Selecting the EWOC Feasibility Quantile

What the Feasibility Quantile Controls

In EWOC, a proposed dose is assessed against uncertainty about the maximum tolerated dose (MTD). If the proposed dose \(d\) exceeds the true MTD \(M\), it is treated as being above the MTD. The EWOC feasibility rule limits the posterior probability of that event. Its threshold, often denoted by \(\alpha\), is therefore a safety-aggressiveness control: smaller values restrict the trial to doses with less posterior probability of exceeding the MTD, while larger values permit more risk under the model.

The threshold does not come from a posterior calculation alone. It is a design choice that must be made in the clinical and operational context. Once the posterior distribution of \(M\) is fixed, changing \(\alpha\) changes which doses satisfy the rule; it does not change that posterior distribution.

The event is written \(M<d\), rather than \(M\leq d\): equality is not classified as a dose above the MTD under this convention. Because \(G(d\mid D)\) is nondecreasing in \(d\), feasible doses form an initial portion of the dose range. When the bound is within that range, doses at or below it satisfy the rule, and doses above it do not. The bound can be understood as a generalized lower-tail quantile of the posterior MTD distribution, with the strict-event convention made explicit.

This is a posterior statement conditional on the data and the model. It is not the probability that a particular patient will experience toxicity, nor is it a frequentist guarantee that the trial will never treat above the true MTD. Its interpretation depends on whether the model’s MTD is a meaningful safety boundary for the trial.

How the Threshold Changes the Bound

The central comparison follows directly from the feasibility condition. If \(\alpha_1\leq\alpha_2\), then every dose feasible at \(\alpha_1\) is also feasible at \(\alpha_2\), because \(G(d\mid D)\leq\alpha_1\) implies \(G(d\mid D)\leq\alpha_2\). Thus the feasible set at the smaller threshold is contained in the set at the larger threshold, and, whenever both bounds are defined, \(b_{\alpha_1}\leq b_{\alpha_2}\). The change need not be smooth: the bound may remain unchanged over a range of thresholds and then jump, particularly when the posterior distribution has point masses.

For a continuous posterior distribution, let \(H(d\mid D)=P(M\leq d\mid D)\) be its usual CDF. With no point masses, \(G(d\mid D)=H(d\mid D)\). If this CDF is strictly increasing across the relevant range and \(\alpha\) is attained at an interior point of the clinical range, the boundary solves \(H(b_\alpha\mid D)=\alpha\). If \(\alpha>H(d_{\max}\mid D)\), every dose on the range is feasible and \(b_\alpha=d_{\max}\); if no dose is feasible, the bound is undefined as specified above. In that case, the feasibility bound is the ordinary \(\alpha\)-quantile of the posterior MTD distribution. This equality is not a safe shortcut for distributions with atoms.

A short justification explains why the strict-event boundary is well behaved on a bounded dose range. As \(d\) increases, the event \(\{M<d\}\) can only grow, so \(G\) is nondecreasing. Also, as doses approach a value \(d\) from below, the events \(\{M<x\}\) increase to \(\{M<d\}\), so their probabilities approach \(G(d\mid D)\). Consequently, if feasible doses approach a finite supremum within the dose range, the limiting boundary itself remains feasible. This left-continuity is why the strict-event rule can include the boundary even when the posterior has a jump there.

Posterior Atoms and Quantile Conventions

A posterior distribution may have point masses, for example when model structure or posterior computation leaves probability concentrated on a finite set of MTD values. At an atom, the usual inverse-CDF quantile and the EWOC feasibility boundary can differ. This does not invalidate the rule; it means that feasibility must be checked using the event specified by the rule.

The example also shows why a reported quantile should include its convention. One common generalized inverse of the usual CDF is \(Q_\alpha^H=\inf\{d:H(d\mid D)\geq\alpha\}\). An EWOC bound, by contrast, is defined through the largest dose satisfying \(G(d\mid D)\leq\alpha\). Under a continuous posterior, these descriptions coincide when the equation \(H(d\mid D)=\alpha\) has a unique solution and that solution lies in the clinical range. If \(H\) is flat at the threshold, they can differ; with atoms, calculate or inspect the strict-event probability directly.

Choosing a Threshold in Practice

The posterior distribution determines what each candidate threshold permits, but it does not by itself determine which threshold is appropriate. Choosing \(\alpha\) requires a protocol-level judgment about the acceptable posterior risk of dosing above the MTD. A smaller \(\alpha\) is more conservative under this particular rule; a larger \(\alpha\) is more permissive. Neither direction is universally correct without considering the trial’s clinical context and objectives.

A disciplined choice separates two tasks. First, specify the acceptable risk threshold before interpreting the resulting bound as a recommendation. Second, calculate the bound from the current posterior and apply any other protocol requirements separately. Those may include dose availability, starting-dose rules, or restrictions on escalation. The feasibility threshold alone does not settle every operational decision.

Diagnostics, Alternatives, and Limitations

Before using a computed boundary, check that the posterior MTD distribution and dose scale are being interpreted consistently. The model must provide a meaningful MTD on an ordered dose scale, and the event \(M<d\) must represent the safety concern the trial intends to control. If dose-response behavior is nonmonotone or the model’s MTD is poorly defined, the feasibility calculation may not reflect the clinically relevant risk. The posterior also depends on the model and its assumptions; a numerically precise quantile does not remove model uncertainty.

For posterior draws \(M^{(1)},\ldots,M^{(B)}\), a direct approximation at a candidate dose \(d\) is:

where \(I\{\cdot\}\) equals 1 when its condition holds and 0 otherwise. This estimate should be checked near the proposed boundary, where Monte Carlo error can affect whether a dose appears to meet the threshold. With discrete posterior draws, use the strict comparison shown: replacing it with \(M^{(b)}\leq d\) changes the calculation when draws equal the candidate dose.

EWOC’s bound is not the same as choosing a dose whose predicted toxicity probability is closest to a target. That kind of decision asks about toxicity at the dose; EWOC as defined here controls posterior probability that the dose lies above the MTD. A fixed dose cap is another alternative, but it does not adapt to the current posterior in the same way. In practice, such restrictions may be combined, with each rule retaining its own interpretation.

A threshold should not be interpreted as a guarantee about realized patient outcomes. It controls a posterior probability under the assumed model and current data. If the model is misspecified, if toxicity is not adequately summarized by the MTD, or if operational rules are applied inconsistently, the nominal feasibility interpretation may not provide the intended protection. Threshold selection is therefore a design judgment, while the posterior calculation is the conditional statistical assessment.