Why the Cutoff Matters
A dose can be judged against a posterior safety rule in two stages. First, the model and accumulated trial data determine the posterior probability that the dose’s toxicity probability exceeds a specified upper limit. Second, a decision rule compares that probability with a cutoff. The cutoff controls how much posterior evidence of exceeding the toxicity limit is tolerated before a dose is excluded.
This tutorial isolates the second stage. We hold the posterior distribution, the toxicity upper limit, and the candidate doses fixed, and vary only the cutoff. This distinction matters: changing the cutoff changes the rule for calling a dose admissible; it does not change the posterior evidence about toxicity. The cutoff should be specified by the trial’s decision policy, not selected after seeing which value produces a preferred dose set.
For example, if \(\phi_U=0.30\), the overdose event is that the dose’s toxicity probability exceeds 0.30. It is not the event that a patient experiences toxicity, nor is it a comparison of the observed toxicity rate with 0.30. The probability \(r_d\) is a posterior probability about the model’s toxicity probability at that dose, conditional on the data and model.
The Effect of Changing the Cutoff
Suppose two cutoffs satisfy \(c_1<c_2\). If a dose is admissible at \(c_1\), then its posterior overdose probability satisfies \(r_d<c_1\). Since \(c_1<c_2\), it follows that \(r_d<c_2\), so the dose is also admissible at \(c_2\). Therefore:
This is a nesting result: raising the cutoff can leave the admissible set unchanged or add doses, but cannot remove an already admissible dose, provided the posterior risks and toxicity limit remain fixed. Lowering the cutoff has the reverse effect: it can remove doses but cannot add them. The set may not change with every adjustment. It changes only when the cutoff passes a posterior overdose probability for at least one candidate dose.
If candidate doses are ordered and the model gives a nondecreasing dose-toxicity relationship for every parameter value represented in the posterior, then their overdose probabilities are also nondecreasing with dose. In that case, the admissible set is a lower segment of the ordered dose list: once a dose fails the rule, every higher dose fails as well. This conclusion depends on the model’s dose ordering, not on the nesting result alone. The nesting result concerns changing the cutoff and holds even if risks across candidate doses are not ordered.
Worked Example: Raising the Cutoff Adds Doses
Worked Example: Raising the Cutoff Adds Doses
Hypothetical setting. Suppose the toxicity upper limit is \(\phi_U=0.30\). For five candidate doses, posterior integration under one fixed model and one fixed data set gives the following posterior overdose probabilities. Treat these values as sufficiently accurate for this illustration.
| Dose | Posterior overdose probability \(r_d\) |
|---|---|
| D1 | 0.04 |
| D2 | 0.12 |
| D3 | 0.24 |
| D4 | 0.31 |
| D5 | 0.43 |
At cutoff \(c=0.20\), apply \(r_d<0.20\). D1 and D2 meet the rule, while D3, D4, and D5 do not. Therefore, \(\mathcal A(0.20)=\{\mathrm{D1},\mathrm{D2}\}\).
At cutoff \(c=0.30\), D1, D2, and D3 meet the rule because \(0.04<0.30\), \(0.12<0.30\), and \(0.24<0.30\). D4 and D5 do not. Thus, \(\mathcal A(0.30)=\{\mathrm{D1},\mathrm{D2},\mathrm{D3}\}\). Raising the cutoff from 0.20 to 0.30 adds D3 without removing either previously admissible dose.
At \(c=0.35\), D4 also qualifies because \(0.31<0.35\), giving \(\mathcal A(0.35)=\{\mathrm{D1},\mathrm{D2},\mathrm{D3},\mathrm{D4}\}\). D5 remains inadmissible because \(0.43\) is not less than \(0.35\). The posterior risks did not change in these calculations; only the cutoff changed.
Boundary Values and Posterior Simulation
The strict inequality makes a boundary value consequential. If \(r_d=c\), the dose fails the rule. If a protocol instead defines admissibility using \(r_d\leq c\), the dose at equality qualifies. Either convention can be stated clearly, but they are not interchangeable. The examples here use \(r_d<c\) throughout.
In practice, posterior overdose probabilities may be estimated from posterior simulation. For \(M\) draws \(\theta^{(1)},\ldots,\theta^{(M)}\), estimate \(r_d\) by counting how many simulated toxicity probabilities exceed \(\phi_U\):
where \(\mathbb I\{\cdot\}\) equals 1 when its condition is true and 0 otherwise. The estimated set uses \(\widehat r_d\) in place of \(r_d\), but a value near the cutoff may be sensitive to Monte Carlo error. In that situation, more draws and simulation diagnostics are important before treating a small numerical difference as a meaningful change in admissibility.
Worked Example: Estimating an Overdose Probability From Posterior Draws
Hypothetical simulation. At one dose, suppose ten posterior draws give toxicity probabilities of 0.12, 0.18, 0.22, 0.27, 0.29, 0.31, 0.34, 0.33, 0.26, and 0.36. Let \(\phi_U=0.30\).
