Prior Predictive Dose-Toxicity Curves

Why Inspect Prior Predictive Curves?

A prior on the BLRM parameters is not merely a mathematical starting point: together with the dose grid, it implies a distribution of toxicity risks at every candidate dose. Inspecting those implied curves before using trial outcomes helps reveal whether the prior expresses clinically credible beliefs about both the toxicity level and how quickly risk may change with dose. A prior can look reasonable when each parameter is considered separately yet imply implausible curves when the parameters act together.

The aim here is to examine the model's implications across the full candidate dose grid, not to select a dose or interpret a posterior distribution. The model and parameter meanings are established in the earlier material on the two-parameter Bayesian logistic regression model and model-specific prior calibration.

Definition: A prior predictive toxicity curve is a summary, across candidate doses, of the distribution of toxicity probabilities implied by the BLRM parameter prior before observing trial outcomes. At each dose it describes prior uncertainty about the probability of a dose-limiting toxicity (DLT) within the trial's specified DLT assessment window.

From the Parameter Prior to a Curve

Let \(d>0\) be a candidate dose and \(d_{\mathrm{ref}}>0\) a reference dose. In the BLRM, the DLT probability \(p(d)\) is modelled by

\operatorname{logit}\{p(d)\}=\alpha+\beta\log\left(\frac{d}{d_{\mathrm{ref}}}\right), \qquad p(d)=\frac{\exp\left\{\alpha+\beta\log(d/d_{\mathrm{ref}})\right\}} {1+\exp\left\{\alpha+\beta\log(d/d_{\mathrm{ref}})\right\}}.

Here \(\alpha\) is the log-odds of toxicity at the reference dose, and \(\beta\) is the change in log-odds per unit increase in log dose. In the usual monotone dose-escalation formulation, the prior puts probability one on \(\beta>0\). The candidate doses must be positive, the reference dose must be stated, and the parameter prior must be a proper joint distribution. The DLT definition, assessment window, and patient population to which the model applies must be held fixed when interpreting the implied risks.

For each prior draw \((\alpha^{(m)},\beta^{(m)})\), calculate \(p^{(m)}(d_j)\) at every dose \(d_j\) in the grid. At a particular dose, summaries such as the prior median and central quantiles describe the distribution of the toxicity probability. Connecting the medians across doses gives a convenient typical curve; plotting the quantiles as well displays uncertainty. These are pointwise summaries: a separate interval at each dose is not automatically a simultaneous band that contains an entire curve with the stated probability.

The prior predictive distribution can also describe outcomes, not just probabilities. If \(Y\) is the number of DLTs among \(n\) patients treated at dose \(d\), and patient outcomes are conditionally independent with common probability \(p(d)\), then \(Y\mid\alpha,\beta\) has a binomial distribution. Before outcomes are observed, its distribution averages that binomial likelihood over the parameter prior:

P(Y=y\mid d) = \int {n\choose y}\,p(d;\alpha,\beta)^y \{1-p(d;\alpha,\beta)\}^{n-y} \,\pi(\alpha,\beta)\,d\alpha\,d\beta.

This mixture includes both binomial outcome variation and uncertainty about the toxicity probability. For assessing curve shape, however, it is usually clearest to first inspect the probability curves themselves.

A Useful Shape Check

The dose-response implication follows directly by differentiating with respect to log dose. Since the derivative of the inverse-logit function at linear predictor \(\eta\) is \(p(1-p)\), and \(\eta=\alpha+\beta\log(d/d_{\mathrm{ref}})\), the derivative is

\frac{d p(d)}{d\log d} = \beta\,p(d)\{1-p(d)\}.

Because \(0<p(d)<1\), a positive slope gives a strictly increasing risk curve. The size of the change depends on both \(\beta\) and the current risk: \(p(1-p)\) is largest at \(p=0.5\) and smaller near zero or one. Thus, inspecting only \(\beta\) does not fully characterize the risk change over a dose grid. This derivative is a quick model-based diagnostic, not evidence that the real dose-toxicity relationship must be logistic.

Worked Example: Comparing Two Slopes Across a Dose Grid

Consider the hypothetical reference dose \(d_{\mathrm{ref}}=10\) mg and suppose the reference-dose risk is fixed at \(p(10\text{ mg})=0.20\). Then the reference-dose odds are \(0.20/(1-0.20)=0.25\), so \(\alpha=\log(0.25)\). Compare two positive slopes, \(\beta=0.15\) and \(\beta=1.5\), at 5, 10, and 20 mg.

