Model-Specific Prior Calibration for BLRM Parameters

Why Calibrate Priors Through Dose and Toxicity

The BLRM parameters are mathematical quantities, but clinical judgment is usually easier to express in terms of questions such as: what toxicity probability seems plausible at the starting dose, and how much might it increase by a higher dose? Prior calibration connects those statements to the model parameters. The goal is not to make a dose recommendation or to assess the model after observing outcomes. It is to choose a prior that encodes defensible beliefs about the model’s intercept and positive dose slope before the trial data are used.

The translation depends on the dose scale and reference dose in the model. A probability judged plausible at one dose cannot be assigned to the intercept without accounting for where that dose lies relative to the reference. The calculations below therefore begin with the model itself and keep the dose units and reference explicit.

Definition: Let $d>0$ denote dose and $d_{\mathrm{ref}}>0$ the model’s reference dose. In the two-parameter BLRM, the probability of a dose-limiting toxicity at dose $d$ is $p(d)$, with $$ \operatorname{logit}\{p(d)\}=\alpha+\beta x(d), \qquad x(d)=\log(d/d_{\mathrm{ref}}), \qquad \beta>0. $$ Here $\operatorname{logit}(p)=\log\{p/(1-p)\}$; $\alpha$ is the log-odds intercept at $d_{\mathrm{ref}}$, and $\beta$ is the slope with respect to log dose. Equivalently, $p(d)=\{1+\exp[-\alpha-\beta x(d)]\}^{-1}$.

At the reference dose $x(d_{\mathrm{ref}})=0$, so $p(d_{\mathrm{ref}})=\operatorname{logit}^{-1}(\alpha)$. At a dose above the reference, $x(d)>0$; with $\beta>0$, the model’s log odds, and therefore its toxicity probability, increase with dose. This is a constraint on the model, not a claim that every observed sequence of cohort toxicity rates must increase.

From Probability Anchors to Parameters

A probability anchor is a dose together with a plausible toxicity probability at that dose. For any anchor $(d_i,p_i)$, calculate $x_i=\log(d_i/d_{\mathrm{ref}})$ and $y_i=\operatorname{logit}(p_i)$. The BLRM says $y_i=\alpha+\beta x_i$. Two distinct dose anchors therefore determine the two parameters, provided their probabilities are between zero and one.

\beta=\frac{y_2-y_1}{x_2-x_1}, \qquad \alpha=y_1-\beta x_1

For ordered doses $d_2>d_1$, $x_2>x_1$. The positive-slope condition is then equivalent to $y_2>y_1$, which is equivalent to $p_2>p_1$, because the logit function is strictly increasing. If the selected higher-dose anchor has a lower toxicity probability, the implied slope is negative and the anchors conflict with this BLRM. A prior that enforces $\beta>0$ cannot make that conflict disappear; the elicitation must be reconsidered or the model assumptions revisited.

Worked Example: Finding a Central Intercept and Slope

Suppose the reference dose is $25$ mg. For a central calibration, clinicians judge toxicity probabilities of $0.20$ at $25$ mg and $0.40$ at $50$ mg plausible. Calculate the corresponding BLRM parameters.

The transformed doses are $x_1=\log(25/25)=0$ and $x_2=\log(50/25)=\log 2=0.6931$. The logit anchors are $y_1=\log(0.20/0.80)=-1.3863$ and $y_2=\log(0.40/0.60)=-0.4055$. Thus $$ \beta=\frac{-0.4055-(-1.3863)}{0.6931-0} =\frac{0.9808}{0.6931}=1.415, \qquad \alpha=-1.3863-(1.415)(0)=-1.3863. $$

The central parameter values are therefore $\alpha=-1.386$ and $\beta=1.415$ per unit of log dose. The intercept reproduces a 20% toxicity probability at the 25 mg reference dose, and the positive slope reproduces 40% at 50 mg. These values specify a central calibration only; they do not, by themselves, specify how uncertain the parameters should be.

For prior calibration, anchor probabilities generally need ranges, not just central values. A range at an anchor describes uncertainty about a plausible toxicity probability; it is not a confidence interval computed from trial data. Transform its endpoints through the logit function, then use the two-anchor equations to examine the parameter values implied by plausible combinations. As the transformation is nonlinear, probability-scale uncertainty should not be treated as if it were already symmetric on the logit scale.

Worked Example: Translating Anchor Ranges into Candidate Marginal Priors

Again let $d_{\mathrm{ref}}=25$ mg. Suppose the elicited plausible probability range at 25 mg is $0.10$ to $0.30$, and at 50 mg it is $0.35$ to $0.65$. Use the corner combinations to find a range of implied $\alpha$ and $\beta$, then construct simple candidate marginal priors whose central 90% intervals match those ranges.

