Managing Dose-Range Boundaries and Extrapolation

Why dose-range boundaries need special attention

A BLRM can calculate a posterior toxicity probability at any positive dose. That mathematical ability does not mean the calculation is equally well supported everywhere. At the lowest or highest candidate dose, there may be no candidate dose on one side to help contextualize a recommendation. More importantly, the candidate dose may lie outside the range actually studied. In that case, the model is extrapolating: its estimate depends on the assumed dose-response form and the posterior for its parameters, rather than on observations at nearby doses.

The distinction matters in dose escalation. A boundary dose can be the next dose considered, the dose currently recommended, or simply the endpoint of the protocol’s candidate grid. These are not interchangeable. The lowest candidate dose need not be the lowest observed dose, and the highest candidate dose need not be supported by observations at or near it. A boundary location alone does not determine whether a dose is safe, toxic, admissible, or appropriate to recommend. Those judgments depend on the posterior evidence and the protocol’s decision rules.

The BLRM at a boundary

For a binary dose-limiting toxicity (DLT) outcome, the two-parameter BLRM represents the probability of a DLT at dose \(d\) as

Here \(d_{\mathrm{ref}}\) is a fixed positive reference dose, \(p(d)\) is the DLT probability at dose \(d\), and \(\alpha\) and \(\beta\) are the intercept and dose slope. The model assumes that, conditional on the dose-specific probabilities, patient DLT outcomes are independent Bernoulli outcomes, and that the log odds change linearly with log dose over the range where the model is used. The model is monotone increasing in dose only when \(\beta>0\); a prior that permits negative values does not guarantee monotonicity in every posterior draw.

Dose-range boundary concerns are therefore about both data support and model assumptions. At a dose beyond the observed range, there is no local observation to check whether the log-linear relationship continues. The posterior still produces an estimate, but its uncertainty reflects uncertainty in the slope as well as the intercept. A strong prior on the slope can make this estimate look precise even when the data contain little information about the extrapolation.

For a given dose, write \(x=\log(d/d_{\mathrm{ref}})\) and \(Z(d)=\alpha+\beta x\). If the posterior for \((\alpha,\beta)\) is approximately bivariate normal, then \(Z(d)\) is normal. Its posterior mean and variance are

The \(x^2\) term shows why uncertainty in the slope can matter more as the dose moves away from the reference dose. The covariance term also matters: depending on the posterior dependence between intercept and slope, it can increase or decrease uncertainty at a particular dose. Thus, posterior variance need not increase steadily with dose. It is not correct to diagnose extrapolation by looking only at the variance or only at the distance from \(d_{\mathrm{ref}}\); compare the dose with the observed range and examine the joint parameter uncertainty.

Using the posterior to assess a boundary dose

The BLRM’s posterior distribution can be used to evaluate uncertainty in toxicity at a boundary. For a fixed overdose threshold \(p_{\mathrm{OD}}\), the event that dose \(d\) exceeds that threshold is equivalent to a threshold event for the linear predictor:

If \(Z(d)\mid\text{data}\) is normal with posterior mean \(m_d\) and standard deviation \(s_d\), then

where \(\Phi\) is the standard normal cumulative distribution function. In practice, this probability is often calculated from posterior draws rather than a normal approximation. Either way, the probability describes posterior uncertainty under the specified model and prior; it is not a direct frequency of future DLTs.

The following calculation also explains why one should inspect the boundary itself, not only a posterior mean toxicity. Applying the logit transformation to a posterior mean of \(\alpha\) and \(\beta\) gives a point estimate, but it does not show how much posterior mass lies above a clinically relevant toxicity threshold.

What the examples imply about boundary decisions

The two examples show that extrapolation can occur at either end of a candidate grid. When the model has a positive dose slope, a lower-than-observed dose will generally have a lower fitted toxicity probability, and a higher-than-observed dose a higher one. But the amount of uncertainty in those predictions depends on the posterior for both parameters and their covariance. The posterior may also be sensitive to prior assumptions, particularly when the data cover a narrow dose range or provide little information about the slope.

A short derivation makes the role of distance explicit. Let \(x=\log(d/d_{\mathrm{ref}})\). Under a bivariate normal posterior for \((\alpha,\beta)\), any linear combination of those parameters is normal. Therefore \(Z(d)=\alpha+x\beta\) has the mean and variance given above. If the posterior covariance is zero, the slope-uncertainty contribution to the variance is \(x^2\operatorname{Var}(\beta\mid\text{data})\), which grows with \(|x|\). With nonzero covariance, the additional term \(2x\operatorname{Cov}(\alpha,\beta\mid\text{data})\) can offset or amplify that growth at some doses. This is why neither “the endpoint must be most uncertain” nor “uncertainty always rises toward the highest dose” is a general theorem.

A boundary dose that was itself studied can still be uncertain. Having observations at a dose provides direct information about that dose’s toxicity probability, but the amount of information depends on the number of treated patients and DLT outcomes. Conversely, an unobserved candidate dose may have a narrow posterior if the model and prior impose strong structure, but that precision is not the same as direct empirical support.

Diagnostics and alternatives

Before acting on an endpoint estimate, verify the dose history and candidate grid, then review the posterior in the context of the observed range. Useful checks include:

• Distance from the observed doses: record whether the proposed dose is bracketed by observed doses and how far it lies beyond the nearest observation on a log-dose scale.
• Posterior slope and dependence: examine the posterior distribution of \(\beta\) and its association with \(\alpha\), rather than relying only on a fitted mean curve.
• Posterior toxicity at the endpoint: inspect the distribution or relevant tail probability for \(p(d)\), not just the inverse-logit of the mean linear predictor.
• Prior sensitivity: check whether reasonable changes to the slope prior substantially change the boundary estimate or posterior probability. Large changes signal that the data provide limited information for that extrapolation.
• Model fit across observed doses: compare fitted probabilities with observed cohort counts. A poor fit within the observed range weakens confidence in extending the same curve beyond it.

These checks do not replace the protocol’s admissibility and allocation rules. They help identify when a recommendation is driven by a model’s extension beyond the data rather than by local observations. If evidence is too weak, possible design responses include restricting the candidate range to doses supported by prior or emerging evidence, requiring additional data before moving farther outward, or using scientifically justified dose-response information. Such choices should be prespecified and clinically justified; changing a range after seeing outcomes can alter the operating characteristics of the design.

A common mistake is to say, “The model predicts a low toxicity rate at the lowest dose, so that dose is certainly safe,” or, at the upper endpoint, “The model returns a value, so the dose is supported.” Both statements confuse a model-based estimate with the strength of evidence. The correction is to state whether the dose is inside the observed range, report posterior uncertainty in context, and apply the protocol’s decision rule without treating extrapolation as direct observation.

Boundary analysis has limits. It cannot verify that the assumed dose-response relationship remains valid beyond the data, and posterior calculations are conditional on the model, prior, endpoint definition, and available follow-up. A sensitivity analysis can reveal dependence on assumptions, but it cannot manufacture information at an unobserved dose. For that reason, boundary estimates should be interpreted as conditional predictions and not as empirical guarantees.