again. The dose assigned to a cohort is known from the trial history, so for a fixed observed assignment it contributes no additional scenario-dependent likelihood factor. This justifies updating the current posterior one completed cohort at a time.
Confirm its assigned dose, cohort size, number of DLTs, and complete observation status.
Use the model probability at that administered dose and the binomial likelihood for the observed DLT count.
Multiply each current posterior weight by its likelihood, sum the weights, and divide by the sum.
For each \(d_j\), sum posterior mass on MTD scenarios strictly below \(d_j\), compare with \(\alpha\), and select the highest feasible dose.
Diagnostics, Alternatives, and Limitations
Several checks catch common implementation errors. Verify that the likelihood uses the probability at the administered dose, not at the scenario’s MTD by default. Check that the exponent of \(p\) is the number of DLTs and the exponent of \(1-p\) is the number without DLT. Confirm that cohort outcomes are not counted twice: after an update, the new posterior becomes the starting weight for the next completed cohort. Finally, compute the feasibility sum from posterior mass below the candidate, not from the candidate’s toxicity probability alone.
One alternative is a simple fixed escalation schedule, which assigns doses according to prespecified cohort outcomes rather than a posterior MTD distribution. That approach can be easier to implement, but it does not use the same model-based posterior calculation. A posterior point estimate of the MTD is another summary, but choosing the dose nearest that estimate alone does not enforce EWOC’s probability bound. EWOC’s characteristic step is the explicit check \(P(M<d_j\mid\text{data})\leq\alpha\) for each candidate.
The calculation is only as credible as its assumptions and model. Incorrect dose-response probabilities, a poorly calibrated prior, dependence among participants’ outcomes, or incomplete follow-up can make the posterior misleading. The finite scenario grid also limits the MTD distribution to the modeled values. Small numerical differences near the feasibility threshold may result from rounding; retain full precision for updating and compare with \(\alpha\) before rounding the reported probabilities. The bound is a design constraint under the posterior model, not a guarantee that an assigned dose is safe in the realized trial.
- Wrong: “The posterior mean MTD is below \(d_3\), so \(d_3\) fails the EWOC rule.”
Correction: Evaluate the posterior probability \(P(M<d_3\mid\text{data})\) and compare that probability with \(\alpha\). A mean alone does not determine the feasibility check. - Wrong: “There were no DLTs, so the next dose must be higher.”
Correction: First update the posterior under all scenarios, then apply the bound to every candidate. The data inform the decision; they do not replace the rule. - Wrong: “A dose is feasible when its modeled toxicity probability is below \(\alpha\).”
Correction: The feasibility bound \(\alpha\) limits posterior probability that the candidate is above the MTD. It is not the target toxicity probability and not a per-participant toxicity probability.
Check Your Understanding
Use the definitions and hypothetical model in this tutorial. Solutions follow the questions.
- For a cohort of four with one DLT at dose \(d_j\), write the likelihood under scenario \(M=d_k\).
- After the first cohort in “Two Successive Cohorts,” what is the approximate posterior probability that \(M<d_3\), and is \(d_3\) feasible at \(\alpha=0.25\)?
- Why does a cohort with no DLTs at \(d_2\) tend to favor higher MTD scenarios in the model table?
- After the second cohort in “Two Successive Cohorts,” is \(d_4\) feasible at \(\alpha=0.25\)? Give the posterior probability used for the check.
Solutions. (1) The likelihood is \({4\choose1}p_{kj}(1-p_{kj})^3=4p_{kj}(1-p_{kj})^3\). (2) The posterior probability is \(0.0217+0.1465=0.1682\), which is at most \(0.25\), so \(d_3\) is feasible. (3) Under the table, higher MTD scenarios have lower DLT probabilities at \(d_2\), and therefore larger likelihoods for zero DLTs; this shifts posterior weight upward without establishing that any scenario is true. (4) No. The posterior probability is \(P(M<d_4\mid\text{data})=1-P(M=d_4\mid\text{data})\approx1-0.3556=0.6444\), which exceeds \(0.25\).