Why Skeleton Construction Matters
A continual reassessment method (CRM) represents toxicity probabilities across ordered dose levels with a dose-toxicity model. Before outcomes are observed, the model needs a set of baseline probabilities, called a skeleton. These values are more than a convenient starting curve: their locations and spacing determine how the model translates a common parameter change into risk changes across doses. A skeleton that is too compressed or too spread out can therefore affect which dose appears closest to the target under plausible model values.
The target toxicity probability, denoted by \(\phi\), is the desired rate used to guide dose selection. Skeleton construction begins by placing one candidate dose at, or near, that target under a reference parameter value. The remaining entries express how toxicity is expected to change as dose increases or decreases. They must be checked as a coherent sequence, not chosen independently without regard to the model that will use them.
Here toxicity is a binary dose-limiting toxicity (DLT) outcome defined for a specified assessment window. The model assumes that each dose has a single probability of DLT for the population represented by the trial, and that the probabilities increase with dose. The power model also assumes that one positive parameter \(a\) changes the entire curve through the same transformation. In particular, it does not give each dose its own freely varying probability.
Because every \(q_j\) lies strictly between zero and one, \(\log(q_j)<0\), and \[ \frac{\partial p_j(a)}{\partial a}=q_j^a\log(q_j)<0. \] Increasing \(a\) therefore lowers every modeled toxicity probability; decreasing \(a\) raises them. For any fixed \(a>0\), the ordering remains monotone: if \(q_i<q_j\), then \(q_i^a<q_j^a\). These are structural properties of this parameterization, not empirical findings from a trial.
Anchoring a Skeleton at the Target
A useful construction sets the skeleton probability at a selected anchor dose \(d_k\) equal to the target: \(q_k=\phi\). A convenient way to specify the other values is to work on the logit scale. Define \(\operatorname{logit}(u)=\log\{u/(1-u)\}\) and \(\operatorname{logistic}(x)=1/(1+e^{-x})\). Choose ordered offsets \(c_1<\cdots<c_J\), with \(c_k=0\), and set
The zero offset guarantees the anchor \(q_k=\phi\). Since the logistic function is strictly increasing, ordered offsets guarantee ordered skeleton probabilities. The offsets are on the log-odds scale; they are not dose increments and do not have units of dose. Their spacing is a design judgment about how quickly toxicity is expected to change across the candidate doses.
Other construction rules are possible, including direct increments on the probability scale. What matters is that the resulting probabilities are valid, ordered, and plausible for the intended dose range. Logit offsets are often convenient because they keep all values between zero and one and make the anchoring calculation explicit.
Worked Example: Building an Anchored Five-Dose Skeleton
Suppose five ordered doses have target toxicity \(\phi=0.25\), with dose 3 as the anchor. Use logit offsets \((-0.4,-0.2,0,0.2,0.4)\). First, \(\operatorname{logit}(0.25)=\log(0.25/0.75)=-1.0986\). Add each offset and apply the logistic function.
For dose 1, \(q_1=\operatorname{logistic}(-1.0986-0.4)=\operatorname{logistic}(-1.4986)\approx0.1826\). For dose 2, \(q_2=\operatorname{logistic}(-1.2986)\approx0.2144\). The anchor is \(q_3=\operatorname{logistic}(-1.0986)=0.2500\). For dose 4, \(q_4=\operatorname{logistic}(-0.8986)\approx0.2893\), and for dose 5, \(q_5=\operatorname{logistic}(-0.6986)\approx0.3321\).
The resulting skeleton is approximately \((0.1826,0.2144,0.2500,0.2893,0.3321)\). It is strictly increasing, all entries are between zero and one, and the reference probability at dose 3 equals the 25% target. These checks follow directly from the construction; they do not establish that the spacing is clinically or statistically well calibrated.
How Spacing Shapes the Model
Under the power model, skeleton spacing controls how separated the modeled dose probabilities remain when \(a\) moves away from one. The transformation is nonlinear: \(p_j(a)=q_j^a\) does not preserve equal differences between probabilities. For \(a>1\), every \(q_j\) is reduced, and for \(0<a<1\), every \(q_j\) is increased. The amount of change depends on the starting value \(q_j\), so a single parameter does not shift every dose by the same number of percentage points.
