From a Posterior Constraint to a Dose on the Grid
EWOC limits the posterior probability that a proposed dose exceeds the maximum tolerated dose (MTD). In practice, a trial usually has a finite set of available doses rather than a continuum from which any dose can be selected. The posterior constraint therefore has two jobs: determine which grid doses are eligible, and specify which one to select if the continuous posterior feasibility quantile falls between grid points.
The rule in this tutorial is to select the highest eligible dose on the prespecified grid. This is the grid-based version of choosing the posterior feasibility quantile. It does not mean that the selected dose is guaranteed safe, or that it has exactly the target toxicity probability. It means that its posterior probability of exceeding the MTD meets the chosen EWOC bound.
The overdose event is \(M<d\): the proposed dose is above the MTD. If the posterior distribution of \(M\) is continuous, \(P(M<d\mid\text{data})\) equals \(P(M\leq d\mid\text{data})\). If the posterior has point masses, the strict inequality defines the event precisely; a protocol should use the same boundary convention throughout.
Assumptions and the Discrete-Grid Rule
Write the available doses in increasing order as \(d_1<d_2<\cdots<d_J\). The usual model-based interpretation assumes that toxicity probability increases with dose, that the dose-toxicity model has a meaningful target crossing defining \(M\), and that the posterior distribution for \(M\) reflects the specified model, prior, and observed data. The candidate doses and the bound \(\alpha\) must also be specified before applying the selection rule.
The probability \(P(M<d_j\mid\text{data})\) cannot decrease as \(d_j\) increases: raising the proposed dose makes it at least as likely that the dose lies above the MTD. Thus, when doses are ordered, eligible doses form an initial segment of the grid. The grid rule can be stated directly:
If the posterior distribution of \(M\) is continuous, let \(q_\alpha\) be its \(\alpha\)-quantile, so that \(P(M\leq q_\alpha\mid\text{data})=\alpha\) when the posterior cumulative distribution is continuous and strictly increasing at that quantile. The feasible grid doses are those at or below \(q_\alpha\), and the selected dose is the largest available dose not exceeding it. Consequently, a grid dose need not equal the continuous quantile: the rule rounds down to the highest available eligible dose rather than rounding to the nearest dose.
Worked Applications
Worked Example: A Normal Posterior and a Quantile Between Grid Doses
Suppose the posterior MTD in a hypothetical trial is \(M\mid\text{data}\sim N(40,10^2)\), with doses measured in mg. Let \(\alpha=0.25\), and let the available grid be 10, 20, 30, 40, and 50 mg. Calculate the posterior overdose probability at each dose and select an EWOC dose.
For this normal posterior, \(P(M<d\mid\text{data})=\Phi((d-40)/10)\), where \(\Phi\) is the standard normal cumulative distribution function. At 30 mg, for example,
The corresponding probabilities are approximately 0.0013 at 10 mg, 0.0228 at 20 mg, 0.1587 at 30 mg, 0.5000 at 40 mg, and 0.8413 at 50 mg. The first three are no greater than 0.25; the last two exceed it. Therefore, 10, 20, and 30 mg are eligible, and the highest eligible dose is 30 mg.
The continuous 25th posterior percentile is \(40+10\Phi^{-1}(0.25)\), approximately \(40+10(-0.6745)=33.26\) mg. There is no 33.26 mg dose on the grid. Selecting 30 mg rounds down to the highest eligible grid point; selecting 40 mg would violate the posterior bound.
Worked Example: Estimating Eligibility From Posterior Draws
In another hypothetical analysis, 1,000 posterior draws of the MTD are used to assess a grid of 5, 10, 15, and 20 mg. The number of draws below each candidate dose is 25, 110, 190, and 315, respectively. With \(\alpha=0.20\), determine the eligible doses and the selection.
Estimate each posterior overdose probability by the fraction of draws for which \(M<d\). For 15 mg, this is \(190/1000=0.190\). The full calculation is:
The first three estimated probabilities are at most 0.20, while the 20 mg estimate is above 0.20. Thus 5, 10, and 15 mg are eligible, and the grid rule selects 15 mg. This calculation is a Monte Carlo estimate of the posterior probabilities; with few draws, simulation error can make an estimate near the bound unstable. More draws or a more accurate posterior calculation can help assess that numerical uncertainty.
Worked Example: Why the Overdose Constraint Takes Priority
Suppose a hypothetical posterior analysis gives the following summaries on a 1, 2, 3, and 4 mg grid. The posterior mean toxicity probabilities are 0.03, 0.07, 0.13, and 0.19; the posterior overdose probabilities are 0.01, 0.06, 0.22, and 0.48. Let the target toxicity probability be \(\theta=0.20\) and let \(\alpha=0.20\). Apply the grid rule.
