Choosing a Prior for the CRM Parameter

Why the Parameter Prior Needs Clinical Interpretation

In the one-parameter power CRM, a fixed dose-toxicity skeleton supplies reference probabilities, while an unknown parameter transforms that skeleton into a curve. Choosing a prior for the parameter therefore does more than specify a distribution on a mathematical scale: it determines which toxicity curves are plausible before observing trial outcomes. A prior that is convenient to write down can still imply curves or cohort outcomes that are difficult to defend clinically.

Let \(p_j\) be the fixed skeleton probability at dose \(j\), with \(0<p_1<\cdots<p_J<1\), and let \(\theta\) be the CRM parameter. The model toxicity probability at dose \(j\), conditional on \(\theta\), is

At \(\theta=0\), the model curve equals the skeleton. A positive \(\theta\) raises the exponent above one and lowers each probability; a negative \(\theta\) puts the exponent between zero and one and raises each probability. The same \(\theta\) acts on every dose, so the prior describes uncertainty about a whole curve, not independent uncertainty at each dose.

A common starting choice is \(\theta\sim N(0,\sigma^2)\). Centering the prior at zero makes the skeleton the curve at the parameter's center, but does not generally make the prior mean toxicity probabilities equal to the skeleton: \(E[q_j(\theta)]\) need not equal \(q_j(E[\theta])=p_j\). The scale \(\sigma\), rather than the center alone, determines how much prior probability lies on substantially higher or lower curves. A normal distribution has unbounded support, so it can also assign some probability to curves that are clinically extreme; their prior-predictive consequences should be checked rather than ignored.

Read the Prior on the Curve Scale

A direct way to assess a candidate prior is to select representative parameter values, calculate their curves at all skeleton doses, and ask whether those curves are plausible. This is a diagnostic translation, not a replacement model. Under the power model, every such curve remains increasing in dose because the skeleton is increasing and the exponent \(\exp(\theta)\) is positive.

For a normal prior, this exercise can use several quantiles of \(\theta\), not only its mean and standard deviation. A curve at a parameter quantile is easy to interpret, but the probability of that one curve is zero for a continuous prior; interpretation should therefore consider ranges of curves and the prior probability assigned to those ranges. The prior can also be assessed through clinically meaningful events, such as the probability that toxicity at a particular dose exceeds a stated threshold.

Prior-Predictive Probabilities

Curve plots show what parameter values imply. Prior-predictive calculations show what the prior implies for data. At one dose, let \(Y\) be the number of patients with a dose-limiting toxicity (DLT) among \(m\) patients treated at that dose. Conditional on \(\theta\), assume patient outcomes are independent Bernoulli variables with common DLT probability \(q_j(\theta)\). Then \(Y\mid\theta\) has a binomial distribution, and the prior-predictive probability is found by averaging the conditional probability over \(\theta\):

where \(\pi(\theta)\) denotes the prior density. For a discrete prior, the integral is replaced by a sum over parameter values and their prior weights. Importantly, patients share the same unknown curve. They are independent conditional on \(\theta\), but outcomes are generally dependent marginally after \(\theta\) is averaged out.

This distinction matters for any cohort outcome. For example, the probability of three DLTs is \(E[q_j(\theta)^3]\), not generally \(\{E[q_j(\theta)]\}^3\). For a continuous prior, prior-predictive probabilities can be calculated by numerical integration or simulation: draw \(\theta\) from the prior, calculate the conditional outcome probability, and average those probabilities over draws. If instead outcomes are simulated, draw one \(\theta\) per simulated trial or cohort and then generate all relevant patient outcomes conditional on that shared value.

A Practical Prior-Assessment Sequence

A candidate prior should be checked in a consistent order. The goal is not to find a uniquely correct prior from the skeleton alone; it is to make the prior implications visible and judge whether they are defensible for the clinical setting.

Calibration and a Short Justification

The prior-predictive calculation has a simple justification. Conditional on \(\theta\), the probability of at least one DLT among \(m\) patients at dose \(j\) is \(g(q_j(\theta))=1-(1-q_j(\theta))^m\). Since \(\theta\) is uncertain before observing outcomes, the law of total probability requires averaging the conditional probability:

For \(m>1\), \(g(q)=1-(1-q)^m\) is concave on \(0<q<1\), because \(g''(q)=-m(m-1)(1-q)^{m-2}<0\). Jensen's inequality therefore gives \(E[g(q_j(\theta))]\leq g(E[q_j(\theta)])\). Thus the plug-in calculation using the mean probability overstates the prior-predictive chance of at least one DLT in this setting, except when the induced probability is effectively constant. The direction of this particular discrepancy follows from concavity; it is not a general license to replace mixture calculations with calculations at a mean.

Diagnostics, Alternatives, and Limitations

Several warning signs indicate that a parameter prior may not be doing what its authors intend:

• Implausible tail curves. A prior can be centered at zero but still assign substantial probability to near-zero or near-one toxicity probabilities at doses where neither is credible.
• Confusion between parameter and probability scales. A symmetric prior on \(\theta\) does not produce symmetric uncertainty around each \(p_j\). The power transformation is nonlinear.
• Predictive checks use an average curve as if it were fixed. This discards uncertainty about the shared parameter and generally gives incorrect cohort probabilities.
• Checks use only one dose. Because a single parameter transforms the entire skeleton, a prior that appears reasonable at one dose may imply implausible values elsewhere.
• Prior and skeleton are treated as interchangeable. The skeleton is the curve at \(\theta=0\); it is not the prior distribution, nor is it necessarily the prior mean curve.

A normal prior is convenient for computation and provides a clear center and scale on the parameter axis. A discrete prior can be easier to elicit when clinicians can compare a small number of plausible curves and assign weights to them. These approaches encode uncertainty differently: the normal prior includes a continuum of parameter values, while the discrete prior places mass only on its listed values. A uniform prior on \(\theta\) is another possible specification, but “uniform” depends on the parameter scale and does not imply uniform uncertainty about toxicity probabilities. Whatever form is used, assess the induced curves and predictive probabilities rather than judging the prior by its name.

These checks do not establish that the power CRM is correctly specified, that the skeleton is clinically accurate, or that the resulting design has acceptable operating characteristics. They assess what the chosen prior says before data are observed. They also cannot remove uncertainty in clinical elicitation: the resulting prior is a structured expression of judgments, not an objective discovery of the true toxicity curve.

Common Mistake