Model-Based Dose Finding: CRM and Extensions · Tutorial 585 of 799
Prior Elicitation for EWOC
Learn to construct and check EWOC dose-toxicity priors from clinically interpretable beliefs about the MTD and toxicity probabilities.
Advanced
10 min read
What You'll Learn
Parameterize the dose-toxicity curve using the MTD and a positive slope.
Translate an elicited median and central interval for the MTD into a lognormal prior.
Calibrate the slope from toxicity probabilities at doses relative to the MTD.
Connect the prior distribution of the MTD to prior overdose probabilities.
Use prior predictive checks to detect implausible dose-toxicity behavior.
Recognize when dependence or parameterization choices undermine an elicited prior.
Why Elicit EWOC Priors in Clinical Terms?
EWOC uses a posterior distribution for the maximum tolerated dose (MTD) when controlling escalation. Its behavior therefore depends not only on the likelihood but also on what the prior says about plausible MTDs and dose-toxicity relationships. A prior specified only as a distribution for abstract regression coefficients can be difficult to review clinically. Eliciting beliefs about the MTD and the steepness of the toxicity curve gives those parameters an interpretable starting point.
The goal here is to construct and assess those priors, not to choose the EWOC feasibility quantile or a dose from a trial grid. The key is to define the model so that the MTD and the slope have direct meanings, then translate clinical judgments into distributions on those quantities.
Definition: Let \(d>0\) be dose, \(\theta\) the prespecified target toxicity probability, and \(\gamma>0\) the dose at which the model toxicity probability equals \(\theta\). In the logistic model
$$
\operatorname{logit}\{\pi(d;\gamma,\beta)\}
=\operatorname{logit}(\theta)+\beta\log(d/\gamma),
\qquad \beta>0,
$$
\(\pi(d;\gamma,\beta)\) is the probability of dose-limiting toxicity at dose \(d\), \(\gamma\) is the MTD under the target definition, and \(\beta\) controls how rapidly toxicity probability changes with log dose.
This parameterization assumes a positive dose scale, a fixed target \(\theta\) strictly between 0 and 1, and a monotone increasing toxicity relationship. The ratio \(d/\gamma\) is dimensionless, so the model is unchanged if both dose values are expressed in another common unit. The positivity restriction on \(\beta\) ensures that \(d<\gamma\) implies \(\pi(d)<\theta\), while \(d>\gamma\) implies \(\pi(d)>\theta\). The model is a working description: it does not assert that real toxicity probabilities follow a logistic curve exactly.
From Clinical Statements to Prior Distributions
A common choice is to model the logarithms of the positive parameters:
$$
\log(\gamma)\sim N(m_\gamma,s_\gamma^2),
\qquad
\log(\beta)\sim N(m_\beta,s_\beta^2).
$$
The opening display states a distribution, not a required independence assumption. Analysts may specify dependence if clinical knowledge supports it; otherwise, independence is a simplifying choice that should be made explicit and examined. Under the lognormal specification, \(\exp(m_\gamma)\) is the median MTD and \(\exp(m_\beta)\) is the median slope. The parameters \(s_\gamma\) and \(s_\beta\) describe uncertainty on the log scale, not on the dose or slope scale.
To translate a central \(100(1-\alpha)\%\) interval \([L,U]\) for \(\gamma\) into a lognormal prior, the interval should be approximately symmetric on the log scale. If it is symmetric, then
$$
m_\gamma=\frac{\log(L)+\log(U)}{2},
\qquad
s_\gamma=\frac{\log(U)-\log(L)}{2z_{1-\alpha/2}},
$$
where \(z_{1-\alpha/2}\) is the corresponding standard normal quantile. When clinicians instead give a median and one uncertainty bound, that information can determine \(m_\gamma\) and, with a stated coverage level, \(s_\gamma\). The elicited interval and its interpretation should be recorded: an informal plausible range is not automatically a formal 90% interval.
Worked Example: Translating an MTD Median and Interval
Suppose a hypothetical team sets \(\theta=0.30\) and elicits a median MTD of 20 mg with a central 90% interval obtained by taking \(\log(\gamma)\sim N(\log(20),0.25^2)\). Find the interval and evaluate toxicity probability at 20 mg when a particular plausible MTD is 30 mg and \(\beta=2.5\).
For a central 90% normal interval, \(z_{0.95}\approx1.645\). Transforming the log-scale endpoints back to dose gives
$$
\left[20\exp(-1.645\times0.25),\;20\exp(1.645\times0.25)\right]
\approx [13.26,\;30.17]\text{ mg}.
$$
For the specified parameter values, the linear predictor at 20 mg is
$$
\operatorname{logit}(0.30)+2.5\log(20/30)
\approx -0.8473-1.0137=-1.8610.
$$
Therefore,
$$
\pi(20;30,2.5)=\frac{1}{1+\exp(1.8610)}\approx0.1346.
$$
Under this particular curve, the estimated toxicity probability at 20 mg is about 13.46%, below the 30% target. This calculation illustrates the model at one parameter value; the prior distribution represents uncertainty over many such curves.
