CRM Decisions Under Skeleton Misspecification

Why Skeleton Shape Matters

The continual reassessment method (CRM) uses a dose-toxicity model to learn which dose is near a target toxicity probability. Its skeleton supplies dose-specific probabilities before trial outcomes are observed. In the one-parameter power CRM, one parameter transforms all skeleton probabilities together. This construction preserves dose ordering, but it also constrains the curve’s shape: a single parameter cannot independently adjust the probability at each dose.

Consequently, an ordering that is clinically credible does not guarantee that the model describes the spacing between dose risks well. If the true curve rises much more sharply, or more gradually, than the skeleton permits, posterior estimates may be systematically displaced. A dose recommendation can then be wrong even when every assumed ordering is correct.

Definition: Let $s_1<\cdots<s_J$ be a fixed CRM skeleton, with $0<s_j<1$, for $J$ ordered doses. In the one-parameter power CRM, the probability of a dose-limiting toxicity (DLT) at dose $j$, conditional on parameter $a>0$, is $$ q_j(a)=s_j^a. $$ The skeleton is order-correct if higher doses have higher toxicity probabilities in the true dose-toxicity curve as well as in the skeleton. It is shape-misspecified when the true probabilities cannot be represented by $s_j^a$ for a common value of $a$ across doses.

The condition $a>0$ ensures that $s_j^a$ increases with $s_j$, so the model preserves the skeleton’s ordering. It does not ensure that the model’s probabilities equal the true probabilities. For example, taking logarithms gives $\log q_j(a)=a\log s_j$: every dose’s log probability is multiplied by the same factor. Thus, the power CRM allows a particular family of curve shapes, not every increasing curve.

Model and Assumptions

Suppose $n_j$ patients have been treated at dose $j$, and $y_j$ of them experienced a DLT. The usual likelihood assumes that outcomes are conditionally independent Bernoulli observations, with a common DLT probability $q_j(a)$ for patients treated at dose $j$. Equivalently, $Y_j$, the DLT count at that dose, has a binomial distribution with sample size $n_j$ and probability $q_j(a)$. The model also assumes that the dose levels are ordered and that the skeleton values are fixed before using the outcomes in the likelihood.

With prior density or mass function $\pi(a)$, the posterior is proportional to the prior times the likelihood:

\pi(a\mid \mathbf{y})\ \propto\ \pi(a) \prod_{j=1}^{J} q_j(a)^{y_j}\{1-q_j(a)\}^{n_j-y_j}.

This likelihood conditions on the doses assigned and the observed outcomes. In a sequential trial, assignments based only on information already observed do not add an unknown-parameter factor to this conditional outcome likelihood. The result still depends on the outcome model being a reasonable description of toxicity at each dose. If follow-up is incomplete, outcomes are dependent in ways not represented by the model, or toxicity risk changes over time, this simple likelihood may be inadequate.

A common CRM decision rule estimates each dose’s toxicity probability by its posterior mean, $\bar q_j=E\{q_j(a)\mid\mathbf{y}\}$, and recommends the eligible dose whose posterior mean is closest to the target $\phi$:

\hat{j}=\arg\min_{j\in\mathcal{E}}|\bar q_j-\phi|.

Here $\mathcal{E}$ is the set of doses still eligible under the trial’s safety exclusions and escalation restrictions. The selection rule does not imply that the selected dose has true toxicity probability closest to $\phi$; that conclusion requires the model’s probability estimates to be adequate.

Worked Example: An Ordered Skeleton With the Wrong Shape

Consider three hypothetical doses with skeleton $s=(0.05,0.20,0.50)$. Suppose, purely for illustration, that the actual DLT probabilities are $p=(0.02,0.09,0.35)$. Both sequences increase with dose, so the ordering is correct. Can one power parameter reproduce all three actual probabilities?

At each dose, a parameter that matches the actual probability would have to be $a_j=\log(p_j)/\log(s_j)$. The three values are approximately $a_1=\log(0.02)/\log(0.05)=1.305$, $a_2=\log(0.09)/\log(0.20)=1.496$, and $a_3=\log(0.35)/\log(0.50)=1.514$.