Four values exceed 0.30: 0.31, 0.34, 0.33, and 0.36. Thus, the simulated estimate is $$ \widehat r_d=\frac{4}{10}=0.40. $$
Using this estimate, the dose fails the strict rule at \(c=0.40\), because \(0.40\) is not less than \(0.40\). It passes at \(c=0.45\), because \(0.40<0.45\). These are decisions based on a deliberately small set of draws: the estimate is coarse and can be substantially affected by simulation variability. With independent posterior draws, the approximate Monte Carlo standard error of this estimated proportion is $$ \sqrt{\frac{\widehat r_d(1-\widehat r_d)}{M}} =\sqrt{\frac{0.40(0.60)}{10}} \approx 0.155. $$ That uncertainty is large relative to the difference between the estimate and either cutoff. A real analysis would use enough well-behaved posterior draws to assess the boundary with suitable precision rather than relying on ten draws.
Worked Example: Equality Excludes a Borderline Dose
Hypothetical setting. With the same fixed upper limit \(\phi_U=0.30\), suppose three doses have posterior overdose probabilities \(0.08\), \(0.25\), and \(0.32\). Consider cutoffs \(c=0.25\), \(c=0.26\), and \(c=0.32\).
At \(c=0.25\), only the dose with risk 0.08 is admissible: the dose with risk 0.25 is at equality and fails the strict rule. At \(c=0.26\), the doses with risks 0.08 and 0.25 are admissible, because both are below 0.26. At \(c=0.32\), the dose with risk 0.32 still fails because equality does not satisfy \(r_d<c\). The set changes only after the cutoff rises above 0.32.
This example shows why reporting only a rounded cutoff comparison can be misleading. If an estimated risk is displayed as 0.32, inspect the unrounded value and the prespecified inequality before deciding how a borderline dose is classified.
What Is Changing—and What Is Not
A cutoff sensitivity analysis holds the posterior risks fixed and asks how the decision set responds to different values of \(c\). This differs from changing the toxicity upper limit or updating the posterior with new data. Each change can affect admissibility, but through a different mechanism.
| Change made | What is held fixed? | What may change? |
|---|---|---|
| Change cutoff \(c\) | Posterior distribution, data, and \(\phi_U\) | The rule applied to each \(r_d\), and therefore the admissible set |
| Change upper limit \(\phi_U\) | Posterior distribution and cutoff | The overdose event and each posterior overdose probability |
| Update with new data | The model and the decision rule, if unchanged | The posterior distribution and potentially each \(r_d\) |
These distinctions also clarify the relationship to a dose-selection rule. Admissibility defines which doses pass the safety criterion; it does not, by itself, specify which passing dose to recommend. A separate selection rule may choose among admissible doses. Changing the cutoff can alter the candidates available to that rule, but it does not determine the final choice on its own.
Diagnostics, Limitations, and Common Mistakes
A useful diagnostic is to tabulate the posterior overdose probability for every candidate dose alongside the cutoff values under consideration. The admissible set can then be checked directly against the inequality. If the assumed dose-toxicity model is monotone, also check that the posterior overdose probabilities are ordered consistently with dose. A violation may indicate simulation error, a coding problem, or a model that does not enforce the assumed ordering.
The nesting result is exact for fixed posterior risks. In a simulation, however, separate runs at different cutoffs should use the same risk estimates or the same posterior draws when comparing sets. If estimates are recomputed with substantial Monte Carlo noise, a dose may appear to leave the set as the cutoff rises. That apparent reversal is not a property of the rule; it signals that the estimates or computations need checking.
- Mistake: treating a higher cutoff as new evidence of safety. A higher cutoff makes the admissibility rule more permissive; it does not make the posterior overdose probability smaller.
- Mistake: changing several inputs and attributing the result to the cutoff. If the data or toxicity upper limit also changed, the posterior risks may have changed. Isolate the cutoff by holding those quantities fixed.
- Mistake: ignoring equality. Under the strict rule, \(r_d=c\) fails. Use the protocol’s stated convention and adequate numerical precision.
- Mistake: assuming every increase adds a dose. The set changes only when the cutoff passes a risk value. Between such values, it stays the same.
The cutoff is a decision-policy parameter, not a parameter estimated from the trial data. Sensitivity analysis can show how strongly a dose set depends on its chosen value, but it cannot establish that one cutoff is universally correct. The appropriate cutoff must be justified in the context of the trial’s safety objectives and prespecified decision policy.
Check Your Understanding
Use the strict rule \(r_d<c\). Assume posterior risks and the toxicity upper limit are fixed unless stated otherwise.
- Three doses have risks 0.10, 0.28, and 0.41. Find the admissible set at \(c=0.30\).
- For the risks in Question 1, what is the set at \(c=0.28\), and why does the middle dose fail?
- If \(c_1<c_2\), can a dose be admissible at \(c_1\) but inadmissible at \(c_2\), when the posterior risks are fixed? Explain.
- Ten posterior draws at a dose yield three toxicity probabilities above the upper limit. Calculate the estimated overdose probability and state whether the dose passes at \(c=0.30\).
Solutions. (1) At \(c=0.30\), 0.10 and 0.28 are below the cutoff, but 0.41 is not, so the first two doses are admissible. (2) At \(c=0.28\), only the dose with risk 0.10 is admissible; the middle dose equals the cutoff and fails the strict inequality. (3) No. If \(r_d<c_1\) and \(c_1<c_2\), then \(r_d<c_2\), so it remains admissible. (4) The estimate is \(3/10=0.30\). It does not pass at \(c=0.30\), because equality does not satisfy \(r_d<c\).