Stopping CRM When No Dose Is Near Target

When a Dose Recommendation Is Not Enough

A CRM can rank candidate doses by how close their estimated toxicity probabilities are to the target. But ranking always produces a closest candidate, even when every candidate is far from the target. If a trial must not recommend an inadequately matched dose, the design needs a separate stopping rule: a prespecified condition under which it stops without selecting a dose.

The distinction matters especially at the ends of a discrete dose grid. If estimated toxicity rises across the grid but remains below target, the highest dose is still the closest. If all estimates exceed target, the lowest dose may be closest. Either boundary can win a ranking without being sufficiently close to the target.

The tolerance \(\delta\) is measured in absolute probability, not as a percentage change relative to \(\phi\). For instance, \(\delta=0.05\) means that an estimated toxicity probability within five percentage points of the target is considered near enough for this rule. The tolerance is a design choice, not a quantity inferred automatically from the CRM.

Assumptions and the Point-Estimate Rule

The rule is applied after updating the CRM with the available dose and toxicity outcomes. It assumes that the target \(\phi\), dose grid, tolerance \(\delta\), model, prior, and rule for calculating \(m_j\) have been specified before the decision. The dose-toxicity model and data must also be meaningful for the decision: for example, the relevant binary toxicity outcome should have been assessed over its defined observation window for the participants whose outcomes are used. An unresolved outcome is not equivalent to an observed absence of toxicity.

The point-estimate rule compares posterior means, not the observed toxicity proportions at each dose. In a small trial, observed proportions can be highly variable, and a dose may have only a few participants. The CRM uses the likelihood and prior under its model to produce posterior toxicity estimates across the dose grid. The posterior means are useful summaries, but they do not describe all posterior uncertainty; a probability-based rule can address uncertainty more directly.

Write \(d_j=|m_j-\phi|\) for the absolute distance from the target at dose \(j\), and let \(d_{\min}=\min_j d_j\). The decision follows directly:

If the rule continues, a common recommendation is the dose minimizing \(d_j\), with a prespecified way to resolve ties. Passing the stopping criterion does not imply that every other trial condition is satisfied; it only says that this particular “no dose near target” rule has not triggered.

Why a Boundary Recommendation Can Be Misleading

Without a stopping condition, a nearest-target rule selects \(\arg\min_j |m_j-\phi|\) from the available grid. This always returns a dose when the grid is nonempty. A stopping rule adds a separate acceptability check: even the minimum distance must be no larger than \(\delta\). The check is therefore not a different way to rank doses; it decides whether to make a recommendation at all.

The following comparison makes the distinction explicit. Both examples use the same target and tolerance, but the second has no candidate close enough.

A dose grid can also have all estimates above target. In that case, a forced nearest-dose rule may select its lowest dose, while a stopping rule may still decline to recommend it if its distance exceeds \(\delta\). “Boundary” describes the dose’s position on the grid; it does not establish that the dose is acceptable. Likewise, stopping for lack of a near-target dose is not the same as choosing a boundary dose.

A Posterior-Probability Version

A point-estimate criterion can hide posterior uncertainty. A probability-based version asks how likely it is, under the fitted CRM posterior, that at least one candidate dose is near target. Define the event that the dose grid contains a near-target dose as \(A=\{\text{there exists a }j\text{ such that }|p_j(\alpha)-\phi|\le\delta\}\), and let \(r=P(A\mid D)\). For a prespecified threshold \(\gamma\), a possible rule is to stop when \(r<\gamma\). The threshold \(\gamma\) expresses how much posterior support the design requires for the existence of a near-target candidate.

This rule and the posterior-mean rule answer different questions. The first asks whether any posterior mean is near target. The second asks whether the posterior probability is sufficiently high that some dose on the grid is near target. A dose may have a posterior mean close to target even when the posterior distribution is broad; conversely, posterior draws may often put some dose near target even if no single posterior mean is within the tolerance. These rules should not be treated as interchangeable.

Derivation, Calibration, and Diagnostics

The point-estimate rule can be understood as adding a threshold to the nearest-dose calculation. First, the nearest-dose calculation gives \(d_{\min}=\min_j |m_j-\phi|\). The acceptable set is \(S=\{j:|m_j-\phi|\le\delta\}\). If \(S\) is empty, then every distance exceeds \(\delta\), so their minimum exceeds \(\delta\). Conversely, if \(d_{\min}\le\delta\), at least one dose attains that minimum and belongs to \(S\). Thus:

This equivalence justifies the stopping condition; it does not justify any particular value of \(\delta\). Investigators should select and document the tolerance in relation to the clinical meaning of a difference in toxicity probability and the available dose grid. They should also specify whether a dose exactly at the tolerance boundary is acceptable: the rule above treats \(|m_j-\phi|=\delta\) as near enough. Posterior-probability rules likewise require a prespecified \(\gamma\), with the boundary convention stated.

At each planned decision, useful checks include verifying the dose labels and target, confirming the posterior calculations use the intended observed outcomes, and inspecting all dose-specific estimates rather than only the selected dose. For the probability rule, verify that posterior draws were generated and counted consistently and that the event is “at least one dose” on the stated grid. Recompute the decision from the displayed values; a reported recommendation inconsistent with the threshold may indicate an implementation or transcription error.

A stopping decision can be sensitive to modest changes in \(\delta\) or \(\gamma\), particularly when the nearest dose lies near the threshold. Report the rule and the distances or posterior probability that triggered it. Such sensitivity is a diagnostic of how borderline the decision is; it is not a reason to change the prespecified threshold after seeing the results. If a stopping decision appears implausible, investigate the data, model fit, and grid rather than silently substituting a different criterion.

Alternatives and Limitations

These are alternative decision rules, not guarantees that the target toxicity rate will be achieved. A CRM model may be an imperfect description of the dose-toxicity relationship, and a finite grid may not contain a dose close to the target even if a suitable dose exists outside that grid. A rule that stops because no candidate is near target answers a limited question about the specified grid and criterion. It does not, by itself, establish why the grid failed to contain a near-target dose.

A common mistake is to reason, “The highest dose is closest to target, so the trial should recommend it.” That conclusion follows only from a forced nearest-dose rule. Under a stopping rule, the correct next step is to compare the minimum distance with the tolerance—or, for the probability version, compare \(r\) with \(\gamma\). If the stopping threshold is crossed, the decision is to stop without a dose recommendation, not to relabel the closest boundary dose as acceptable.