Why Dose-Skipping and Overdose Control Are Separate Rules
A model-based dose-finding method can identify a dose that best matches a target toxicity rate even when that dose is several levels above the highest dose tested so far. Posterior overdose control adds a safety screen: it can rule out a dose when the posterior probability that its toxicity rate exceeds the target is too large. But passing that screen does not, by itself, mean the dose may be reached in one escalation.
A dose-skipping restriction addresses that separate operational question. During escalation, a cohort may not be assigned above the dose level immediately following the highest level already administered. The restriction limits the next action; it does not change the fitted model, the posterior probabilities, or the target-dose criterion. A protocol therefore needs both rules when it requires both controlled posterior risk and stepwise escalation.
For each dose \(d_j\), write \(\pi_j\) for its true dose-limiting toxicity (DLT) probability and \(p_T\) for the target toxicity probability. Given the accumulated data \(\mathcal D_n\), define its posterior overdose probability as \(R_j=P(\pi_j>p_T\mid\mathcal D_n)\). If the prespecified maximum acceptable posterior overdose probability is \(\eta\), dose \(d_j\) passes overdose control when \(R_j\leq\eta\).
A common model-based criterion is to choose the dose whose posterior mean toxicity probability is closest to \(p_T\). Let \(\hat\pi_j=E(\pi_j\mid\mathcal D_n)\). Under this criterion, first form \(\mathcal A(h)\), then choose \(j\in\mathcal A(h)\) minimizing \(|\hat\pi_j-p_T|\). A protocol may instead specify another model-based ranking rule. The restriction logic is unchanged: only doses passing both rules may be considered, and the protocol must state how ties are resolved.
Assumptions and the Decision Logic
The rule presumes an ordered, prespecified dose grid and an identified highest administered level \(h\). Posterior risks must come from the trial’s specified model and current data. When the model assumes toxicity probability is nondecreasing with dose, posterior summaries should reflect that model assumption; monotonicity is not a substitute for checking each candidate’s overdose probability. If the fitted model does not impose monotonicity, do not infer one dose’s safety from another dose’s result.
The examples below assume that relevant DLT follow-up is complete and that the posterior summaries shown are outputs of the specified model. This isolates the decision rule: pending outcomes or incomplete observation require an appropriate model and protocol procedure before updating the summaries.
Find the highest dose level ever administered, \(h\). Under a one-level cap, no assignment may exceed \(h+1\).
For each dose, compare \(R_j=P(\pi_j>p_T\mid\mathcal D_n)\) with the prespecified \(\eta\). Retain only doses with \(R_j\leq\eta\).
Keep doses satisfying both rules, then apply the protocol’s dose-selection criterion to that set. Do not replace an excluded recommendation with an unfiltered dose.
This sequence follows directly from the decision constraints. A dose is assignable under these two rules exactly when it belongs to both sets: \(j\in\mathcal N(h)\) and \(j\in\mathcal S\). Therefore, the admissible set is their intersection. Minimizing a selection score over the full grid and filtering afterward is not equivalent: the minimum might be excluded, and the remaining best eligible dose would not have been identified.
Worked Decisions
Worked Example: A Safe Nonadjacent Recommendation
Suppose a hypothetical trial has five ordered dose levels, \(p_T=0.25\), and \(\eta=0.25\). The highest administered level is \(h=2\). The model summaries are:
| Level \(j\) | \(\hat\pi_j\) | \(R_j\) | \(|\hat\pi_j-p_T|\) |
|---|---|---|---|
| 1 | 0.06 | 0.01 | 0.19 |
| 2 | 0.13 | 0.03 | 0.12 |
| 3 | 0.20 | 0.10 | 0.05 |
| 4 | 0.24 | 0.20 | 0.01 |
| 5 | 0.32 | 0.45 | 0.07 |
With the target-proximity criterion, the unrestricted model recommendation is level 4 because \(|0.24-0.25|=0.01\), the smallest distance. It passes overdose control because \(0.20\leq0.25\). Nevertheless, the one-level cap permits no assignment above \(h+1=3\). The safety-passing doses within the cap are levels 1, 2, and 3; level 4 is not an available action this round.
Among the allowed doses, level 3 is closest to target: \(|0.20-0.25|=0.05\), compared with 0.12 for level 2 and 0.19 for level 1. The decision is therefore level 3. Posterior safety of level 4 does not override the escalation restriction; it means only that the dose passes the stated overdose screen.
Worked Example: The Adjacent Dose Fails the Safety Screen
In a second hypothetical trial, let \(p_T=0.25\), \(\eta=0.25\), and \(h=3\). Suppose the model’s target-proximity recommendation is level 5. Its summaries, together with those of nearby doses, are:
| Level \(j\) | \(\hat\pi_j\) | \(R_j\) | \(|\hat\pi_j-p_T|\) |
|---|---|---|---|
| 2 | 0.10 | 0.02 | 0.15 |
| 3 | 0.18 | 0.12 | 0.07 |
| 4 | 0.23 | 0.31 | 0.02 |
| 5 | 0.26 | 0.48 | 0.01 |
The no-skipping cap is \(h+1=4\), so level 5 is excluded by the escalation rule. Level 4 is adjacent and within that cap, but it fails overdose control because \(0.31>0.25\). Level 3 passes because \(0.12\leq0.25\), and level 2 also passes. Comparing the eligible doses, level 3 has distance 0.07, which is smaller than level 2’s distance of 0.15. Thus level 3 is selected.