At 5 mg, the dose ratio is \(0.5\). With \(\beta=0.15\), the odds are \(0.25(0.5)^{0.15}=0.22531\), giving risk \(0.22531/(1+0.22531)=0.1839\). At 20 mg, the ratio is 2; the odds are \(0.25(2)^{0.15}=0.27739\), giving risk \(0.27739/(1+0.27739)=0.2172\). This is a shallow curve across a fourfold dose range.

For \(\beta=1.5\), the 5 mg odds are \(0.25(0.5)^{1.5}=0.08839\), giving risk \(0.08839/(1+0.08839)=0.0812\). The 20 mg odds are \(0.25(2)^{1.5}=0.7071\), giving risk \(0.7071/(1+0.7071)=0.4142\). Both slopes preserve monotonicity, but the second implies a much larger change in toxicity risk across the grid.

This fixed-parameter comparison isolates the effect of slope while holding reference risk constant. It is not, by itself, a prior predictive analysis: a prior predictive check must also account for uncertainty in the intercept, slope, and any dependence between them.

Summarizing Prior Curves

A simulation-based check is straightforward: draw parameter pairs from the specified joint prior, transform each pair into a curve across the grid, and summarize the resulting values at each dose. A plot should identify the dose units, reference dose, DLT window, and the prior summaries shown. A table can be useful when the grid is small or when exact numerical values matter.

State the grid and model.
List the positive candidate doses, name \(d_{\mathrm{ref}}\), and state the BLRM parameterization and DLT assessment window.

Generate prior parameter draws.
Draw paired values from the joint prior, preserving any specified dependence and the constraint \(\beta>0\).

Transform each draw to risks.
For every draw, compute \(p(d_j)\) at every candidate dose using the inverse-logit model.

Inspect the grid summaries and curves.
Review pointwise quantiles, representative individual curves, and endpoint risks for clinically implausible levels, near-flatness, or abrupt changes.

Worked Example: A Small Prior Predictive Curve Summary

For a transparent illustration, define a hypothetical discrete joint prior with four equally likely parameter pairs. Take \(d_{\mathrm{ref}}=10\) mg. The pairs are chosen to have reference risks of 0.10 or 0.30 and slopes of 0.5 or 1.5. This deliberately small prior is for arithmetic illustration; a practical analysis would generally use enough draws from its specified prior to estimate summaries reliably.

Prior draw	\(p(10\text{ mg})\)	\(\alpha=\operatorname{logit}\{p(10\text{ mg})\}\)	\(\beta\)
A	0.10	-2.1972	0.5
B	0.10	-2.1972	1.5
C	0.30	-0.8473	0.5
D	0.30	-0.8473	1.5

At 5 mg, \(\log(5/10)=-0.6931\). For draw A, the linear predictor is \(-2.1972+0.5(-0.6931)=-2.5438\), so \(p(5\text{ mg})=0.0729\). Applying the same calculation to B, C, and D gives 0.0378, 0.2326, and 0.1316, respectively. Ordered from lowest to highest, these risks are 0.0378, 0.0729, 0.1316, and 0.2326; the equal-weight median is \((0.0729+0.1316)/2=0.1023\), and the range is 0.0378 to 0.2326.

At 10 mg, the four risks are 0.10, 0.10, 0.30, and 0.30. Their median is 0.20 and their range is 0.10 to 0.30. At 20 mg, where \(\log(20/10)=0.6931\), the four risks are 0.1357, 0.2390, 0.3774, and 0.5479. The median is \((0.2390+0.3774)/2=0.3082\), and the range is 0.1357 to 0.5479.

These summaries show both increasing typical risk and substantial uncertainty across the grid. They do not imply that every prior curve has the pointwise median shape: the median at each dose may come from a different draw. Plotting the four individual curves alongside the pointwise summaries makes that distinction visible.

Diagnosing Implausible, Flat, or Abrupt Curves

A prior predictive plot is a diagnostic of the prior-model combination over the doses actually being considered. Review the lower and upper grid endpoints as carefully as the reference dose. Check whether the prior assigns meaningful probability to risks near zero or near one where these would be implausible, and whether it permits a large jump between neighboring candidate doses. A curve that is nearly flat may be reasonable in some settings, but it may also indicate that the prior concentrates \(\beta\) too close to zero to encode the intended dose response.

An abrupt change should be judged against the actual dose spacing and clinical expectations, not against an arbitrary universal cutoff. A large slope can imply a sharp rise in risk over a small dose increment; a high reference-dose risk can also put the upper end of the grid close to one even when the slope is more moderate. Inspecting only a central curve may hide a tail of prior draws with problematic risks. In addition to pointwise quantiles, examine individual draws that generate the most extreme grid values and check whether those outcomes arise from plausible joint parameter values.

