Dose-Skipping Constraints in CRM Escalation

Why a CRM Recommendation May Not Be the Next Dose

A continual reassessment method (CRM) uses accumulating dose-limiting toxicity (DLT) data to estimate toxicity across ordered dose levels. Its model can recommend a dose several levels above or below the dose just given. That recommendation is a model-based target, not automatically the permitted dose for the next cohort. A trial protocol may restrict transitions so that participants are not exposed to an abrupt escalation or an otherwise disallowed dose jump.

The key operational distinction is between what the fitted model recommends and what the transition rules allow next. Applying a restriction does not erase the CRM estimate or change the posterior. It narrows the set of doses that can be assigned at the next decision. Keeping these steps separate makes the decision auditable: the model’s preferred dose can be recorded alongside the constrained action.

Definition: Suppose the trial has ordered dose levels \(d_1,\ldots,d_J\), and the current cohort received \(d_c\). Let \(\hat p_j^{CRM}\) be the CRM posterior mean DLT probability at dose \(d_j\), given the data available at the decision, and let \(\phi\) be the target DLT probability. Under the closest-to-target rule, the unconstrained recommendation is the dose minimizing \(\lvert \hat p_j^{CRM}-\phi\rvert\) across the doses considered by the protocol. A transition restriction then limits which dose may be assigned next.

This definition assumes that the dose levels have a meaningful order and that the CRM provides toxicity estimates on that scale. The usual interpretation also relies on a prespecified target and a prespecified way to handle ties. For the examples below, ties are resolved in favor of the lower dose. The examples treat the posterior mean estimates as increasing with dose, as expected under the monotone dose-toxicity model used for this setting.

Writing the Transition Rule Precisely

An adjacent-dose rule allows a transition only to the current dose or one of its immediate neighbors. With dose indices, the allowed set before any separate safety screen is

\mathcal{A}_{adj}(c)=\{j:1\leq j\leq J,\ |j-c|\leq 1\}.

A no-skipping escalation rule is often stated more narrowly: if the next assignment is an escalation, it may not exceed the next dose level above the current one. Its upward restriction is \(j\leq c+1\). This rule alone does not necessarily restrict downward moves; a protocol may permit a direct de-escalation to a lower dose. By contrast, a symmetric adjacent-dose rule limits both upward and downward moves. These are related but not identical policies, so the protocol must specify which applies.

A transition rule also needs an initial-dose convention. Before a dose has been administered, “adjacent to the current dose” is undefined. A protocol may specify a starting dose directly, or define a separate initial escalation procedure. Once a current dose exists, the rule can be applied using its index.

Safety eligibility is a separate filter, not a substitute for the transition rule. For example, a protocol may deem dose \(d_j\) eligible only when its posterior probability of exceeding a prespecified overdose boundary is below a prespecified threshold. The next dose must satisfy both the transition restriction and any eligibility requirement. If those sets have no overlap, the rules do not authorize a dose merely because the CRM has a preferred level.

Constrained Recommendation and Its Justification

Let \(\mathcal{E}\) be the set of doses meeting the protocol’s safety eligibility criteria. Under a symmetric adjacent rule, the feasible next-dose set is \(\mathcal{F}(c)=\mathcal{A}_{adj}(c)\cap\mathcal{E}\). Under an escalation-only no-skipping rule, the transition set is defined by that rule instead. The constrained recommendation minimizes the same target-distance criterion over the feasible set:

j^*_{\text{next}}=\arg\min_{j\in\mathcal{F}(c)} \left|\hat p_j^{CRM}-\phi\right|.

This follows directly from the decision being made. The CRM criterion ranks candidate doses according to how close their estimated toxicity is to the target. The protocol first removes doses that cannot be assigned at this decision; among the remaining doses, the dose with the smallest criterion value is selected. Thus, an unconstrained optimum outside \(\mathcal{F}(c)\) cannot be assigned next under the stated rule, even if it remains the model’s preferred target.

If \(\mathcal{F}(c)\) is empty, the minimization has no solution. This is not a reason to silently choose a dose outside the rules. The trial must follow its prespecified action for this situation, such as holding enrollment, using a defined stopping criterion, or convening the designated review process. Those actions must come from the protocol; they cannot be inferred from the CRM target-distance calculation alone.

