Sensitivity of EWOC to Its Feasibility Bound

Why Vary the Feasibility Bound?

EWOC uses a bound on posterior risk to limit how far above the maximum tolerated dose (MTD) a trial assigns participants. The bound is a design choice: a more conservative value can restrict assignment to lower doses, while a larger value permits more posterior probability that the assigned dose exceeds the MTD. Assessing sensitivity means varying this bound while keeping the prior, toxicity target, observed data, and dose-assignment rules fixed. It does not mean refitting the model or changing the evidence.

This distinction matters in practice. If a small change in the bound causes the recommended dose to jump several levels, the assignment is sensitive to that choice. If a dose remains selected across a broad range of plausible bounds, the decision is more robust to this particular design choice. Neither finding establishes that the posterior model is correct; sensitivity analysis isolates one source of variation rather than resolving all uncertainty.

The strict inequality \(M<d_j\) describes assignment above the MTD; a dose equal to the MTD is not counted as exceeding it. The posterior distribution of \(M\) is computed from the model, prior, target, and data. In this tutorial, those ingredients stay fixed, so each \(r_j\) stays fixed as \(\xi\) changes. Because the events \(\{M<d_j\}\) are nested as dose increases, the probabilities satisfy \(r_1\leq r_2\leq\cdots\leq r_J\).

Assignment Thresholds and Monotonicity

Each candidate dose has a fixed threshold \(r_j\): it becomes feasible at precisely the bounds \(\xi\geq r_j\). Since EWOC selects the highest feasible dose, the assignment changes only when \(\xi\) reaches one of these thresholds. Between consecutive distinct thresholds, the selected dose stays the same. If two or more doses have the same threshold, the rule can move past them together.

To justify the result, let \(F(\xi)=\{j:r_j\leq\xi\}\) be the set of feasible dose indices. If \(j\in F(\xi_1)\), then \(r_j\leq\xi_1\leq\xi_2\), so \(j\in F(\xi_2)\). Thus \(F(\xi_1)\subseteq F(\xi_2)\). When the sets are nonempty, their largest indices obey \(\max F(\xi_1)\leq\max F(\xi_2)\). If the first set is empty, the rule stops; the second set is either also empty or permits an assignment. This argument relies only on the fixed probabilities and the rule of selecting the highest feasible dose.

For a strictly increasing sequence of thresholds, dose \(d_j\), for \(j<J\), is selected when \(r_j\leq\xi<r_{j+1}\); the highest dose \(d_J\) is selected when \(\xi\geq r_J\). If the lowest candidate dose is not feasible, stopping occurs when \(\xi<r_1\). Equal thresholds can make some of these intervals empty. These boundaries make sensitivity easy to summarize: the threshold list, rather than a long list of arbitrary trial runs, fully describes how assignments vary with \(\xi\).

Stopping Is Also Sensitive to the Bound

The stopping rule is part of the same threshold structure. If even the lowest candidate dose has \(r_1>\xi\), no dose is feasible. Increasing \(\xi\) to \(r_1\) makes that lowest dose feasible; the decision then changes from stopping to assigning. This is a consequence of the specified feasibility rule, not evidence that the posterior model has changed. In particular, stopping at a given bound should not be interpreted as proof that every dose is toxic; it means only that no allowed candidate dose satisfies this EWOC constraint at that bound.

Assessing Robustness and Diagnosing Problems

A useful sensitivity report gives the fixed \(r_j\) values, the assignment intervals they imply, and the range of bounds considered plausible for the design. If the entire range lies within one assignment interval, the dose choice is robust to changes in \(\xi\) over that range. If the range crosses one or more thresholds, report which assignments occur and where the changes happen. A single recommended bound can still be required by the protocol, but the threshold analysis makes the consequence of that choice explicit.

The main alternative is to select a dose by how closely its estimated toxicity probability matches the target, as in target-probability CRM. That criterion and EWOC answer different decision questions: target matching emphasizes proximity to the target, whereas EWOC imposes a bound on the posterior probability that the MTD lies below the assigned dose. A dose can be close to the target estimate yet fail the EWOC feasibility constraint. Sensitivity to \(\xi\) should therefore be evaluated using the EWOC rule itself, rather than inferred from a target-matching recommendation.

Several checks help diagnose an analysis that may be going wrong:

• Check cumulative ordering. Since \(d_j\) increase, the events \(M<d_j\) are nested. A reported sequence with \(r_{j+1}<r_j\) signals a calculation, indexing, or implementation error.
• Check the fixed inputs. If the posterior probabilities change across the sensitivity runs, something besides the feasibility bound has changed, contrary to the intended comparison.
• Check the equality rule. This tutorial treats \(r_j=\xi\) as feasible. Using a strict inequality instead changes decisions exactly at threshold values.
• Check the candidate set. A protocol restriction can remove doses from consideration. The highest feasible dose then means the highest among the allowed candidates, not necessarily every dose on the original grid.

Sensitivity analysis has limits. It does not assess sensitivity to the prior, toxicity model, target, or observed data; those require separate analyses. Nor does a broad interval of \(\xi\) with an unchanged assignment guarantee a safe or effective dose. It establishes only that the discrete assignment is insensitive to the feasibility bound over that interval, conditional on the fixed posterior and candidate set.