Since these values differ, no single $a$ gives all three actual probabilities. For instance, matching dose 2 requires $a\approx1.496$. That gives $q_1(a)=0.05^{1.496}\approx0.0113$, below the hypothetical actual value $0.02$, while $q_3(a)=0.50^{1.496}\approx0.354$, close to the actual value $0.35$. The skeleton has the correct ordering, but the power family cannot match this curve at every dose.

How Misspecification Enters a Recommendation

The posterior combines the assumed curve with the observed outcomes. It does not learn an unrestricted toxicity probability at every dose: it updates $a$, then maps that one parameter through the fixed skeleton. If the skeleton’s shape is wrong, outcomes at one dose can move the estimates at other doses in a way that is appropriate for the assumed family but not for the true curve.

The following finite prior makes that mechanism transparent. It is a hypothetical calculation, not a recommended default prior. Consider skeleton $s=(0.05,0.20,0.50)$, target $\phi=0.25$, and a prior assigning probability $1/3$ to each of $a=0.5,1,2$. Suppose three patients are treated at dose 2 and one experiences a DLT. At dose 2 the modeled probabilities are $0.20^{0.5}$, $0.20$, and $0.20^2$ for those three parameter values.

Worked Example: Posterior Update After One DLT

For one DLT and two non-DLT outcomes at dose 2, the binomial likelihood at a candidate value $a$, omitting the common binomial coefficient, is $q_2(a)\{1-q_2(a)\}^2$. Substitution gives:

\begin{aligned} L(0.5)&=\sqrt{0.20}(1-\sqrt{0.20})^2\approx0.1366563,\\ L(1)&=0.20(1-0.20)^2=0.1280000,\\ L(2)&=0.20^2(1-0.20^2)^2=0.0368640. \end{aligned}

Because the three prior probabilities are equal, the posterior probabilities are proportional to these unrounded likelihoods. Their sum is $0.3015203$, so the posterior probabilities for $a=0.5,1,2$ are approximately $0.4532,0.4245,0.1223$, respectively.

At dose 1, the three modeled probabilities are $\sqrt{0.05}$, $0.05$, and $0.05^2$. Weighting by the posterior probabilities gives $\bar q_1\approx0.453224\sqrt{0.05}+0.424515(0.05)+0.122260(0.0025)=0.1229$. At dose 2, the corresponding calculation is $\bar q_2\approx0.453224\sqrt{0.20}+0.424515(0.20)+0.122260(0.04)=0.2925$. At dose 3, it is $\bar q_3\approx0.453224\sqrt{0.50}+0.424515(0.50)+0.122260(0.25)=0.5633$.

The absolute distances from the target $0.25$ are approximately $0.1271$, $0.0425$, and $0.3133$. Therefore, if all three doses are eligible, this rule recommends dose 2. The calculation is internally consistent with the assumed power CRM, but by itself it cannot establish that dose 2 is truly closest to the target.

Distinguishing Data Variability From Shape Failure

An unusual cohort outcome is not, by itself, evidence that the skeleton shape is wrong. Even under a correct model, random outcomes can look surprising, particularly in small cohorts. Conversely, a shape-misspecified model may produce plausible-looking outcomes while its estimated probabilities remain biased. The distinction matters: changing the skeleton in response to every small fluctuation can make decisions unstable, while treating persistent lack of fit as random noise can leave a systematic error unaddressed.

Worked Example: A Plausible Cohort Can Still Lead to a Misleading Dose

Use the hypothetical true probabilities $p=(0.02,0.09,0.35)$ from the earlier example and target $\phi=0.25$. The distances from the target under that true curve are $0.23$ at dose 1, $0.16$ at dose 2, and $0.10$ at dose 3. Thus, dose 3 is the closest of these doses to the target under the hypothetical truth. Yet the CRM calculation with the stated skeleton and cohort data recommends dose 2.