Eliciting the Slope From Toxicity Judgments
The slope \(\beta\) is easiest to discuss through the change in toxicity probability across a stated dose ratio. For example, if clinicians can judge toxicity probability at half the MTD, the model implies
$$
\beta=
\frac{\operatorname{logit}(\theta)-\operatorname{logit}\{\pi(\gamma/2)\}}
{\log(2)}.
$$
This follows by substituting \(d=\gamma/2\) into the model and using \(\log(1/2)=-\log(2)\). More generally, for a dose \(d<\gamma\) and an elicited probability \(q=\pi(d)\),
$$
\beta=
\frac{\operatorname{logit}(\theta)-\operatorname{logit}(q)}
{\log(\gamma/d)}.
$$
The formula converts a probability judgment into a slope; it does not make an uncertain judgment exact. If the elicited probability is uncertain, elicit a range or distribution and translate that uncertainty into a slope distribution.
Worked Example: Calibrating Slope From a Probability at Half the MTD
In a hypothetical setting, let the target be \(\theta=0.30\). Clinicians judge that toxicity probability at half the MTD is about 0.10. Calculate the implied slope. Then suppose their plausible range for that probability is 0.08 to 0.14 and construct an approximate central 90% lognormal prior for \(\beta\).
Using the point judgment,
$$
\beta=\frac{\operatorname{logit}(0.30)-\operatorname{logit}(0.10)}{\log(2)}
=\frac{-0.8473-(-2.1972)}{0.6931}
\approx1.947.
$$
At probability 0.08, the same calculation gives
$$
\beta_{\text{low probability}}
=\frac{-0.8473-\operatorname{logit}(0.08)}{\log(2)}
=\frac{-0.8473-(-2.4423)}{0.6931}
\approx2.301.
$$
At probability 0.14 it gives
$$
\beta_{\text{high probability}}
=\frac{-0.8473-\operatorname{logit}(0.14)}{\log(2)}
=\frac{-0.8473-(-1.8153)}{0.6931}
\approx1.397.
$$
A lower toxicity probability at half the MTD implies a steeper curve, explaining why the endpoints reverse. Treating 1.397 and 2.301 as approximate central 90% limits on \(\beta\), a lognormal approximation has
$$
m_\beta\approx\frac{\log(1.397)+\log(2.301)}{2}\approx\log(1.793),
\qquad
s_\beta\approx\frac{\log(2.301)-\log(1.397)}{2(1.645)}\approx0.152.
$$
Thus one approximate prior is \(\log(\beta)\sim N(\log(1.793),0.152^2)\). This translation assumes the team’s probability range is a central 90% range and treats \(\gamma\) as fixed for the calibration. If uncertainty in \(\gamma\) matters, it must also be included when checking the induced toxicity probabilities.
What the MTD Prior Implies About Overdose
The parameterization gives a direct interpretation of the prior probability that a fixed dose exceeds the target toxicity level. For any fixed \(d>0\), and with \(\beta>0\),
$$
\pi(d;\gamma,\beta)>\theta
\quad\Longleftrightarrow\quad
d>\gamma
\quad\Longleftrightarrow\quad
\gamma<d.
$$
Consequently, before observing trial outcomes,
$$
P\{\pi(d;\gamma,\beta)>\theta\}=P(\gamma<d).
$$
The probability is determined by the MTD prior in this model, not by the slope prior. The slope still matters greatly for the magnitude of toxicity probabilities away from the MTD and for how sharply those probabilities change with dose.
For a lognormal MTD prior, the prior probability on the right can be calculated directly:
$$
P(\gamma<d)=\Phi\left(\frac{\log(d)-m_\gamma}{s_\gamma}\right),
$$
where \(\Phi\) is the standard normal cumulative distribution function. This identity is a useful elicitation check: clinicians can be asked whether the resulting prior probability that a proposed dose exceeds the target is plausible. It is a prior implication, not a recommendation about which dose EWOC should select.
Worked Example: Checking an Implied Prior Probability
Suppose the hypothetical MTD prior is \(\log(\gamma)\sim N(\log(24),0.2465^2)\), with dose measured in mg. Calculate the prior probability that an 18 mg dose is above the MTD.
By the identity above,
$$
P(\gamma<18)=\Phi\left(\frac{\log(18)-\log(24)}{0.2465}\right)
=\Phi\left(\frac{\log(0.75)}{0.2465}\right)
=\Phi(-1.167)\approx0.122.
$$
Thus, under this prior and model, the prior probability that 18 mg has toxicity probability above the 30% target is about 12.2%. This statement concerns uncertainty before observing trial data; it is not the probability that the model or prior is correct.
