Why a Dose-Toxicity Skeleton Is Needed
In a model-based dose-finding design, the available toxicity data may be sparse when a decision is required. The continual reassessment method (CRM) addresses this by representing toxicity probabilities across ordered dose levels with a model. Before data can update that model, the design needs a reference pattern for how toxicity is anticipated to change with dose. That pattern is the dose-toxicity skeleton.
Let \(d_1,\ldots,d_J\) denote the trial's dose levels, ordered from lowest to highest. For each dose, the outcome is whether a participant experiences a dose-limiting toxicity (DLT) under the protocol's defined assessment window. The target rate, denoted by \(\tau\), is the toxicity probability the design aims to approach. A skeleton does not replace that target: it provides a set of dose-specific reference probabilities against which a model can be anchored.
The values \(q_j\) are design inputs, not estimates calculated from the trial's observed DLT counts. They are also not probabilities assigned to the unknown model parameter. Their role is to encode a reference dose-toxicity shape. A model-based method can then use a parameterized curve that reproduces the skeleton at a designated reference setting and changes the curve as information accumulates. The specific model and its updating calculations are separate questions.
The skeleton rests on several substantive conditions. Dose levels must have a meaningful ordering; the DLT definition and assessment window must be the same at every level; and, as a design assumption, the anticipated toxicity probability should not decrease as dose increases. These conditions do not claim that every observed sample proportion will rise with dose. With small cohorts, observed proportions can fluctuate substantially.
How the Skeleton Anchors a Model
The anchoring idea can be stated without choosing a particular CRM parameterization. Suppose a model assigns a toxicity probability \(r_j(\theta)\) to dose \(d_j\), where \(\theta\) represents the model's unknown parameter or parameters. At a designated reference setting \(\theta_0\), the model is anchored by \(r_j(\theta_0)=q_j\) for every dose \(j\). The skeleton therefore specifies the curve's reference values; evidence from participants can inform where the fitted curve lies relative to that reference.
A skeleton is consequential because it influences the model's starting representation of toxicity. If its values are too low or too high, or if its spacing poorly reflects plausible changes across doses, the model may be poorly anchored for the clinical setting. This does not mean the skeleton is a fixed claim about the true probabilities. It is an explicit, reviewable design assumption that should be scientifically defensible.
Monotonicity can be justified as a property of the model as well as of the skeleton. For example, consider a family of curves \(r_j(\theta)=g(q_j,\theta)\). If, for each fixed \(\theta\), the function \(g(\,\cdot\,,\theta)\) is increasing in its first argument, then \(q_j<q_{j+1}\) implies \(r_j(\theta)<r_{j+1}(\theta)\). Thus the model preserves the dose ordering. A monotone skeleton alone does not guarantee that every possible model family preserves this ordering; the chosen model must also be checked.
This reasoning establishes an ordering property, not the correctness of the numerical probabilities. Monotonicity is usually a scientifically reasonable structural assumption for increasing doses of the same treatment, but it may be a poor description if dose levels differ in other ways or if the treatment has nonstandard toxicity patterns. The trial team must judge the clinical plausibility of the skeleton as well as its mathematical order.
Worked Example: Building a Monotone Skeleton
Suppose a hypothetical trial plans five increasing dose levels and has a target DLT rate of \(\tau=0.25\). After considering the treatment and the dose range, the team proposes the following reference probabilities: \(q=(0.05,0.12,0.25,0.40,0.58)\). Assess the proposal.
First, check that every value is a probability strictly between zero and one. The smallest is \(0.05>0\), and the largest is \(0.58<1\). Next, check the dose ordering: \(0.05<0.12<0.25<0.40<0.58\). The skeleton is strictly increasing and therefore encodes the assumed ordering.
The successive increases are \(0.12-0.05=0.07\), \(0.25-0.12=0.13\), \(0.40-0.25=0.15\), and \(0.58-0.40=0.18\). These differences are not required to be equal: a dose-toxicity relationship can rise more sharply in one part of the dose range than another. Here the target rate equals \(q_3\), so the third dose is the reference level whose anticipated probability is exactly the target. That fact describes the skeleton; it is not a claim that the true DLT probability at that dose is known to be 0.25.
The numerical order and target alignment are internally coherent. The team must still check whether the probabilities are clinically plausible for these particular dose levels and the specified DLT window. If, for example, the highest dose is expected to be substantially more toxic than a 0.58 probability suggests, the upper part of the skeleton deserves review.
Choosing and Comparing Candidate Skeletons
There is no universal numerical spacing that is correct for all trials. A useful construction starts with the actual ordered dose levels and the clinical judgment about how toxicity is expected to change across them. The skeleton should make that judgment explicit. Some designs may place a reference probability near the target rate; others may not. What matters is that the values and the dose-toxicity pattern are considered plausible for the intended setting, not that a particular dose is forced to equal \(\tau\).
Candidate skeletons can be compared by examining their level, spacing, and relation to the target. Large jumps imply a sharper anticipated change between adjacent levels than small jumps. A sequence clustered far below the target describes a different reference pattern from one that places several doses near or above it. These are properties to scrutinize, not mechanical reasons to accept or reject a skeleton. In particular, apparent precision to several decimal places should not be mistaken for strong clinical knowledge.
Worked Example: Comparing Two Plausible Skeletons
For a hypothetical four-dose trial with target rate \(\tau=0.20\), the team considers skeleton A, \(q^{(A)}=(0.04,0.10,0.22,0.40)\), and skeleton B, \(q^{(B)}=(0.08,0.16,0.27,0.43)\). Both vectors are strictly increasing and contain probabilities between zero and one. Which features differ?