This decision cannot be described as “move to the next dose because it is adjacent.” Adjacency permits consideration of level 4 under the no-skipping rule, but the separate safety screen still excludes it.
Worked Example: A Stricter Threshold Changes the Allowed Set
Suppose a hypothetical trial has \(h=2\), \(p_T=0.25\), and posterior summaries \(\hat\pi_2=0.14\), \(R_2=0.04\); \(\hat\pi_3=0.20\), \(R_3=0.16\); and \(\hat\pi_4=0.24\), \(R_4=0.22\). The model’s unrestricted target-proximity recommendation is level 4, whose distance from target is \(|0.24-0.25|=0.01\). The escalation cap is level 3.
If \(\eta=0.20\), level 4 first fails the no-skipping cap, while level 3 passes both rules because \(0.16\leq0.20\). Its distance is \(|0.20-0.25|=0.05\), so level 3 is selected over level 2, whose distance is 0.11.
Now keep the posterior summaries fixed but tighten the threshold to \(\eta=0.10\). Level 3 no longer passes because \(0.16>0.10\). Level 2 passes both restrictions, so it is selected from the remaining candidates. Tightening \(\eta\) changes the safety-passing set; it does not change the no-skipping boundary or the posterior summaries. This is why the threshold and the escalation cap must be reported separately.
Diagnostics, Alternatives, and Limitations
A useful decision record shows \(h\), \(p_T\), \(\eta\), each candidate’s \(R_j\), the no-skipping boundary, the intersection of the two rules, and the selection criterion. This makes it possible to identify which rule determined the action. Recheck that \(h\) refers to the highest level actually administered, not merely the most recently assessed dose or the current cohort’s dose. Also verify that posterior risks and model summaries use the same current data and model.
Several errors are especially consequential:
- Confusing “safe” with “reachable.” A nonadjacent dose can pass \(R_j\leq\eta\) and still be barred by the escalation cap.
- Confusing “adjacent” with “safe.” A dose at level \(h+1\) is not automatically admissible; it must also pass the posterior threshold.
- Treating the cap as a required escalation. The rule sets a maximum permitted level, not a command to move upward. The model criterion may favor a lower eligible dose.
- Filtering after selecting. If the preferred dose is excluded, do not stop without comparing the remaining doses that pass both rules. Apply the ranking criterion to the intersection.
- Using an outdated boundary. If another cohort has since received a higher dose, the value of \(h\) and hence the cap may have changed.
The main alternative is to use a model-based target-dose rule without a separate escalation cap. That can permit a larger jump if the preferred dose passes the specified safety criterion, but it does not implement a no-skipping policy. EWOC feasibility bounds also express posterior safety through a bound on the probability that a proposed dose exceeds the MTD; they are a different dose-selection formulation, not a substitute definition of the one-level cap. These approaches should not be blended informally: the trial protocol must specify the model criterion, the safety rule, and any operational restrictions.
A no-skipping rule can slow escalation and may delay reaching a dose near the target. Conversely, it limits the size of an upward step when the model’s preferred dose is nonadjacent. Neither feature guarantees that the selected dose is optimal, and posterior overdose control is only as reliable as the model, prior, and data supporting its posterior probabilities. The rule also does not resolve what to do if no dose belongs to the intersection; that circumstance requires a prespecified procedure rather than an improvised assignment.
Check Your Understanding
For each question, identify the relevant restriction and give the decision. Assume the target-proximity criterion unless stated otherwise.
- The highest administered level is \(h=1\). Level 3 is the closest dose to target and has \(R_3=0.05\), with \(\eta=0.20\). Is level 3 assignable under a one-level cap?
- Let \(h=3\) and \(\eta=0.20\). Level 4 has \(R_4=0.21\), while level 3 has \(R_3=0.08\). Which level passes both rules, assuming level 3 is the best target-proximity candidate among passing doses?
- With \(h=2\), two doses pass overdose control: level 2 has distance 0.08 from target and level 3 has distance 0.03. Which dose is selected under a one-level cap?
- Explain why lowering \(\eta\) can change the selected dose even though \(h\) and the posterior summaries remain fixed.
Solutions. (1) No. The cap is \(h+1=2\), so level 3 is excluded even though it passes overdose control. (2) Level 4 is adjacent but fails because \(0.21>0.20\); level 3 passes both restrictions and is selected under the stated condition. (3) Level 3 is within the cap, and both doses pass safety control; its distance 0.03 is smaller than 0.08, so select level 3. (4) A lower threshold can remove doses whose posterior overdose probabilities exceed the new \(\eta\). The intersection of the safety-passing set and the unchanged no-skipping set then changes, so the best remaining candidate may differ.