Worked Example: An Upward Recommendation That Skips Two Levels

Suppose five ordered doses are under consideration, the current cohort received \(d_1\), and the target DLT probability is \(\phi=0.25\). For this hypothetical decision, the CRM posterior mean DLT probabilities are 0.05, 0.12, 0.22, 0.38, and 0.61 at \(d_1\) through \(d_5\), respectively. Assume the protocol permits upward moves of at most one level and that \(d_2\) passes all separate safety eligibility checks.

The target distances are \(\lvert0.05-0.25\rvert=0.20\), \(\lvert0.12-0.25\rvert=0.13\), \(\lvert0.22-0.25\rvert=0.03\), \(\lvert0.38-0.25\rvert=0.13\), and \(\lvert0.61-0.25\rvert=0.36\). The unconstrained CRM recommendation is \(d_3\), because its distance, 0.03, is the smallest.

From current dose \(d_1\), the no-skipping escalation rule permits at most \(d_2\). Therefore \(d_3\), although it is the model’s preferred dose, is not permitted for the next cohort. Among the allowed upward options, \(d_2\) is the target-seeking escalation (and it has distance 0.13); the next assignment is \(d_2\), assuming the protocol’s other conditions are met. The model estimates are unchanged: the difference between \(d_3\) as the unconstrained target and \(d_2\) as the permitted next step should be documented.

Worked Example: A Nonadjacent Downward Recommendation

Consider five ordered doses with the current cohort at \(d_5\), target \(\phi=0.25\), and hypothetical CRM posterior mean DLT probabilities 0.04, 0.12, 0.24, 0.46, and 0.68. Suppose there is no additional safety exclusion for this calculation.

The target distances at \(d_1,\ldots,d_5\) are 0.21, 0.13, 0.01, 0.21, and 0.43. The unconstrained CRM recommendation is \(d_3\), with distance \(\lvert0.24-0.25\rvert=0.01\). This is a two-level downward move from \(d_5\).

If the protocol uses a symmetric adjacent-dose rule, the feasible dose indices are \(\{4,5\}\). Their distances are 0.21 and 0.43, so the constrained recommendation is \(d_4\). If instead the protocol uses only an escalation-only no-skipping rule and permits unrestricted de-escalation, that rule does not by itself prohibit a move from \(d_5\) to \(d_3\). The action therefore depends on the precise protocol language, not on the phrase “no skipping” interpreted without its direction and scope.

Worked Example: Applying Transition and Safety Restrictions Together

Suppose the current dose is \(d_2\), the target is \(\phi=0.25\), and the hypothetical CRM posterior mean DLT probabilities at \(d_1,\ldots,d_5\) are 0.06, 0.13, 0.23, 0.39, and 0.58. The corresponding target distances are 0.19, 0.12, 0.02, 0.14, and 0.33, so the unconstrained recommendation is \(d_3\).

Assume a symmetric adjacent-dose rule. The transition set is \(\{d_1,d_2,d_3\}\). Now suppose the protocol additionally requires \(P(p_j>0.30\mid\text{data})\leq0.25\), and the hypothetical posterior probabilities at \(d_1,d_2,d_3\) are 0.01, 0.05, and 0.08, respectively. All three transition-eligible doses pass this screen. Comparing their target distances, \(d_3\) is the constrained recommendation, with distance 0.02.

If the posterior probability at \(d_3\) were instead 0.31, that dose would fail the stated safety requirement because \(0.31>0.25\). The feasible set would then contain only \(d_1\) and \(d_2\), and \(d_2\) would be selected because \(0.12<0.19\). The safety screen changes the feasible set; it does not change the target distance or the CRM posterior estimate.

Diagnostics, Alternatives, and Limitations

A reliable implementation should make the sequence of decisions visible. At each cohort decision, record the current dose, target, model-based estimates, unconstrained recommendation, transition-eligible doses, safety-eligible doses, and final action. This makes discrepancies easy to audit and helps detect a rule implemented in the wrong direction. In particular, confirm whether the restriction applies only to escalation or also to de-escalation, and whether remaining at the current dose is permitted.