A short derivation makes the role of the skeleton especially clear. Taking logs gives \(\log p_j(a)=a\log q_j\). Since \(-\log q_j>0\), taking the log of the negative log yields
Thus, on the \(\log\{-\log(p)\}\) scale, changing \(a\) adds the same amount, \(\log(a)\), at every dose. The spacing between doses on this transformed scale comes entirely from the skeleton. The model is consequently quite structured: altering one parameter moves the curve as a whole but cannot independently correct a poor relative spacing between adjacent doses.
When a model-based selection rule compares candidate doses with target \(\phi\), the skeleton affects which candidates are close to the target over the range of plausible \(a\). The example below illustrates this at a fixed \(a\), only to isolate the effect of spacing. It is not a posterior calculation and does not incorporate observed outcomes or parameter uncertainty.
Worked Example: Comparing Tight and Wide Spacing
Keep \(\phi=0.25\) and dose 3 as the anchor, and evaluate both skeletons at \(a=1.4\). A tight logit-offset choice uses \((-0.4,-0.2,0,0.2,0.4)\), giving the skeleton from the previous example. The modeled probabilities are obtained by raising each entry to 1.4: \(0.1826^{1.4}\approx0.0925\), \(0.2144^{1.4}\approx0.1158\), \(0.25^{1.4}\approx0.1436\), \(0.2893^{1.4}\approx0.1762\), and \(0.3321^{1.4}\approx0.2137\). The transformed sequence is approximately \((0.0925,0.1158,0.1436,0.1762,0.2137)\).
The absolute differences from the target are approximately \(0.1575,0.1342,0.1064,0.0738,\) and \(0.0363\). For this fixed parameter value, dose 5 is closest to the target, at about 21.37% toxicity. The anchor dose is no longer the closest because the common exponent has lowered its modeled probability to about 14.36%.
For a wider skeleton, use offsets \((-1.2,-0.6,0,0.6,1.2)\). The baseline probabilities are approximately \((0.0912,0.1547,0.2500,0.3779,0.5253)\). Raising these to 1.4 gives approximately \((0.0350,0.0734,0.1436,0.2560,0.4060)\); the fourth probability, calculated before rounding as approximately \(0.25602\), is about \(0.00602\) above the target. The fifth probability is about \(0.1560\) above target. The other three absolute differences are about \(0.2150,0.1766,\) and \(0.1064\). Dose 4 is therefore closest to the target in this wider-skeleton illustration.
The contrast is the point: with the same target and exponent, the tight skeleton makes dose 5 closest, while the wider skeleton makes dose 4 closest. A skeleton is therefore not a neutral list of starting values. Its spacing, combined with the model form, shapes the dose-toxicity curve and can affect decisions.
Construction Choices and Checks
A practical construction starts by identifying the target rate and the candidate dose intended to anchor it. Then choose the direction and scale of offsets, calculate the probabilities without premature rounding, and inspect the full sequence. The clinical context matters: if the dose range spans large changes in exposure, a skeleton with extremely small adjacent differences may fail to represent meaningful risk changes. Conversely, a widely spaced skeleton may imply sharp jumps between adjacent doses.
State the target toxicity rate \(\phi\) and the candidate dose \(d_k\) to which it will be attached at \(a=1\). If no candidate dose is intended to equal the target at the reference value, identify the closest bracketing doses instead.
For logit offsets, require \(c_1<\cdots<c_J\) and \(c_k=0\). Make the magnitude of the offsets reflect a considered expectation about how rapidly risk changes over the dose range.
Verify \(0<q_1<\cdots<q_J<1\), the anchor value, and the adjacent differences. Calculate \(q_j^a\) over representative positive values of \(a\) to see whether the curve behaves plausibly away from the reference point.
The check over representative \(a\) values is a sensitivity assessment, not a substitute for prior calibration. A skeleton specifies the reference curve, while a prior distribution for the model parameter specifies how much uncertainty is assigned to curves around it. The earlier material on prior predictive calibration addresses that broader task. A sensible-looking skeleton alone does not guarantee sensible prior predictive toxicity probabilities.
