Prior Predictive Calibration for Dose-Toxicity Models

Why Calibrate Before Observing Outcomes?

A dose-toxicity model can be mathematically well behaved and still imply implausible risks before any trial data are collected. For example, a prior might place substantial probability on toxicity being common at the lowest candidate dose, or on toxicity remaining negligible throughout the entire dose range. Prior predictive calibration makes these implications visible. It is a check on what the model and its prior jointly say about toxicity across doses, not a way to use future trial outcomes.

The focus here is a logistic model for a binary dose-limiting toxicity (DLT) outcome. Let \(Y_i=1\) denote a DLT for participant \(i\), and \(Y_i=0\) otherwise. Let \(x(d)\) be the centered dose covariate, increasing with dose, with \(x(d_{\mathrm{ref}})=0\) at a chosen reference dose. Centering makes the intercept interpretable at that reference dose.

Here the model and its assumptions are

The Bernoulli likelihood assumes that, conditional on the parameters and assigned dose, each participant's DLT outcome has the stated probability. For the usual likelihood across participants, outcomes are conditionally independent. The logistic model maps any real linear predictor to a probability strictly between zero and one. If \(\beta>0\) and \(x(d)\) increases with dose, the model implies increasing toxicity risk. A prior that allows \(\beta\leq 0\) also allows flat or decreasing toxicity curves; that may be scientifically implausible for the intended setting and should be checked rather than overlooked.

What to Examine in the Prior Predictive Distribution

A single prior mean risk does not describe the full range of implications. At each candidate dose, useful summaries include the median and quantiles of the risk across prior draws, the prior mean risk, and the prior probability that risk exceeds a clinically specified target. If the target toxicity rate is \(\phi\), one such check is \(P_{\pi}\{p(d\mid\alpha,\beta)>\phi\}\). This is a probability over parameters under the prior; it is not a probability that the target or model is true.

The joint curves matter as well as the pointwise summaries. A set of curves can reveal whether the prior implies plausible ordering across doses, whether most curves reach the target within the candidate range, and whether some draws imply excessive toxicity at the lowest dose. Since each parameter draw produces one whole curve, it is also useful to inspect curves together rather than treating each dose independently.

For a positive slope, the dose covariate at which a particular parameter draw reaches target risk \(\phi\) follows directly from the model:

This calculation translates intercept and slope draws into target-crossing locations. If \(x_{\phi}\) is below the covariate value at the lowest candidate dose, that curve is already above target at the bottom of the range. If it is above the value at the highest candidate dose, that curve does not reach target within the range. A crossing location is not defined by this expression when \(\beta=0\), and a negative slope reverses the interpretation.

Calibrating Priors to a Target and Dose Range

Calibration is an iterative process: specify a prior, generate prior parameter draws, transform each draw into a curve over the candidate doses, and assess the resulting risks against clinical expectations. If the implications are unsuitable, change the prior and repeat the check. The model structure and candidate dose range should be fixed for the check; changing the range can change whether curves appear to reach a target.

At the reference dose, the intercept has a particularly direct interpretation: \(p(d_{\mathrm{ref}})=\operatorname{logit}^{-1}(\alpha)\). If a central prior risk \(q\) is desired there, the corresponding intercept center is \(\operatorname{logit}(q)=\log\{q/(1-q)\}\). The spread of the intercept prior controls uncertainty in reference-dose risk. The slope prior then controls how rapidly risk changes with the centered dose covariate. These are different calibration tasks: matching a plausible reference risk does not by itself ensure plausible risk at other doses.

A short derivation clarifies how the target risk constrains both parameters. The logistic model gives \(\operatorname{logit}\{p(d)\}=\alpha+\beta x(d)\). Setting \(p(d)=\phi\) and applying the logit transformation yields \(\operatorname{logit}(\phi)=\alpha+\beta x(d)\). For \(\beta>0\), rearranging gives \(x_{\phi}=\{\operatorname{logit}(\phi)-\alpha\}/\beta\). Thus, even with an intercept centered to a plausible risk, a broad or large slope prior can make target crossings fall outside the candidate range or create steep changes between adjacent doses.

Simulation, Diagnostics, and Alternatives

In practice, draw many parameter vectors from the joint prior, calculate \(p(d)\) for every candidate dose using each vector, and summarize the resulting risks and curves. Preserve the joint parameter draws: independently simulating an intercept at each dose would not represent the specified dose-toxicity model or its correlated curve behavior. For any dose, an empirical estimate of prior exceedance probability is the fraction of draws whose risk exceeds \(\phi\). With independent prior draws, this fraction estimates the prior probability \(P_{\pi}\{p(d)>\phi\}\); Monte Carlo precision improves as the number of draws increases.

Common warning signs include a substantial probability of high toxicity at the lowest dose, little probability of reaching the target anywhere in the candidate range, implausibly sharp risk increases across adjacent doses, or frequent decreasing curves when increasing toxicity is expected. Check the slope prior, the intercept's reference-dose interpretation, the dose scaling and candidate range, and any dependence between intercept and slope. A broad intercept prior can produce extreme risks throughout the range; an overly concentrated or small slope prior can make curves nearly flat. These patterns are not automatically errors, but should be reconciled with clinical knowledge and the intended trial setting.

One alternative is to elicit plausible toxicity probabilities at selected doses and use them to inform parameter priors. This is often easier to discuss clinically, but the probabilities at different doses must be coherent with the assumed curve shape and their uncertainty must be translated into a joint parameter distribution. Another option is to assign priors directly to intercept and slope using expert judgment, then use prior predictive curves to test their consequences. Neither approach removes the need to check the implied curves. Prior predictive calibration is distinct from evaluating the posterior after trial outcomes become available: it identifies prior implications before those outcomes can influence the analysis.

A useful limitation to keep in mind is that calibration does not prove the logistic model is correct. The check is conditional on the chosen dose covariate, link function, candidate range, and prior. It also cannot determine a clinically acceptable exceedance probability by itself. Those judgments require substantive input. Finally, pointwise risk quantiles summarize uncertainty at each dose, but do not show which risks occur together along a single curve; inspect sampled curves or target-crossing locations when joint behavior matters.