Two Rules, Two Questions
EWOC and the target-probability CRM rule use the same posterior information about toxicity, but they ask different questions of it. EWOC asks whether a dose is sufficiently likely to be at or below the maximum tolerated dose (MTD). The target-probability CRM rule asks which dose has an estimated toxicity probability closest to the target rate. A safety constraint and a target-matching criterion are not interchangeable: one dose can match the target on average while retaining substantial posterior probability of exceeding it.
Let the available doses be ordered \(d_1<d_2<\cdots<d_J\), let \(p_j\) denote the probability of dose-limiting toxicity at dose \(d_j\), and let \(p_T\) be the target toxicity probability. Write \(D\) for the trial data observed so far. The posterior distribution of the toxicity curve, conditional on \(D\), is the shared input to both rules. We assume here that toxicity probability is nondecreasing with dose in each posterior curve, so that a higher dose does not have lower toxicity probability.
The calculations below use the standard posterior-mean version of target-probability CRM: it selects the dose whose posterior mean toxicity probability is closest to \(p_T\). Other estimators or additional clinical restrictions can be used in practice, but they are not part of this comparison. For EWOC, let \(\alpha_F\) denote the feasibility bound. A dose is feasible when its posterior probability of being at or below the MTD is at least \(1-\alpha_F\); among feasible doses, the usual discrete-grid rule selects the highest dose. If no dose passes, the design must follow a prespecified stopping or other safety policy rather than treating an infeasible dose as feasible.
Formal Rules and Their Relationship
The EWOC condition is a posterior constraint, not an optimization of distance from the target. Its probability has a direct event interpretation: it is the posterior probability that dose \(d_j\) does not exceed the MTD. The CRM rule summarizes the posterior toxicity distribution at each dose by its mean and compares that number with the target. The posterior mean alone does not reveal how much posterior mass lies above \(p_T\).
The link between EWOC’s event and the MTD index follows directly from its definition. Under monotonicity, if \(\xi\geq j\), then dose \(d_j\) is at or below the largest dose satisfying \(p_j\leq p_T\). Conversely, if \(d_j\) is at or below the MTD, that largest qualifying index must be at least \(j\). Thus the events are equivalent for every posterior curve, and therefore have equal posterior probabilities:
This identity justifies computing EWOC feasibility either from posterior draws of \(p_j\) or by counting how often the sampled MTD index is at least \(j\). It does not make the CRM rule equivalent to EWOC. In general, the mean \(E[p_j\mid D]\) cannot determine \(P(p_j\leq p_T\mid D)\): different posterior distributions can have the same mean but very different probabilities of exceeding the target.
Worked Example: CRM Selects a Dose That EWOC Does Not Pass
Suppose there are three doses, the target is \(p_T=0.25\), and the posterior has two possible toxicity curves. With posterior probability 0.60 the probabilities at \(d_1,d_2,d_3\) are \(0.03,0.12,0.24\); with probability 0.40 they are \(0.10,0.28,0.45\). These curves are nondecreasing with dose, as assumed. Let \(\alpha_F=0.20\), so EWOC requires feasibility of at least \(0.80\).
For CRM, take the posterior mean at each dose. At \(d_1\), it is \(0.60(0.03)+0.40(0.10)=0.058\); at \(d_2\), it is \(0.60(0.12)+0.40(0.28)=0.184\); and at \(d_3\), it is \(0.60(0.24)+0.40(0.45)=0.324\). Their absolute distances from the target are \(0.192\), \(0.066\), and \(0.074\), respectively. The closest is \(d_2\), so CRM selects \(d_2\).
For EWOC, the MTD indices in the two posterior states are 3 and 1: in the first state all three doses are at or below 0.25; in the second only \(d_1\) is. The feasibility probabilities at \(d_1,d_2,d_3\) are therefore \(1.00,0.60,0.60\). Only \(d_1\) reaches the required \(0.80\), so EWOC selects \(d_1\). The rules differ because the posterior mean toxicity at \(d_2\) is close to target, while only 60% of posterior curves place that dose at or below the MTD.
Worked Example: EWOC Selects a Higher Dose Than CRM
Now use two doses, target \(p_T=0.25\), and feasibility bound \(\alpha_F=0.25\), requiring posterior feasibility of at least 0.75. In 75% of posterior states, the toxicity probabilities at \(d_1,d_2\) are \(0.23,0.24\). In the remaining 25%, they are \(0.23,0.60\). Both curves are nondecreasing.
The posterior mean at \(d_1\) is \(0.75(0.23)+0.25(0.23)=0.230\). At \(d_2\), it is \(0.75(0.24)+0.25(0.60)=0.330\). The distances to target are \(0.020\) at \(d_1\) and \(0.080\) at \(d_2\), so target-probability CRM selects \(d_1\).
At \(d_2\), 75% of posterior states have toxicity probability at or below 0.25, so its EWOC feasibility probability is exactly 0.75. At \(d_1\), feasibility is 1.00. Both doses pass; EWOC selects the highest passing dose, \(d_2\). The recommendation depends on the stated feasibility bound: here EWOC permits a 25% posterior probability of being above the MTD. This example also shows why a posterior mean above target does not automatically make a dose infeasible under a probability constraint.
Worked Example: Correctly Calculating Both Rules From Posterior States
Consider three posterior states with probabilities 0.50, 0.30, and 0.20. Their toxicity probabilities at doses \(d_1,d_2,d_3\) are, respectively, \((0.05,0.16,0.28)\), \((0.08,0.22,0.40)\), and \((0.12,0.30,0.50)\). Let \(p_T=0.25\) and \(\alpha_F=0.25\). Each state is monotone in dose.