Prior Predictive Checks and Failure Modes
A prior should be reviewed through the dose-toxicity curves it generates, not only through marginal summaries of \(\gamma\) and \(\beta\). Draw parameter values from the proposed joint prior, calculate \(\pi(d;\gamma,\beta)\) at clinically relevant doses, and inspect the resulting probabilities and curves. Ask whether the implied MTD range, probabilities at low and high doses, and degree of curve steepness are compatible with clinical knowledge. These are prior predictive checks: they assess the consequences of the prior before outcomes are observed.
Check the dose scale. A prior elicited in mg must be converted consistently if doses are represented in another unit. The ratio \(d/\gamma\) is dimensionless, but mixing units changes its value.
Check tail behavior. A lognormal prior has positive support but can assign non-negligible probability to very small or very large MTDs. Review quantiles and implied overdose probabilities at actual candidate doses.
Check probability behavior. Simulated curves should remain increasing and probabilities should remain between 0 and 1. The model guarantees these features for \(\beta>0\), but the curves may still be clinically implausibly steep or shallow.
Check joint assumptions. Independence between \(\gamma\) and \(\beta\) is not guaranteed by the parameterization. If plausible lower MTDs are also thought to have steeper curves, an independent prior may fail to represent that belief.
Check elicitation precision. A single probability at one dose ratio identifies a slope only when the target and MTD are treated as known. Uncertainty in those quantities propagates to uncertainty in \(\beta\).
A practical elicitation process separates the questions: first define the target \(\theta\) and elicit plausible MTD values; next elicit toxicity probabilities at dose ratios such as one-half or twice the MTD to inform steepness; then assess the implied curves and revise the priors with clinical reviewers. If probability judgments at several dose ratios cannot be represented reasonably by one logistic curve, that is evidence to revisit the model or its prior structure rather than force a precise fit.
Comparing Elicitation Strategies
The MTD-and-slope parameterization is not the only way to specify a prior. Direct priors on regression intercept and slope can be convenient computationally, but the intercept is less directly interpretable and its meaning depends on dose coding. A CRM skeleton with a prior on its model parameter can be appropriate for that model, but it does not automatically express beliefs about the MTD in the same way. Eliciting toxicity probabilities at several fixed doses can be intuitive, although those probabilities must be coherent with a monotone curve and a common dose-toxicity model.
Approach
Strength
Review point
MTD and positive slope
Parameters have direct clinical interpretations.
Check how uncertainty in both parameters shapes curves.
Regression coefficients
May align directly with software parameterization.
Interpretation can change with dose coding and centering.
Probabilities at fixed doses
Clinicians can often judge probabilities directly.
Ensure the judgments are mutually compatible with the model.
No parameterization removes the need for sensitivity analysis. If reasonable alternative priors materially change the implied curves or posterior MTD distribution, that sensitivity should be visible to the design team. A prior is a structured expression of uncertainty and clinical judgment, not a substitute for trial data or a guarantee of safe decisions.
Common Mistake
Wrong reasoning: “The median MTD is 20 mg, so the toxicity probability at 20 mg is exactly the target for every curve represented by the prior.”
Correction: For each individual parameter draw, \(\pi(\gamma;\gamma,\beta)=\theta\). But if the dose is fixed at 20 mg and \(\gamma\) varies across the prior, then 20 mg is not the MTD for every draw. The prior distribution of toxicity probability at 20 mg can therefore be spread above and below \(\theta\), even when 20 mg is the median of the MTD prior. Examine that distribution directly rather than equating the median MTD with a fixed dose whose toxicity probability is known.
Key takeaway: Elicit EWOC priors on quantities clinicians can interpret, transform the judgments carefully, and validate the resulting joint prior by examining dose-toxicity curves and implied probabilities at relevant doses.
Check Your Understanding
Try each problem, then compare your work with the solutions.
With target \(\theta=0.25\), MTD \(\gamma=40\) mg, and \(\beta=2\), calculate the model toxicity probability at 20 mg.
A lognormal MTD prior has median 30 mg and log-scale standard deviation 0.20. Give its central 90% interval.
With target \(\theta=0.30\), toxicity probability at half the MTD is elicited as 0.12. Find the implied slope.
Explain why \(P\{\pi(d)>\theta\}=P(\gamma<d)\) requires the positive-slope restriction in this model.
Solutions. (1) The predictor is \(\operatorname{logit}(0.25)+2\log(20/40)=-1.0986-1.3863=-2.4849\). Thus \(\pi(20)=1/\{1+\exp(2.4849)\}\approx0.07692\), or about 7.69%. (2) The interval is \(30\exp(\pm1.645\times0.20)\) mg, approximately \([21.59,41.69]\) mg. (3) \(\beta=[\operatorname{logit}(0.30)-\operatorname{logit}(0.12)]/\log(2)=[-0.8473-(-1.9924)]/0.6931\approx1.652\). (4) Since \(\log(d/\gamma)>0\) exactly when \(d>\gamma\), a positive \(\beta\) makes the linear predictor exceed \(\operatorname{logit}(\theta)\) exactly for \(d>\gamma\). If \(\beta\) could be negative, the direction would reverse, so \(\gamma\) would not have the stated MTD interpretation.