The overdose probabilities at 1 and 2 mg meet the constraint, but those at 3 and 4 mg do not, since \(0.22>0.20\) and \(0.48>0.20\). The eligible set is therefore \(\{1,2\}\) mg, and the selected dose is 2 mg.
The posterior mean toxicity probability at 4 mg, 0.19, is numerically closer to the target 0.20 than the mean at 2 mg, 0.07. But closeness of a posterior mean toxicity probability to the target is not the EWOC eligibility criterion. Selecting 4 mg on that basis would disregard its 0.48 posterior overdose probability. Under the stated rule, the overdose constraint is applied first, and selection is made only from eligible doses.
Why the Highest Eligible Dose Is the Grid Version of EWOC
For a continuous dose choice, the EWOC feasibility bound identifies a posterior quantile of the MTD. On a discrete grid, there may be no dose at that quantile. The highest-eligible rule follows from the ordering of both the doses and their overdose probabilities.
Write the fixed grid as \(d_1<\cdots<d_J\).
For each dose, calculate \(P(M<d_j\mid\text{data})\), or estimate it from posterior draws.
Retain only doses whose overdose probability is at most \(\alpha\).
Choose its largest dose. The monotonicity of the posterior cumulative distribution means a larger eligible dose cannot have a smaller overdose probability than a smaller one.
The justification for step 4 is the monotonicity of cumulative probabilities. For \(d_j<d_k\), the event \(\{M<d_j\}\) is contained in \(\{M<d_k\}\), so
Therefore, once a dose fails the bound, every higher grid dose also fails it. If at least one dose is eligible, the highest eligible dose is the most aggressive available grid choice that still satisfies the specified posterior constraint. This argument depends on applying the same posterior distribution, overdose definition, and \(\alpha\) at every grid point.
Diagnostics, Alternatives, and Limitations
Before making a selection, check that the grid is correctly ordered, that all dose units are consistent, and that the probabilities increase or remain constant across the grid. A decrease across increasing doses can indicate a coding error, mismatched posterior draws, or calculations based on different model quantities. When probabilities come from simulation, report enough precision to see how close the selected and next-higher doses are to \(\alpha\); a rounded value alone can obscure a boundary decision.
The highest-eligible rule is not the only possible policy for a finite grid. A protocol could instead choose the eligible dose whose posterior mean toxicity probability is closest to \(\theta\), or use another prespecified utility criterion. Those are different selection rules: they must still respect the overdose bound if that bound is required, and they can select a lower eligible dose than the highest-eligible rule. Simply choosing the grid dose whose estimated toxicity probability is nearest to \(\theta\), without first applying the bound, is not the EWOC rule described here.
The grid also limits what the rule can achieve. A wide gap between doses can make the selected dose appreciably lower than the posterior feasibility quantile, while the next grid point may exceed the bound. The rule does not create an off-grid dose or establish that the selected dose has toxicity probability exactly \(\theta\). Its safety interpretation is conditional on the dose-toxicity model and posterior being adequate. If no grid dose meets the bound, there is no eligible dose to select by this rule; a protocol needs a prespecified action rather than relaxing the constraint implicitly.
Check Your Understanding
Use the posterior overdose constraint and the highest-eligible-dose rule in each question.
- If \(M\mid\text{data}\sim N(32,8^2)\), the grid is 16, 24, 32, and 40 mg, and \(\alpha=0.20\), which dose is selected?
- Posterior draws give estimated overdose probabilities 0.06, 0.18, and 0.31 at 5, 10, and 15 mg. With \(\alpha=0.20\), identify the eligible doses and selected dose.
- Why is choosing the grid dose whose posterior mean toxicity probability is closest to \(\theta\) not, by itself, the EWOC selection rule?
- For increasing grid doses, the posterior overdose probabilities are 0.05, 0.20, and 0.45. If \(\alpha=0.20\), is the second dose eligible, and what is selected?
Solutions. (1) The standardized values at 16 and 24 mg are \(-2\) and \(-1\), giving overdose probabilities \(\Phi(-2)=0.0228\) and \(\Phi(-1)=0.1587\). At 32 mg the probability is 0.50, so the highest eligible dose is 24 mg. (2) The 5 and 10 mg doses are eligible because 0.06 and 0.18 are at most 0.20; the selected dose is 10 mg. (3) EWOC first requires the posterior overdose probability to meet its bound. A dose close to the target in posterior mean toxicity can still violate that constraint. (4) Yes. Eligibility uses “at most” \(\alpha\), so the second dose, with probability 0.20, is eligible. The highest eligible dose is the second dose.