The posterior mean at \(d_1\) is \(0.50(0.05)+0.30(0.08)+0.20(0.12)=0.025+0.024+0.024=0.073\). At \(d_2\), it is \(0.50(0.16)+0.30(0.22)+0.20(0.30)=0.080+0.066+0.060=0.206\). At \(d_3\), it is \(0.50(0.28)+0.30(0.40)+0.20(0.50)=0.140+0.120+0.100=0.360\). Their distances from 0.25 are \(0.177\), \(0.044\), and \(0.110\). CRM therefore selects \(d_2\).
The MTD indices in the three states are 2, 2, and 1. The resulting feasibility probabilities are \(P(\xi\geq1\mid D)=1.00\), \(P(\xi\geq2\mid D)=0.50+0.30=0.80\), and \(P(\xi\geq3\mid D)=0\). Since the required feasibility is 0.75, EWOC selects \(d_2\), the highest passing dose. The methods agree here, but for distinct reasons: CRM’s posterior mean is closest to target at \(d_2\), while EWOC finds that \(d_2\) passes its posterior constraint and \(d_3\) does not.
Assumptions, Diagnostics, and Practical Differences
These rules require a posterior distribution from a specified dose-toxicity model and a clearly stated target \(p_T\). The examples also require a dose ordering and monotone toxicity curves, so that the MTD index and the event equivalence used for EWOC are well defined. In a model where monotonicity is not imposed, the MTD can still be defined as the highest dose with \(p_j\leq p_T\), but a lower dose could then be above target while a higher dose is below it. In that case, interpreting feasibility solely as “dose \(d_j\) is below the MTD” needs care, and the assumed dose-response structure should be examined.
In computation, posterior simulation provides a straightforward diagnostic. For each posterior draw, record the toxicity probability at every dose and the corresponding MTD index. Check that sampled curves obey the model’s intended ordering, estimate each \(E[p_j\mid D]\) by averaging toxicity probabilities, and estimate EWOC feasibility by the fraction of draws with \(\xi\geq j\). Monte Carlo uncertainty can matter near the feasibility cutoff: if the estimated probability is close to \(1-\alpha_F\), more posterior draws may be needed before classifying the dose reliably. Also check whether the selected dose lies at the grid boundary. CRM choosing the highest dose can mean the grid has no dose whose estimated toxicity is closer to target; EWOC choosing its highest feasible dose means only that no higher grid dose passes the specified constraint.
| Feature | Target-probability CRM | EWOC |
|---|---|---|
| Quantity compared across doses | Posterior mean toxicity and its distance from \(p_T\) | Posterior probability that the dose is at or below the MTD |
| Selection logic | Choose the dose with the smallest target distance | Keep doses passing the feasibility bound; choose the highest passing one |
| Primary interpretation | Which dose has estimated toxicity closest to target? | Which doses have enough posterior support for being no higher than the MTD? |
| What the rule alone does not guarantee | A bound on posterior probability of exceeding the target | That the selected dose’s posterior mean is closest to target |
Neither rule removes dependence on the model, prior information, or observed data. A sparse data set can leave posterior toxicity estimates and MTD probabilities strongly influenced by the prior. A posterior mean can conceal a broad or skewed distribution, and an EWOC feasibility estimate can be unstable near its cutoff. A dose grid can also be coarse: the closest available target match may still be clinically distant from \(p_T\), or the highest feasible dose may be far below it. These rules specify different selection criteria; they do not by themselves establish that a trial is safe or that a dose is clinically optimal.
Common Mistake
The earlier material on EWOC feasibility bounds and CRM dose selection provides the context for these criteria. The practical decision is not to declare one rule universally preferable, but to state which criterion governs the recommendation and to report the other quantity when it helps clarify the trade-off.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- With target \(p_T=0.20\), suppose the posterior MTD index has probabilities \(P(\xi=1\mid D)=0.20\), \(P(\xi=2\mid D)=0.50\), and \(P(\xi=3\mid D)=0.30\). If \(\alpha_F=0.25\), which doses pass EWOC feasibility?
- At a dose, posterior toxicity probability is 0.10 with probability 0.70 and 0.50 with probability 0.30. Find its posterior mean and its probability of being above a target of 0.20. Why are these different summaries?
- In one sentence, distinguish the decision question asked by target-probability CRM from the question asked by EWOC.
- Name one assumption used to equate EWOC feasibility with \(P(\xi\geq j\mid D)\), and explain what it contributes.
Solutions. (1) Feasibility is 1.00 for \(d_1\), \(P(\xi\geq2\mid D)=0.50+0.30=0.80\) for \(d_2\), and \(P(\xi\geq3\mid D)=0.30\) for \(d_3\). The required probability is \(1-\alpha_F=0.75\), so \(d_1\) and \(d_2\) pass; EWOC selects \(d_2\). (2) The mean is \(0.70(0.10)+0.30(0.50)=0.22\). The probability of being above 0.20 is 0.30. The mean averages toxicity values, while the probability counts posterior mass on one side of the target. (3) CRM selects the dose with posterior mean toxicity closest to the target; EWOC selects the highest dose whose posterior probability of being at or below the MTD meets its feasibility requirement. (4) Monotonicity of dose-toxicity curves makes the event that dose \(d_j\) is at or below the MTD equivalent to \(\xi\geq j\), allowing feasibility to be computed from posterior MTD indices.