Worked Example: Finding a Prior Implication That Needs Review

Suppose a hypothetical prior draw has \(p(10\text{ mg})=0.60\) and \(\beta=1.5\), with \(d_{\mathrm{ref}}=10\) mg. The reference log-odds are \(\alpha=\log(0.60/0.40)=0.4055\). At 40 mg, the dose ratio is 4 and \(\log(4)=1.3863\), so the linear predictor is \(0.4055+1.5(1.3863)=2.4849\). The implied risk is \(\exp(2.4849)/(1+\exp(2.4849))=0.9231\).

At 5 mg, the dose ratio is 0.5 and the linear predictor is \(0.4055+1.5(-0.6931)=-0.6342\), giving risk \(\exp(-0.6342)/(1+\exp(-0.6342))=0.3465\). The draw therefore implies an increase from about 34.7% at 5 mg to about 92.3% at 40 mg.

Whether those risks are implausible depends on the clinical context and candidate grid; there is no universal numerical cutoff for this diagnostic. If the high-dose implication conflicts with prespecified clinical expectations, review the joint prior on reference risk and slope rather than simply deleting the dose from the plot. The prior predictive check identifies a setting to reconsider; it does not establish which alternative prior is appropriate.

How This Check Differs From Alternatives

A fixed-parameter scenario, such as the comparison of two slopes above, is useful for explaining how the model behaves but does not represent uncertainty under a prior. A prior predictive curve incorporates the full joint parameter prior. Conversely, marginal prior summaries for \(\alpha\) and \(\beta\) alone do not show the dose-specific toxicity implications, especially when the parameters are dependent. The transformation from parameters to risk is nonlinear, so in general the curve formed from average parameter values is not the average prior predictive curve.

Prior predictive curves are also distinct from posterior toxicity curves: the former use the prior before trial outcomes, while the latter condition on observed data. The latter is a separate analysis. Here the practical question is whether the proposed prior, through the specified BLRM, yields a credible range and shape of risks across the candidate grid before outcomes can influence the assessment.

The check has limitations. It cannot validate the logistic dose-response form, prove that the clinical assumptions are correct, or account for model features that were not included. Its conclusions apply only to the chosen grid, reference dose, DLT definition, assessment window, and population. If the dose grid changes, repeat the inspection; curves that appear reasonable at the reference dose may behave very differently at more distant doses.

Common Mistakes

Checking the reference dose only. A plausible reference risk does not ensure plausible risks at the lowest and highest candidate doses. Transform prior draws across the full grid.
Treating a positive slope as proof of a plausible curve. Positivity enforces increasing risk in this model, but a steep positive slope can still imply an abrupt or clinically implausible increase.
Using the curve at mean parameters as the prior predictive mean curve. The inverse-logit transformation is nonlinear, so transform each joint prior draw first and summarize the resulting risks.
Reading pointwise intervals as whole-curve guarantees. Pointwise summaries describe uncertainty dose by dose; they do not alone give a simultaneous probability statement about an entire curve.
Changing a prior based on one extreme without investigating it. Identify which parameter combinations generate the extreme risk and assess whether they are plausible under the intended joint prior.

Key takeaway: Generate toxicity risks from joint prior draws at every candidate dose, then assess the whole grid for plausible levels, meaningful dose response, and unacceptable abruptness. The prior predictive curve is a diagnostic of prior implications, not a posterior result or a dose recommendation.

Check Your Understanding

Answer each question, then compare your reasoning with the concise solutions.

If \(p(d_{\mathrm{ref}})=0.25\), what is \(\alpha\) in the stated BLRM parameterization?
Why does \(\beta>0\) imply an increasing risk curve, and what does it fail to guarantee?
Why might a prior predictive median curve differ from the curve evaluated at the prior mean parameters?
What is the key distinction between a pointwise 90% interval at each grid dose and a 90% interval for an entire curve?

Solutions. (1) \(\alpha=\operatorname{logit}(0.25)=\log(0.25/0.75)=\log(1/3)\approx-1.099\), the reference-dose log-odds. (2) The derivative is \(\beta p(d)\{1-p(d)\}\), which is positive when \(\beta>0\); this guarantees monotonicity under the model, not clinical plausibility or a suitably gradual increase. (3) The inverse-logit is nonlinear, so transforming average parameters generally does not equal averaging or taking the median of the transformed prior draws. (4) A pointwise interval concerns the risk at one dose at a time; it does not by itself state that the whole dose-response curve is contained in the intervals with 90% probability.