Several patterns can signal that the procedure is being applied incorrectly or that the operational policy needs review:

The recorded next dose is not in the protocol-defined transition set.
A dose that fails the safety screen is selected because it is closest to the target.
The reported “CRM recommendation” is the constrained dose, with no record of the distinct unconstrained target.
A team treats an escalation-only rule as if it automatically limits downward moves, or assumes a symmetric adjacent rule when the protocol allows wider de-escalation.
The feasible set is empty, but a dose is nevertheless selected without a prespecified action for that case.

The main alternative is to use the unconstrained model recommendation directly, assigning the dose closest to the target without an adjacency limit, subject to whatever safety rules the protocol specifies. This can move more quickly toward the model’s target but permits larger transitions. Another approach is a rule-based escalation design with fixed movement rules rather than a model-selected target. That may be operationally simpler, but it does not use the CRM’s current dose-toxicity estimates in the same way. These approaches differ in the decision rule; none removes the need to define safety eligibility and the protocol’s response to an empty feasible set.

An adjacency restriction is an operational safeguard, not evidence that the model’s nonadjacent estimate is wrong. It can slow movement toward a target, and the dose assigned next may be less close to the target than the unconstrained recommendation. Conversely, the restriction does not guarantee safety: the adjacent dose may still be too toxic under the model or may fail a separate overdose criterion. Its effect is limited to controlling transitions as specified. The model, target, safety screen, and transition rule should therefore be reviewed as distinct parts of the decision.

Common Mistake

Pitfall: “The CRM recommends \(d_4\), so the next cohort should receive \(d_4\); applying a no-skipping rule would alter the model.” This confuses a statistical recommendation with a protocol-constrained action. The posterior and the unconstrained recommendation remain as calculated. The rule restricts the next-cohort choice to a feasible set, and the closest-to-target dose is selected within that set. If no dose is feasible, follow the protocol’s specified action rather than overriding the restriction.

Key takeaway: Calculate and report the unconstrained CRM target first. Then apply the exact transition rule and any separate safety eligibility criteria to form the feasible next-dose set. The permitted next dose is the best target-seeking option within that set, not necessarily the unconstrained CRM target.

Check Your Understanding

Use the stated target-distance rule and distinguish the transition policy in each question.

The current dose is \(d_1\). Posterior mean DLT probabilities at \(d_1,d_2,d_3,d_4\) are 0.08, 0.18, 0.27, and 0.44, with target \(\phi=0.25\). Under an upward no-skipping rule, which dose is the unconstrained recommendation, and which dose may be assigned next?
The current dose is \(d_4\). The unconstrained recommendation is \(d_2\). Does an escalation-only no-skipping rule, by itself, prohibit this de-escalation? What would a symmetric adjacent-dose rule permit?
The current dose is \(d_2\), and a symmetric adjacent rule permits \(d_1,d_2,d_3\). Their target distances are 0.10, 0.06, and 0.02. If \(d_3\) fails the protocol’s safety eligibility screen, which dose is selected?
Under a symmetric adjacent rule, the current dose is \(d_3\), but both \(d_2\) and \(d_3\) fail the applicable safety screen. The only other dose passing the screen is \(d_1\). Is \(d_1\) a feasible next dose under the stated rule, and what should happen if the feasible set is empty?

Solutions: (1) The target distances are 0.17, 0.07, 0.02, and 0.19, so the unconstrained recommendation is \(d_3\). From \(d_1\), the upward no-skipping rule permits only an escalation as high as \(d_2\), so the next dose is \(d_2\), assuming it passes other protocol checks. (2) No. An escalation-only rule restricts upward transitions and does not, by itself, prohibit a downward move from \(d_4\) to \(d_2\). A symmetric adjacent rule permits only \(d_3\) or \(d_4\). (3) After excluding \(d_3\), compare the remaining distances: 0.10 at \(d_1\) and 0.06 at \(d_2\). Select \(d_2\). (4) \(d_1\) is not feasible because \(\lvert1-3\rvert=2>1\), even though it passes the safety screen. If no dose satisfies both restrictions, the protocol’s prespecified response applies; a dose outside the feasible set must not be assigned by this calculation.