Why Calibrate Around the Target?
A dose-toxicity skeleton is not merely a list of increasing probabilities. Its values set the reference curve for the CRM: under the one-parameter power model, the parameter moves that curve while preserving its dose ordering. A skeleton can be monotone and still be poorly calibrated. For example, plausible parameter values might put every dose far below the target, or might make only the highest dose reach it. Calibration checks whether reasonable curves place the target toxicity rate near doses that are clinically plausible.
The target toxicity rate is denoted by \(\phi\), and the skeleton probability at dose \(j\) by \(\alpha_j\). The power CRM model is \(p_j(\beta)=\alpha_j^{\exp(\beta)}\). Here \(\beta\) is a scalar model parameter, and \(\beta=0\) reproduces the skeleton exactly. Calibration examines where the model reaches \(\phi\) as \(\beta\) varies. It does not by itself select a dose for participants, determine a prior for \(\beta\), or establish that the model is clinically adequate.
Target-Crossing Parameter
Assume \(0<\alpha_j<1\) for every dose and \(0<\phi<1\). The skeleton should be increasing across ordered doses, so \(0<\alpha_1<\cdots<\alpha_J<1\). The power-model transformation uses one common \(\beta\), and the same model form applies at every dose. These assumptions ensure probabilities are defined and preserve the dose ordering.
To find the parameter value where dose \(j\) has toxicity probability \(\phi\), set the model probability equal to the target and solve. Taking natural logarithms twice gives:
Both logarithms in the ratio are negative, so the ratio is positive and the outer logarithm is defined. The sign also has a direct interpretation. Since \(\log(\alpha_j)<0\), the derivative \(dp_j(\beta)/d\beta=p_j(\beta)\exp(\beta)\log(\alpha_j)\) is negative: increasing \(\beta\) lowers the modeled toxicity probability. Thus, if \(\alpha_j<\phi\), the crossing is at a negative \(\beta\), because the probability must rise from its reference value. If \(\alpha_j>\phi\), the crossing is at a positive \(\beta\), because it must fall. If \(\alpha_j=\phi\), the crossing is at \(\beta=0\).
Worked Example: Finding the Crossing for One Dose
Suppose the target is \(\phi=0.25\), and dose 2 has skeleton probability \(\alpha_2=0.10\). Find the parameter value at which the power CRM assigns dose 2 a 25% toxicity probability.
Substitute the target and skeleton value into the crossing formula:
At \(\beta=0\), dose 2 is below the target at 10%. A negative value of \(\beta\) raises its modeled probability, so the negative crossing is consistent with the model’s direction. The ratio is kept unrounded until the final result; rounding it first can change the last reported digit.
Choosing a Reference Dose and Spacing
A useful starting point is to decide which dose or neighboring doses should be near \(\phi\) when \(\beta=0\). This is a clinical and design judgment, not a mathematical requirement. If a particular dose is the natural reference for the target, setting its skeleton probability near \(\phi\) makes its target crossing near zero. The adjacent skeleton probabilities then express how rapidly the reference toxicity curve rises across the dose grid.
A single crossing is not enough to judge spacing. Consider the following hypothetical five-dose skeleton with target \(\phi=0.25\):
| Dose | Skeleton probability \(\alpha_j\) | Target-crossing \(\beta_j^*\), approximately |
|---|---|---|
| 1 | 0.05 | -0.772 |
| 2 | 0.10 | -0.507 |
| 3 | 0.20 | -0.149 |
| 4 | 0.30 | 0.141 |
| 5 | 0.45 | 0.552 |
For example, at dose 3 the calculation is \(\beta_3^*=\log\{\log(0.25)/\log(0.20)\}\approx-0.1493\). At dose 4 it is \(\beta_4^*=\log\{\log(0.25)/\log(0.30)\}\approx0.1410\). Their crossings lie on opposite sides of zero and relatively close to it. This indicates that the target is near the reference curve between those doses, rather than requiring a large parameter shift to reach either one. Whether that placement is clinically appropriate still depends on the actual dose grid and expected toxicity.
Worked Example: Checking Curves Across a Plausible Parameter Range
For the same hypothetical skeleton \((0.05,0.10,0.20,0.30,0.45)\), suppose the design team wants to inspect the curves at \(\beta=-0.5\) and \(\beta=0.5\). These values are illustrative calibration points, not a claim that the parameter must lie in this range. Calculate the probabilities and identify the dose nearest the 25% target at each point.
At \(\beta=-0.5\), \(\exp(\beta)=0.6065\). For dose 2, for example, \(p_2(-0.5)=0.10^{0.6065}=\exp\{0.6065\log(0.10)\}\approx0.2474\). Applying the same formula to all doses gives approximately \((0.1625,0.2474,0.3767,0.4818,0.6160)\). Dose 2 is nearest to 0.25; its absolute difference is \(0.25-0.2474=0.0026\).
At \(\beta=0.5\), \(\exp(\beta)=1.6487\). For dose 5, \(p_5(0.5)=0.45^{1.6487}=\exp\{1.6487\log(0.45)\}\approx0.2681\). The complete vector is approximately \((0.0072,0.0225,0.0704,0.1374,0.2681)\). Dose 5 is nearest to target, with absolute difference \(0.2681-0.25=0.0181\).
The exercise shows how a common parameter shift changes the whole curve: negative \(\beta\) raises the modeled probabilities, while positive \(\beta\) lowers them. It also shows that the target can move from a lower dose to a higher dose as the curve shifts. In actual calibration, the parameter values inspected should be justified as representing plausible curves for the clinical setting; they are not automatically appropriate simply because they are symmetric around zero.
