Why a Feasibility Bound Is Needed
In model-based dose finding, the estimated MTD is uncertain. A dose may be near the posterior center of the MTD distribution while still having substantial posterior probability of being above the MTD. EWOC (Escalation With Overdose Control) makes that uncertainty operational: it restricts the next assigned dose according to the posterior probability that the dose exceeds the MTD.
Let \(D\) denote the trial data available at an interim decision, and let \(M\) denote the MTD under the dose-toxicity model. The posterior distribution of \(M\), as covered in the earlier material on the posterior distribution of the MTD, summarizes uncertainty about the dose at which the toxicity probability reaches the trial's target. EWOC uses this posterior distribution to screen proposed doses. The bound is a restriction on assignment, not by itself a rule for choosing one dose from all eligible doses.
The event in the definition concerns the unknown MTD, not the outcome for an individual patient. It is therefore not the probability that a patient assigned dose \(d\) will experience toxicity. That patient-level probability is the dose-toxicity probability at \(d\); the EWOC probability instead measures posterior uncertainty about whether \(d\) lies above the MTD.
The notation \(\alpha_{\mathrm{EWOC}}\) is used here for the feasibility bound to distinguish it from a conventional significance level. It is a design choice governing the assignment restriction, not a type I error rate and not a probability that a null hypothesis is true.
Assumptions and the Shape of the Restriction
The rule presumes a meaningful posterior distribution for the MTD. In the usual setting, dose-toxicity probability is modeled as increasing with dose, and the MTD is defined as the dose at which that probability reaches the prespecified target. The model, prior, target, and observed outcomes together determine the posterior distribution of \(M\). If the model does not define an MTD over the relevant dose range, the feasibility calculation has no well-defined quantity to use.
For a continuous posterior distribution of \(M\), write its conditional cumulative distribution function as \(F_M(d\mid D)=P(M\leq d\mid D)\). Then \(P(M<d\mid D)=F_M(d\mid D)\), because a continuous distribution assigns probability zero to \(M=d\). The feasibility bound is the posterior \(\alpha_{\mathrm{EWOC}}\)-quantile of \(M\): doses at or below that quantile pass the probability constraint, subject to the dose range under consideration. If the posterior has point masses, the distinction between \(M<d\) and \(M\leq d\) can matter; calculate the event in the definition rather than silently treating the distribution as continuous.
The reason feasible doses form a lower portion of the dose range is elementary. If \(d_1\leq d_2\), then every posterior state with \(M<d_1\) also has \(M<d_2\). Thus the events are nested, and their posterior probabilities cannot decrease as dose increases.
Consequently, once a dose violates the bound, any higher dose also violates it, provided the dose is on the same ordered scale and the same posterior distribution is used. This is the key operational feature: EWOC imposes an upper restriction on the next dose. It does not, on its own, say whether to assign the largest feasible dose, a lower feasible dose, or a dose chosen by another trial rule.
Calculating and Applying the Bound
Worked Example: A Continuous Posterior MTD Distribution
Suppose a hypothetical interim analysis gives \(U=M/100\sim\operatorname{Beta}(2,2)\), so the posterior MTD lies between 0 and 100 mg. The trial uses a feasibility bound of \(\alpha_{\mathrm{EWOC}}=0.25\). Is a proposed dose of 30 mg feasible, and approximately where is the bound?
For \(0\leq u\leq1\), a \(\operatorname{Beta}(2,2)\) variable has cumulative distribution function \(P(U\leq u\mid D)=3u^2-2u^3\). Therefore, for a dose \(d\) in mg,
At 30 mg, the posterior overdose probability is \(3(0.30)^2-2(0.30)^3=3(0.09)-2(0.027)=0.216\). Since \(0.216\leq0.25\), 30 mg is feasible.
To locate the bound, solve \(3u^2-2u^3=0.25\). The solution in the interval from 0 to 1 is approximately \(u=0.326\), giving \(d=100(0.326)=32.6\) mg. As a check, the probability at 35 mg is \(3(0.35)^2-2(0.35)^3=0.28175\), which exceeds 0.25. Thus the bound is about 32.6 mg: 30 mg passes, whereas 35 mg does not.
This calculation illustrates the distinction between a posterior summary and an assignment restriction. The posterior mean or median MTD describes the center of the posterior distribution; the feasibility bound uses a lower posterior quantile when the rule limits the probability that the proposed dose is above the MTD. Those summaries answer different questions.
Worked Example: How a Tighter Bound Changes Feasibility
Use the same hypothetical posterior distribution, \(M/100\sim\operatorname{Beta}(2,2)\). Compare feasibility bounds of 0.10 and 0.30. Evaluate a proposed dose of 30 mg under each.
For a bound of 0.10, solve \(3u^2-2u^3=0.10\). The solution is approximately \(u=0.196\), so the dose bound is about 19.6 mg. At 30 mg, the overdose probability remains \(0.216\), which is greater than 0.10; 30 mg is not feasible under this tighter restriction.