At the reference dose, $\alpha$ is the logit of the probability there. Hence $\operatorname{logit}(0.10)=-2.197$ and $\operatorname{logit}(0.30)=-0.8473$, giving an intercept range of $[-2.197,-0.8473]$. At 50 mg, $x_2=\log 2=0.6931$, and the transformed probability range is $\operatorname{logit}(0.35)=-0.6190$ to $\operatorname{logit}(0.65)=0.6190$. The smallest slope uses the lowest upper-dose logit and the highest reference-dose logit: $$ \beta_{\min}=\frac{-0.6190-(-0.8473)}{0.6931}=0.3291. $$ The largest slope uses the highest upper-dose logit and the lowest reference-dose logit: $$ \beta_{\max}=\frac{0.6190-(-2.197)}{0.6931}=4.063. $$ Every corner combination has a positive slope, so these stated ranges are consistent with the model’s positivity constraint.

As one simple marginal approximation, choose $\alpha\sim N(\mu_\alpha,\sigma_\alpha^2)$ with the range as its central 90% interval. Since a central 90% normal interval is approximately $\mu_\alpha\pm1.645\sigma_\alpha$, $$ \mu_\alpha=\frac{-2.197+(-0.8473)}{2}=-1.522,\qquad \sigma_\alpha=\frac{-0.8473-(-2.197)}{2(1.645)}=0.410. $$ For a positive slope, a candidate is $\log(\beta)\sim N(\mu_{\log\beta},\sigma_{\log\beta}^2)$. Matching the corner-derived slope range as its central 90% interval gives $$ \mu_{\log\beta}=\frac{\log(0.3291)+\log(4.063)}{2}=0.145,\qquad \sigma_{\log\beta}=\frac{\log(4.063)-\log(0.3291)}{2(1.645)}=0.764. $$ The implied median slope is $\exp(0.145)=1.156$. These are candidate marginal distributions, not a uniquely determined prior: they use the anchor ranges as calibration targets, and separate marginal priors do not guarantee that the two anchor probabilities jointly have the elicited uncertainty pattern.

Why Reference Dose and Dependence Matter

If neither anchor is at the reference dose, the intercept must be recovered rather than read off directly. For two anchors, the parameters are linear transformations of the two logit probabilities. This makes their dependence visible: uncertainty in anchor probabilities can induce dependence between $\alpha$ and $\beta$, even if the anchor uncertainties themselves were specified independently.

Worked Example: Anchors on Either Side of the Reference Dose

Let $d_{\mathrm{ref}}=20$ mg, and suppose central anchors are $p(10\text{ mg})=0.15$ and $p(40\text{ mg})=0.45$. Find the implied intercept and slope, and check that both anchors are reproduced.

The transformed doses are $x_1=\log(10/20)=-0.6931$ and $x_2=\log(40/20)=0.6931$. The logits are $y_1=\log(0.15/0.85)=-1.735$ and $y_2=\log(0.45/0.55)=-0.2007$. Therefore $$ \beta=\frac{-0.2007-(-1.735)}{0.6931-(-0.6931)} =\frac{1.534}{1.386}=1.106, \qquad \alpha=-1.735-(1.106)(-0.6931)=-0.968. $$

Substitution at 10 mg gives $-0.968+(1.106)(-0.6931)=-1.735$, whose inverse logit is $0.15$. At 40 mg it gives $-0.968+(1.106)(0.6931)=-0.201$, whose inverse logit is $0.45$. The anchors straddle the reference dose, so neither probability alone equals the intercept’s inverse logit.

The potential dependence can be shown directly. Write $\Delta x=x_2-x_1>0$. Then $\beta=(y_2-y_1)/\Delta x$ and $\alpha=(x_2y_1-x_1y_2)/\Delta x$. If the anchor logits have independent uncertainties, with variances $v_1$ and $v_2$, the covariance is $$ \operatorname{Cov}(\alpha,\beta) =\frac{-x_2v_1-x_1v_2}{(\Delta x)^2}. $$

For example, if the first anchor is at the reference dose, $x_1=0$, then this covariance is $-v_1/\Delta x$, which is negative when the reference-dose anchor is uncertain. A high value of that anchor raises the intercept but reduces the slope needed to connect it to an independently uncertain second anchor. This is one reason that independently chosen priors for intercept and slope may not preserve the uncertainty clinicians expressed at the anchor doses.

Choosing and Checking a Prior

The anchor calculations do not dictate a single prior family. They provide a way to judge whether proposed distributions imply sensible parameter values and dose-specific probabilities. A normal prior for $\alpha$ permits any real intercept. A lognormal prior for $\beta$ has support only on positive slopes. Other positive-support distributions can also be used, provided their calibration is explicit and their implications are checked.

