Why Dose Skipping Needs Its Own Rule
A dose-escalation algorithm often lists candidate doses in increasing order. Moving from one level to the next is not the same decision as jumping over an untried level. A jump changes which dose is assigned next, what evidence is collected, and how much uncertainty remains about the dose-response relationship. The fact that a proposed destination passes an overdose screen is important, but does not by itself settle whether the protocol permits the jump.
The earlier material on posterior overdose probabilities and the admissible dose set supplies the safety quantities used here. This tutorial focuses on how to translate those quantities into explicit rules for skips during escalation or de-escalation. Any numerical thresholds below are hypothetical; a real protocol must specify its own thresholds and operational conditions in advance.
The cutoff \(\phi\) defines what toxicity probability counts as overdose; \(\eta\) limits the posterior probability that the dose exceeds that cutoff. These are different quantities. For example, \(\phi=0.30\) and \(\eta=0.25\) mean that a dose is eligible under this particular screen when the posterior probability that its DLT probability exceeds 30% is at most 25%. Admissibility is a model-based criterion, not a guarantee that the dose is safe or that a patient will not experience a DLT.
A complete decision rule therefore has two layers. First, the posterior calculation determines which doses meet the overdose constraint. Second, the protocol specifies which moves among those doses are allowed, using conditions such as the current dose, the direction and size of the move, and any required review. A destination can be admissible while a move to it is prohibited.
Monotonicity and the Shape of the Admissible Set
Suppose the dose-toxicity model imposes monotonicity: for every posterior draw, the DLT probability cannot decrease as dose increases. In symbols, \(p_1\leq p_2\leq\cdots\leq p_K\). Then an overdose at a lower dose implies an overdose at every higher dose in that same draw. This gives a useful justification for the usual ordered shape of the admissible set.
Thus, with a monotone model and a common cutoff and probability limit, admissible doses form a lower-dose prefix: if \(d_k\) is admissible, all lower levels are admissible; if \(d_k\) is inadmissible, all higher levels are inadmissible. This is a property of the model and criterion, not a universal fact about observed DLT proportions.
This implication must not be used when the fitted model does not impose monotonicity, when the doses have been analyzed under different safety criteria, or when estimates are too imprecise to classify reliably. In those situations, the protocol should state how discordant results are handled rather than letting a numerical irregularity create an unplanned route to a higher dose.
Write the Move Rule Separately From the Safety Screen
Before enrollment, a protocol should say whether escalation is ordinarily limited to the next dose level, whether any specified exception allows an upward skip, and what conditions govern a downward skip. A useful specification identifies the eligible destination, the posterior safety requirement, the maximum number of levels that may be crossed, and what happens if the required evidence is incomplete or numerically uncertain.
For each candidate dose, compute \(q_k=P(p_k>\phi\mid D)\) using the prespecified model and data available at the decision. Mark \(d_k\) admissible only if it meets the protocol limit.
Record the current level, the proposed destination, direction, and number of levels crossed. Do not infer permission to skip from admissibility alone.
Check the destination against the overdose constraint and apply the protocol’s separate limits on move size, review, and supporting evidence. For a downward skip, verify that the landing dose is admissible and that the rule allows passing over intervening levels.
If posterior computation is unstable, data are incomplete under the protocol, or the admissibility pattern conflicts with model assumptions, follow the prespecified review or fallback rule rather than treating a borderline result as definitive.
Worked Decisions With Hypothetical Posterior Results
Worked Example: An Upward Skip to an Admissible Dose
Suppose the ordered doses are \(d_1,\ldots,d_5\), the current dose is \(d_2\), and the hypothetical protocol uses \(\phi=0.30\) and \(\eta=0.25\). After a cohort of three patients at \(d_2\), with no DLTs observed, the model is updated. For illustration, posterior simulation gives the following counts among 2,000 draws in which the dose-specific DLT probability exceeds 0.30.
| Dose | Draws with \(p_k>0.30\) | Estimated \(q_k\) | Admissible? |
|---|---|---|---|
| \(d_2\) | 120 | \(120/2000=0.0600\) | Yes |
| \(d_3\) | 220 | \(220/2000=0.110\) | Yes |
| \(d_4\) | 420 | \(420/2000=0.210\) | Yes |
| \(d_5\) | 660 | \(660/2000=0.330\) | No |
The admissibility calculation is \(q_4=420/2000=0.210\leq0.25\), so \(d_4\) passes the overdose screen. But moving from \(d_2\) directly to \(d_4\) skips \(d_3\). If the protocol permits escalation by at most one level, the move is prohibited; the highest allowed next level is \(d_3\). If a protocol explicitly allows a two-level upward move under specified conditions, \(d_4\) meets the numerical overdose condition here, but the remaining conditions still have to be checked. The posterior calculation alone does not authorize the jump.
Worked Example: A Downward Skip to Escape an Inadmissible Level
Now suppose the current dose is \(d_5\) after a hypothetical cohort of three patients in which all three experienced a DLT. The same illustrative values, \(\phi=0.30\) and \(\eta=0.25\), are used. Posterior simulation gives the following results:
| Dose | Draws with \(p_k>0.30\) out of 1,000 | Estimated \(q_k\) | Admissible? |
|---|---|---|---|
| \(d_2\) | 40 | \(40/1000=0.0400\) | Yes |
| \(d_3\) | 180 | \(180/1000=0.180\) | Yes |
| \(d_4\) | 760 | \(760/1000=0.760\) | No |
| \(d_5\) | 930 | \(930/1000=0.930\) | No |
For example, \(q_4=760/1000=0.760>0.25\), so \(d_4\) fails the screen; \(q_3=180/1000=0.180\leq0.25\), so \(d_3\) passes. A rule that permits de-escalation by more than one level when the immediate lower level is inadmissible could allow a move from \(d_5\) to \(d_3\), skipping \(d_4\). The landing dose meets the overdose criterion, and the pattern is consistent with monotonic dose-toxicity. The protocol must still specify that this size of downward move is allowed; the table does not establish that every patient or every trial should be assigned to \(d_3\).