Worked Example: A Different Target and Uneven Offsets
Consider four ordered doses, target \(\phi=0.30\), and dose 2 as anchor. Choose offsets \((-1,0,0.25,1)\). Since \(\operatorname{logit}(0.30)=\log(0.30/0.70)\approx-0.8473\), the probabilities are
The sequence is valid and increasing, and the second dose is anchored at 30%. At \(a=1.5\), the modeled probabilities are approximately \((0.0503,0.1643,0.2115,0.3948)\), from \(0.1362^{1.5}\), \(0.3000^{1.5}\), \(0.3550^{1.5}\), and \(0.5381^{1.5}\), respectively. Their absolute distances from 0.30 are approximately \((0.2497,0.1357,0.0885,0.0948)\). Dose 3 is closest to the target: its modeled toxicity is about 21.15%, compared with about 39.48% at dose 4.
This calculation also shows why anchoring is a reference condition rather than a guarantee of later selection. The anchor probability is 0.30 at \(a=1\), but at \(a=1.5\) the model lowers that dose to about 16.43%. The neighboring doses and their spacing determine which modeled probability is then nearest 30%.
Diagnostics, Alternatives, and Limitations
Several checks can reveal a skeleton that is not doing its intended job:
- Ordering failure: if \(q_j\leq q_{j-1}\), the skeleton does not encode strictly increasing toxicity over ordered doses. Check dose order, offset order, and calculations.
- Invalid probabilities: a value at or outside zero and one is incompatible with the ordinary power construction and may indicate an arithmetic or implementation error.
- Unintended anchor: if the intended anchor does not equal \(\phi\) at \(a=1\), check that its offset is exactly zero and that the same target was used throughout.
- Uninformative compression or extreme spread: compare adjacent gaps and evaluate \(q_j^a\) over plausible parameter values. Nearly indistinguishable risks or abrupt jumps may be inconsistent with the intended dose range.
- Overinterpretation: a monotone skeleton is not evidence that true toxicity is monotone, and a target-matching entry does not establish that the dose is safe or optimal.
A common alternative is to use an ordered-dose logistic model, where dose covariates and model parameters determine the curve. Such a model can express a different pattern of dose changes and is not identical to the power model. Another option is to construct probabilities by direct probability-scale increments. Each approach has its own structural assumptions. Comparisons should therefore use the implied toxicity curves, not just the numerical values of parameters or the apparent smoothness of the baseline sequence.
The CRM power model has important limitations. It forces all dose probabilities to move together through one parameter, so data may not be able to represent local irregularities in the dose-response curve. Skeleton values are design inputs, and poor choices can influence model behavior, especially with sparse information. The parameter transformation also does not make \(a\) a dose effect: \(a\) is a model parameter, while the \(q_j\) values correspond to the ordered doses.
Finally, skeleton construction is only one part of dose-finding design. A selected dose must also be judged against safety constraints, the target definition, and the trial's decision rules. The calculations here explain how the reference curve is constructed and how its spacing affects modeled probabilities; they do not replace those broader design considerations.
Check Your Understanding
Answer each question using the definitions and calculations above. Solutions are included below.
- For target \(\phi=0.20\), an anchor dose has offset \(c=0\). What is its skeleton probability, and why?
- For \(q=0.25\), calculate the modeled probability \(q^2\) and state whether it is above or below the skeleton value.
- A four-dose skeleton is \((0.10,0.28,0.24,0.60)\). Identify the construction problem.
- At a fixed positive \(a\), why does a strictly increasing skeleton remain strictly increasing after the power transformation?
Solutions. (1) The probability is \(q=\operatorname{logistic}\{\operatorname{logit}(0.20)+0\}=0.20\), because a zero offset preserves the target. (2) \(0.25^2=0.0625\), below 0.25 because raising a number strictly between zero and one to a power greater than one reduces it. (3) The third entry, 0.24, is below the second entry, 0.28, so the skeleton is not strictly increasing; check its ordering or calculation. (4) For \(a>0\), the function \(x\mapsto x^a\) is strictly increasing for \(x>0\), since its derivative is \(a x^{a-1}>0\). Therefore \(q_i<q_j\) implies \(q_i^a<q_j^a\).