One DLT among three patients at dose 2 is not an implausible observation under this true curve. Its probability is $3(0.09)(1-0.09)^2=3(0.09)(0.91)^2=0.223587$, or approximately $0.2236$.

This example does not show that the data identify dose 3 as the correct recommendation: three outcomes at one dose provide little information about the full curve. It shows the more limited point that a plausible outcome, processed through an order-correct but shape-misspecified model, can yield a model-based recommendation that differs from the dose nearest target under the stipulated true probabilities.

Diagnostics, Alternatives, and Limits

Shape misspecification is hard to diagnose from sparse early-phase data because few doses may have been observed and each may have only a small number of patients. Useful checks should therefore be interpreted cautiously and alongside clinical knowledge.

Check the skeleton against plausible curve shapes. Ask whether the implied spacing between adjacent probabilities is credible, not only whether the probabilities increase. A correct rank order is necessary for this CRM specification but does not validate its shape.
Compare observed counts with model implications. Posterior predictive summaries can reveal repeated discrepancies between observed DLT counts and counts generated from the fitted model. With small, adaptively assigned cohorts, apparent discrepancies can be chance variation; doses with little or no data cannot be checked directly.
Look for persistent patterns, not isolated outcomes. For example, repeated observations suggesting a much steeper rise in risk than the model can represent warrant review. A single cohort should not be treated as proof of a shape failure.
Evaluate operating behavior before a trial. Simulations under a range of plausible increasing true curves can show whether the design often selects a dose far from the true target-nearest dose or exposes too many patients to overly toxic doses. Such results depend on the simulated curves and trial rules; they are not guarantees about an individual trial.

The principal alternative is to improve the assumptions rather than to rely on ordering alone. Clinician elicitation and calibration can make skeleton probabilities and their spacing more credible. A more flexible dose-toxicity model can represent a wider range of curves, but greater flexibility also demands adequate information and careful specification. A conservative starting dose, safety exclusions, and no-escalation restrictions can limit exposure while uncertainty remains; they do not repair an incorrect curve shape or ensure a correct final selection. No model eliminates uncertainty, and no diagnostic can prove that the true dose-toxicity curve belongs to the assumed family.

Common pitfall: “The skeleton is increasing, so the CRM must preserve the correct dose-toxicity relationship.” The first clause establishes only an ordering constraint. A one-parameter power transformation preserves that order, but fixes how probabilities at different doses move together. If their spacing is wrong, estimated probabilities and the target-based recommendation can still be distorted.

The practical implication is to treat a CRM recommendation as conditional on its model and decision rule. The recommendation is the dose closest to the target according to the fitted probabilities among eligible doses, not a model-free statement about the true safest or most effective dose. This distinction is especially important when the posterior is concentrated: concentration under a restricted model does not rule out systematic shape error.

Check Your Understanding

Answer each question using the assumptions and examples in this tutorial. Brief solutions follow the questions.

Why does $a>0$ preserve the ordering of a strictly increasing skeleton in the power CRM?
For a skeleton value $s=0.25$ and parameter $a=2$, calculate the modeled toxicity probability.
Suppose the true probabilities at three increasing doses are $0.03,0.12,0.40$, while the skeleton is $0.05,0.20,0.50$. What condition must hold for the power CRM to represent all three probabilities exactly?
In the posterior example, what dose is selected by the stated rule, and what does that selection not establish?

Solutions. 1. For $0<s_j<s_k<1$ and $a>0$, raising both to the positive power preserves their order: $s_j^a<s_k^a$. 2. $q=0.25^2=0.0625$, or 6.25%. 3. The three ratios $\log(p_j)/\log(s_j)$ must be equal to the same positive value of $a$; if they differ, one power parameter cannot match all three. 4. Dose 2 is selected because its posterior mean probability, $0.2925$, is closest to $0.25$. This establishes only the decision under the stated model, data, eligibility set, and rule; it does not establish that dose 2 is truly nearest the target.