Worked Example: A Borderline Estimate Is Not a Reliable Skip Signal
Consider a hypothetical posterior calculation near the cutoff \(\eta=0.25\), using 1,000 independently generated simulation draws per dose. The estimated overdose probabilities at \(d_2,d_3,d_4,d_5\) are 0.140, 0.247, 0.243, and 0.300. The slight decrease from \(d_3\) to \(d_4\) is inconsistent with the exact monotone ordering expected under a monotone model, and the estimate at \(d_3\) is close to the admissibility limit.
For \(\hat q_3=0.247\), the approximate Monte Carlo standard error is $$ \sqrt{\frac{0.247(1-0.247)}{1000}} =\sqrt{0.000186} \approx 0.0136. $$ A rough upper endpoint using 1.96 standard errors is \(0.247+1.96(0.0136)\approx0.274\), above the limit 0.25. This is a numerical-precision check, not a posterior credible interval for the dose-toxicity probability. The apparent pass at \(d_3\) is too close to the cutoff to treat as robust on these computations. Before making a skip decision, the team should investigate the nonmonotone estimates and improve or verify the posterior calculation. It should not select a higher destination based on the small apparent difference between 0.247 and 0.243.
Diagnostics, Alternatives, and Limitations
A skip rule is only as dependable as its model output and its implementation. Useful checks include confirming that the same \(\phi\) and \(\eta\) were applied at every dose, verifying that the data used are the data available at the decision time, and checking that the reported posterior probabilities are stable to additional simulation. Under a monotone model, investigate a materially nonmonotone sequence of \(q_k\) values: possible causes include a coding or indexing error, inconsistent computation, or excessive Monte Carlo error. Tiny reversals from separate finite simulation runs may be numerical noise, but they should not be converted into protocol permission to skip.
If \(q_k\) is near \(\eta\), a binary admissible/inadmissible label can conceal substantial numerical uncertainty. A protocol can address this by requiring a prespecified computational precision, a review when estimates are borderline, or a conservative fallback such as not escalating until the calculation is resolved. The method for assessing numerical precision should be specified; an approximate Monte Carlo standard error describes simulation error, not uncertainty about the unknown toxicity probability itself.
| Approach | What it requires | Main trade-off |
|---|---|---|
| One-level escalation | Move only to the immediately higher level when allowed by the full protocol | Preserves an orderly sequence of dose experience, but may take more cohorts to reach higher levels |
| Protocolized upward skip | Destination passes the overdose screen and every additional skip condition is met | Can move past an untried level, but that level remains unobserved and permission depends on the prespecified rule |
| Protocolized downward skip | Landing dose is admissible and the rule permits a larger de-escalation | Can avoid an inadmissible intervening level, but requires a clear rule for choosing the landing dose |
| Stop when no dose is admissible | No candidate dose meets the prespecified admissibility criterion | Avoids assigning a dose that fails the screen, but does not itself select a dose or authorize a skip |
The admissibility screen controls one defined posterior risk; it does not optimize all trial objectives, establish that the dose is effective, or remove uncertainty caused by sparse data or model misspecification. A monotone model makes the ordering argument above available, but that assumption should be checked against the scientific setting and made explicit. A skip also leaves less direct experience at the passed-over dose, which can matter for later decisions even when the destination is admissible.
Common mistake: “The higher destination is admissible, so skipping directly to it is safe and permitted.” This conflates a posterior safety condition with a complete allocation rule. The correction is to verify the destination’s \(q_k\), then separately apply the prespecified direction, step-size, and review requirements. Equally, finding an inadmissible intermediate dose does not justify jumping upward over it: under monotonicity, a still-higher dose cannot pass the same criterion.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(q_k=0.18\) and \(\eta=0.20\), is \(d_k\) admissible under the definition used here? What does that say about permission to skip to it?
- Under posterior-draw monotonicity, explain why an admissible \(d_{k+1}\) cannot have an inadmissible \(d_k\) under a shared cutoff and probability limit.
- A protocol allows downward skips only when the landing dose is admissible. The immediate lower dose has \(q=0.40\), and a dose two levels below has \(q=0.15\), with \(\eta=0.25\). What additional fact is needed before the two-level move is allowed?
- Why should a small nonmonotone reversal in estimated \(q_k\) values be investigated before it guides a skip?
Solutions. (1) Yes, because \(0.18\leq0.20\). This establishes only that the dose passes the overdose screen; the protocol must separately permit a skip to it. (2) Monotonicity gives \(p_k\leq p_{k+1}\) in every posterior draw, so \(\{p_k>\phi\}\subseteq\{p_{k+1}>\phi\}\), hence \(q_k\leq q_{k+1}\). If \(d_{k+1}\) is admissible, \(q_{k+1}\leq\eta\), and therefore \(q_k\leq\eta\) as well. (3) The protocol must explicitly allow a two-level downward move, and any other specified conditions must be met; admissibility of the landing dose alone is not authorization. (4) Under the monotone model the exact probabilities should be ordered. The reversal may reflect simulation error or a computation problem, so its size and numerical reliability must be checked rather than interpreted as a meaningful safety advantage.