For a bound of 0.30, the equation \(3u^2-2u^3=0.30\) has solution \(u\approx0.363\), giving a bound of about 36.3 mg. At 30 mg, \(0.216\leq0.30\), so 30 mg is feasible under this less restrictive bound. At 35 mg the probability is \(0.28175\), also no greater than 0.30.
The calculation shows how the bound acts: a smaller allowed posterior overdose probability moves the maximum feasible dose downward. Choosing the value of the bound requires a design judgment; the calculation itself does not establish which value is appropriate.
Using Posterior Draws
In a fitted Bayesian model, the posterior distribution of \(M\) may be represented by simulation draws rather than a closed-form distribution. If \(M^{(1)},\ldots,M^{(B)}\) are posterior draws, estimate the overdose probability at dose \(d\) by counting the draws below \(d\) and dividing by the total number of draws. Compare that estimate with the feasibility bound. This estimates the same posterior quantity as direct integration, but adds Monte Carlo error.
Worked Example: Checking a Proposed Dose With Posterior Draws
Suppose a hypothetical analysis produces 10,000 posterior draws of the MTD, measured in mg. Of these, 1,200 are below 20 mg, 2,450 are below 30 mg, and 4,100 are below 40 mg. The feasibility bound is 0.25. Which of these doses pass the estimated constraint?
At 20 mg, the estimated posterior overdose probability is \(1{,}200/10{,}000=0.120\), so 20 mg passes. At 30 mg, it is \(2{,}450/10{,}000=0.245\), also below 0.25, so 30 mg passes the estimated constraint. At 40 mg, it is \(4{,}100/10{,}000=0.410\), so 40 mg fails. A proposed assignment of 40 mg would therefore be excluded by this bound; 30 mg would pass the feasibility screen.
The estimate at 30 mg is close to the cutoff. Its approximate Monte Carlo standard error is \(\sqrt{0.245(1-0.245)/10{,}000}\approx0.00430\). That is a measure of simulation precision, not uncertainty in the posterior probability itself. Because the estimated value is close to the decision threshold, more posterior draws or a more conservative numerical check may be appropriate before making the assignment decision.
Diagnostics, Alternatives, and Limitations
A feasibility calculation is only as reliable as the posterior distribution used to obtain it. Check that the dose-toxicity model supports a meaningful MTD, that the posterior simulation has converged adequately, and that the estimated probability is stable enough for the decision being made. For simulation-based calculations, inspect mixing and effective sample size, and recompute the relevant tail probability with additional draws when the estimate is near the cutoff. A small Monte Carlo standard error cannot correct a misspecified model or an unsuitable prior.
EWOC is also distinct from a rule that selects the dose whose estimated toxicity probability is closest to the target. Such a target-matching rule ranks doses by closeness to the target; EWOC first imposes a posterior safety eligibility condition. Passing the condition does not mean a dose is optimal, and failing it does not quantify every aspect of clinical risk. A protocol may also include safety exclusions or limits on escalation. Those restrictions are separate: a dose must satisfy all applicable rules, not just the EWOC bound.
The main limitations follow from what the bound measures. It depends on the posterior distribution of the MTD, and so can be sensitive to modeling choices and sparse data. It is not a guarantee that toxicity will not occur, nor does it protect against every form of model misspecification. If the dose-toxicity relationship is not monotone or the target crossing is not uniquely defined, the usual single-MTD interpretation may fail. Finally, the bound only screens assignments; it does not resolve what to do if no candidate dose satisfies the constraint. That response must be specified by the trial design.
Check Your Understanding
Answer each question, then compare your work with the solutions.
- If \(P(M<d\mid D)=0.18\) and the feasibility bound is 0.20, does the dose pass the EWOC constraint? State what the probability means.
- For the scaled \(\operatorname{Beta}(2,2)\) posterior in the examples, calculate the posterior overdose probability at 40 mg and assess feasibility when the bound is 0.25.
- Explain why a dose above a dose that fails the feasibility constraint must also fail, when both are evaluated under the same posterior.
- A posterior-draw estimate of the overdose probability is 0.248, based on 1,000 draws, with a bound of 0.25. Give one reason to check the estimate before relying on it.
Solutions. (1) Yes. Since \(0.18\leq0.20\), it passes; the value is the posterior probability, given the interim data, that the MTD is below the proposed dose. (2) At 40 mg, the probability is \(3(0.40)^2-2(0.40)^3=0.48-0.128=0.352\). Since \(0.352>0.25\), it fails. (3) For a higher dose \(d_2\) than \(d_1\), the event \(M<d_1\) is contained in \(M<d_2\), so the posterior probability cannot decrease. (4) The estimate is close to the cutoff and is based on a finite number of draws, so Monte Carlo error could affect the comparison; additional draws or a stability check can clarify the numerical estimate.