The successive increases for A are \(0.10-0.04=0.06\), \(0.22-0.10=0.12\), and \(0.40-0.22=0.18\). For B they are \(0.16-0.08=0.08\), \(0.27-0.16=0.11\), and \(0.43-0.27=0.16\). Both have a larger increase toward the upper end, though their exact spacing differs.
Relative to the target, the nearest reference value in A is \(0.22\), with absolute difference \(|0.22-0.20|=0.02\). In B it is \(0.16\), with difference \(0.04\), rather than \(0.27\), whose difference is \(0.07\). Thus A places a reference value closer to the target, while B has a lower anticipated probability at its second dose. Neither comparison establishes which skeleton is better. The decision depends on whether the anticipated probabilities and spacing fit the clinical knowledge for the actual dose levels; closeness to the target alone is not sufficient.
Checking and Revising an Inconsistent Skeleton
A basic diagnostic is to list the values in dose order and verify \(0<q_1<\cdots<q_J<1\). Also inspect adjacent differences \(q_{j+1}-q_j\). A negative difference directly conflicts with the monotone assumption; an unusually large difference may be intentional but merits confirmation. Very small differences can make adjacent dose levels nearly indistinguishable in the reference pattern, which may or may not match the clinical expectation.
When elicited values violate the order, do not silently sort them. Sorting would attach probabilities to different dose levels and therefore change the clinical input. Instead, check for a transcription error, confirm the dose ordering and DLT definition, and ask whether the elicited probabilities reflect a genuine non-monotone concern or uncertainty. If a monotone sequence is required, a transparent adjustment can be used as a diagnostic starting point, then reviewed clinically.
Worked Example: Diagnosing a Reversal in Elicited Values
For five hypothetical ordered doses, a team records \(q=(0.07,0.24,0.19,0.42,0.61)\). The third value is below the second, so the sequence is not monotone. As a simple equal-weight diagnostic, find the closest nondecreasing sequence by minimizing the sum of squared changes from the elicited values.
The violation is between \(0.24\) and \(0.19\). Pooling these two values gives their average: \((0.24+0.19)/2=0.215\). Replacing both by \(0.215\) gives \((0.07,0.215,0.215,0.42,0.61)\), which is nondecreasing. The pooled values minimize the squared changes for this violating pair: for any common replacement \(x\), the relevant sum is \((x-0.24)^2+(x-0.19)^2\). Differentiating gives \(4x-0.86=0\), so \(x=0.215\).
This is a mathematical correction, not a clinical verdict. Equal probabilities at adjacent doses may be acceptable if the levels are expected to have nearly identical toxicity, but may be unsuitable if the design requires distinct reference probabilities. The team should revisit the two elicited values and document any final adjustment. It should not claim that \(0.215\) is a newly observed or objectively known toxicity probability.
What a Skeleton Does Not Do
The skeleton is one input to a model-based design; it is not a complete decision procedure. It does not provide observed toxicity evidence, dictate a dose assignment on its own, or guarantee that the fitted model will be accurate. Nor is it a list of dose-specific probabilities known to be true. Trial data and the specified model are needed to estimate toxicity as the trial proceeds.
This separates skeleton-based modelling from rule-based escalation. A rule-based design specifies actions from prespecified outcome-count rules. A skeleton instead supplies reference probabilities to a model-based method. The two approaches therefore represent dose-toxicity information differently. Neither removes the need for safety monitoring, a clear DLT definition, or careful attention to the trial population and assessment window.
A practical failure mode is to treat a skeleton as harmless because it is only a starting pattern. A poorly chosen reference pattern can be consequential, particularly when data are limited. Another is to regard a strictly increasing list as sufficient validation: the order can be correct while the probabilities themselves are implausible. Review both the ordering and the clinical interpretation of the values, and record why the chosen pattern is defensible.
Specify ordered dose levels, the DLT definition, the assessment window, and the target rate.
Assign one anticipated toxicity probability to each dose, using clinical judgment appropriate to the treatment and dose range.
Check probability bounds, monotonicity, adjacent differences, and the position of the target rate relative to the proposed values.
Resolve reversals or implausible spacing with clinical input, and preserve the rationale for the final skeleton.
Check Your Understanding
Answer each question using the definitions and checks developed above. Solutions follow the questions.
- A three-dose skeleton is \((0.06,0.18,0.35)\). Is it strictly increasing, and what does that tell you?
- For a target rate of \(0.30\), a four-dose skeleton is \((0.04,0.12,0.26,0.48)\). Which reference value is closest to the target, and what can you conclude from that alone?
- Why should a team not silently sort a proposed vector \((0.08,0.31,0.25,0.52)\)?
- In one sentence, distinguish a skeleton from observed DLT proportions.
Solutions. (1) Yes: \(0.06<0.18<0.35\). It encodes increasing anticipated toxicity across the ordered doses, but does not establish that the probabilities are true. (2) \(0.26\) is closest because \(|0.26-0.30|=0.04\); this identifies the closest skeleton value, not a dose assignment or a guarantee about the true rate. (3) Sorting would attach probabilities to different dose levels and alter the clinical input; the ordering, elicitation, and any transcription errors should be reviewed instead. (4) A skeleton is a prespecified reference vector, whereas observed DLT proportions are calculated from trial participants and may fluctuate from dose to dose.