A practical calibration process is:

Fix the dose scale.
State $d_{\mathrm{ref}}$, dose units, and the meaning of $x(d)=\log(d/d_{\mathrm{ref}})$. Changing the reference dose changes the numerical intercept.

Elicit interpretable anchors.
Choose at least two doses and state central probabilities and plausible ranges. For ordered doses, check that the higher-dose probability range supports a higher toxicity probability.

Transform and solve.
Apply the logit to probabilities and use the two-anchor equations. Assess the implied ranges, including whether all plausible combinations respect $\beta>0$.

Specify candidate distributions.
Choose a distribution for $\alpha$ and a positive-support distribution for $\beta$, or a joint distribution if dependence is important. Match the distributions to the elicited locations and spread.

Check the implications at the anchors.
Use parameter values or parameter draws to calculate $p(d)$ at each elicited dose. Compare the resulting probabilities with the intended central values and plausible ranges, and revise if they do not agree.

The final check is essential: matching separate parameter means or quantiles does not automatically match the intended dose-specific probabilities. In particular, because the inverse logit is nonlinear, the toxicity probability at a dose evaluated using mean parameter values need not equal the prior mean toxicity probability at that dose. This calibration check is distinct from evaluating data or making a dose decision.

Diagnostics, Alternatives, and Limitations

Several warning signs indicate that calibration needs attention. First, if $d_2>d_1$ but a central anchor has $p_2\leq p_1$, the implied central slope is nonpositive. Second, if the elicited probability ranges allow many combinations with $p_2<p_1$, the corner calculations may imply negative slopes even though the model excludes them. Third, a very small separation between anchor doses makes $\Delta x$ small, so even modest differences in anchor logits can imply a wide range of slopes. Finally, if the resulting prior assigns substantial probability to toxicity probabilities that stakeholders regarded as implausible at an anchor dose, the chosen parameter distributions are not adequately calibrated.

Two broad approaches are useful to distinguish. Parameter-first calibration chooses distributions for $\alpha$ and $\beta$ directly, then checks what they imply at clinically meaningful doses. It is convenient when prior parameter information already exists, but the clinical meaning of parameter-scale spread can be difficult to communicate. Anchor-first calibration starts with plausible dose-probability statements and transforms them to parameter values. This is usually easier to elicit in clinical language, but the induced parameter distributions can be correlated and may not have a convenient standard form. A joint prior can represent that dependence; independent marginal priors are simpler but should be treated as an approximation and checked against the anchors.

A further limitation is that two anchors determine a straight line on the logit-versus-log-dose scale. They cannot test whether this functional form is clinically adequate over the full dose range. Anchors also depend on the chosen reference dose, the dose scale, and the clinical population to which the judgments apply. Calibration is therefore a transparent way to express assumptions, not a guarantee that those assumptions are correct. It should document the elicited judgments, parameter transformation, prior family, and resulting dose-specific implications.

Common Mistake

Wrong reasoning: “The plausible toxicity probability at the reference dose is 20%, so the intercept prior should be centered at $0.20$.”
Correction: The intercept is on the log-odds scale. A 20% reference-dose probability corresponds to $\alpha=\operatorname{logit}(0.20)=\log(0.20/0.80)=-1.386$, not $0.20$. More generally, transform probabilities with the logit and account for dose offset before assigning parameter priors.

Key takeaway: Calibrate the BLRM by expressing plausible dose-specific toxicity probabilities on the logit scale, solving for $\alpha$ and $\beta$, and checking the resulting prior at the anchor doses. Enforce $\beta>0$, but do not assume independent parameter priors preserve the elicited joint beliefs.

Check Your Understanding

Answer each question, then compare with the solutions below.

At the reference dose, a central toxicity probability is $0.25$. What intercept does this imply?
For two ordered doses, the transformed dose values are $x_1=-0.4$ and $x_2=0.6$, while the anchor logits are $y_1=-1.2$ and $y_2=-0.7$. Find the implied slope and intercept. Is the slope positive?
Why can independent uncertainty in two anchor logits induce dependence between the intercept and slope?
What calibration problem arises if the plausible probability at the higher of two doses is lower than at the lower dose?

Solutions. (1) $\alpha=\operatorname{logit}(0.25)=\log(0.25/0.75)=-1.099$. (2) $\beta=(-0.7-(-1.2))/(0.6-(-0.4))=0.5$, and $\alpha=-1.2-(0.5)(-0.4)=-1.0$; the slope is positive. (3) Both parameters are functions of both anchor logits: $\beta$ depends on their difference, while $\alpha$ is a weighted combination. Variation in an anchor can therefore change both parameters together. (4) For ordered doses it implies a nonpositive slope for those anchor values, conflicting with the BLRM constraint $\beta>0$; revisit the anchors or the model assumptions rather than ignoring the conflict.