Worked Example: Comparing Two Candidate Reference Placements
Suppose dose 3 is considered the clinically natural reference dose for a hypothetical five-dose study with target \(\phi=0.25\). Candidate A assigns it \(\alpha_3=0.20\), while candidate B assigns it \(\alpha_3=0.40\). Which candidate puts dose 3 closer to the target at \(\beta=0\), and what parameter shift makes each candidate reach the target there?
At \(\beta=0\), the modeled probabilities equal the skeleton probabilities. Candidate A is 0.05 below target, whereas candidate B is 0.15 above target. The crossing values are:
Candidate A therefore puts the target crossing closer to the reference curve at dose 3. Its negative crossing raises the probability from 0.20 to 0.25. Candidate B needs a larger positive shift to lower 0.40 to 0.25. This comparison alone does not establish that A is preferable: the remaining dose probabilities must also be plausible and appropriately spaced. If clinical judgment instead makes a higher reference toxicity at dose 3 more credible, candidate B may be more defensible despite its more distant crossing.
A Calibration Workflow and Its Checks
Calibration is an iterative check of a proposed skeleton, not a mechanical optimization with a universally correct answer. A practical workflow is:
State \(\phi\), order the doses, and identify which dose or neighboring doses clinicians consider plausible locations for target-level toxicity.
Specify probabilities strictly between zero and one, increasing over dose. Explain the clinical reasoning behind their levels and gaps.
For each dose compute \(\beta_j^*=\log\{\log(\phi)/\log(\alpha_j)\}\). Check both the values and their signs against whether each \(\alpha_j\) is below or above \(\phi\).
Choose clinically justified values of \(\beta\), calculate \(p_j(\beta)\) across the dose grid, and check which doses are near target. Look for curves that place target-level toxicity only at implausible extremes.
If reference levels or gaps create implausible crossings, revise the skeleton and repeat the checks. Record the rationale, including uncertainty about the plausible curves.
Several diagnostics are especially useful. A target crossing far outside the parameter values regarded as plausible indicates that the relevant skeleton probability may be poorly placed. If all crossing values lie on one side of zero, the reference curve places the target consistently above or below its dose probabilities; this may be intentional, but should be checked against the clinical rationale. Very large gaps between adjacent crossing values can mean the skeleton skips a broad range of possible target locations. Conversely, nearly identical adjacent probabilities may make dose distinctions weak. These are warning signs to investigate, not automatic rejection rules.
The model also imposes a limitation: every dose probability is transformed by the same one-parameter power relationship. Calibration can assess whether this family of curves is plausible, but cannot make a restrictive model flexible. If clinicians believe toxicity rises unusually sharply over only part of the dose range, or if the skeleton’s pattern is inconsistent with that belief, a well-chosen reference level alone will not resolve the mismatch.
Common Pitfalls and Alternatives
- Reversing the crossing sign. Because \(p_j(\beta)\) decreases with \(\beta\), a dose with \(\alpha_j<\phi\) reaches the higher target at a negative crossing, not a positive one. A dose with \(\alpha_j>\phi\) crosses at a positive value.
- Treating \(\beta=0\) as a dose recommendation. It means the model curve equals the skeleton. It does not mean the dose should be assigned, nor does it account for observed outcomes or posterior uncertainty.
- Checking only the dose nearest the target at the reference curve. The shifted curves matter too. Inspect several clinically plausible \(\beta\) values and all doses, not just the skeleton at \(\beta=0\).
- Choosing spacing by arithmetic difference alone. Equal differences between skeleton probabilities do not imply equal gaps in target-crossing \(\beta\). The logarithmic crossing formula is nonlinear, so inspect the crossing values directly.
Direct clinician elicitation of dose-specific toxicity probabilities is a useful basis for a skeleton, while calibration translates those elicited values into consequences for the power CRM. A different approach is to specify a small set of plausible complete toxicity curves and derive or adjust skeleton values to represent them. Neither approach removes the need for clinical judgment. Calibration is a check on the implications of the chosen skeleton, not a substitute for elicitation and not a guarantee against model misspecification.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For target \(\phi=0.30\) and skeleton probability \(\alpha_j=0.15\), should the target crossing be positive or negative? Explain using the direction of the power CRM curve.
- A dose has \(\alpha_j=\phi\). What is its target-crossing value, and what does that mean for the reference curve?
- For \(\phi=0.25\), compare the target-crossing values for skeleton probabilities 0.20 and 0.40. Which crossing is negative, and why?
- Why is checking the target-nearest dose at \(\beta=0\) insufficient for calibrating the skeleton?
Solutions. (1) The crossing is negative because \(0.15<0.30\), and a negative \(\beta\) raises \(p_j(\beta)\) from its reference value. (2) The crossing is \(\beta_j^*=0\); the reference skeleton already assigns that dose the target probability. (3) The crossings are \(\log\{\log(0.25)/\log(0.20)\}\approx-0.1493\) and \(\log\{\log(0.25)/\log(0.40)\}\approx0.4141\), respectively. The first is negative because its skeleton probability is below target; the second is positive because its skeleton probability is above target. (4) Shifting \(\beta\) changes probabilities at every dose, so a different dose may be closest to target, or target-level toxicity may occur only at an implausible extreme. Inspecting several plausible curves reveals these effects.