From Clinical Judgments to a Prior on a Treatment Effect
For a continuous trial endpoint, clinicians often express uncertainty in directly interpretable terms: a treatment may add a few symptom-free days, reduce pain by a certain number of points, or improve a functional score. A prior distribution translates such judgments into a probability model on the treatment-effect scale. Its parameters should retain the clinical units, so they can be checked and explained rather than treated as abstract tuning constants.
The elicitation must first settle the direction of the effect. In this tutorial, let \(\delta\) denote the treatment-arm mean minus the control-arm mean, measured in the endpoint's original units. We orient the endpoint so that larger values are better. Thus, a positive \(\delta\) favors treatment, while a negative \(\delta\) favors control. If the original endpoint is one where smaller values are better, reverse the contrast or explicitly change the sign convention before asking for judgments.
The prior mean describes the center of the elicited distribution; it need not be zero or equal to the minimum effect considered clinically important. The prior standard deviation describes uncertainty about the effect itself. It is not the standard deviation of individual patient outcomes, nor the standard error of a trial estimate.
Assumptions and Elicitation Questions
A normal prior is most defensible when uncertainty about the effect can reasonably be represented by a continuous, approximately symmetric distribution on the chosen scale. It has support over the entire real line, so it assigns some probability to effects in both directions and to arbitrarily large magnitudes. The assigned probability in implausible regions must be small enough to be clinically acceptable. If substantial asymmetry, hard bounds, or unusually heavy tails are essential to the judgment, a normal distribution may not be an adequate summary.
Ask the questions in endpoint units. A useful sequence is to agree on a central effect, identify a lower and upper value that bound a stated proportion of plausible effects, and ask how likely it is that the effect exceeds a clinically meaningful threshold. The threshold, written \(\delta_M\), is the smallest effect regarded as meaningful for the decision at hand. It is not automatically the effect expected to occur.
A phrase such as “a plausible range from 2 to 8 points” is incomplete until its probability content is clarified. It might mean that almost all effects lie in the range, or that the range contains 90% of the prior distribution. These statements imply different uncertainty. Record both the endpoints and the elicited coverage probability. Likewise, “likely to help” must be made quantitative by specifying whether “help” means \(\delta>0\) or \(\delta>\delta_M\).
Deriving Parameters from a Central Plausible Range
Suppose the elicited interval \([L,U]\) is intended to contain a central proportion \(1-\alpha\) of a normal prior. The standard normal quantile \(z_{1-\alpha/2}\) satisfies \(P(Z\le z_{1-\alpha/2})=1-\alpha/2\) for \(Z\sim N(0,1)\). The lower and upper quantiles of the treatment-effect prior are then \(\mu_\delta-z_{1-\alpha/2}\tau_\delta\) and \(\mu_\delta+z_{1-\alpha/2}\tau_\delta\). Matching these to \(L\) and \(U\) gives:
This conversion assumes that the elicited range is central and symmetric around the prior mean. The resulting mean is the midpoint, and the standard deviation is the half-width divided by the relevant normal quantile. For a central 90% interval, \(z_{0.95}\approx1.645\); for a central 95% interval, \(z_{0.975}\approx1.960\). The same numerical range therefore yields a smaller standard deviation when it is said to contain 95% rather than 90% of the prior.
Worked Example: Converting a 90% Range in Days
Suppose the endpoint is the number of days alive and free of symptoms during a 28-day period, with larger values better. The treatment effect is the treatment mean minus the control mean, in days. A clinical group gives 3 days as its central expectation and says that a central 90% plausible range is from -2 to 8 days. Assume the group accepts a symmetric normal representation.
The interval midpoint is \((-2+8)/2=3\) days, agreeing with the stated central expectation. Its half-width is \(8-3=5\) days. For a central 90% normal interval, \(z_{0.95}=1.645\), so:
The elicited prior is therefore \(\delta\sim N(3,3.040^2)\), with mean 3 days and standard deviation 3.040 days. To check its implication for a meaningful benefit of at least 4 days, standardize the threshold:
Under this prior, the probability that the mean treatment effect exceeds 4 additional symptom-free days is about 37.1%. This is a probability statement about the prior distribution, not a claim that an observed trial has established benefit. The 90% range extends below zero, so it also allows effects favoring control.
Using a Probability of Meaningful Benefit
A probability judgment can determine the prior standard deviation when a prior center is also specified. For a threshold \(\delta_M\), suppose the elicited probability is \(q=P(\delta>\delta_M)\), and the prior mean \(\mu_\delta\) is already agreed. Standardizing the threshold gives \(q=1-\Phi((\delta_M-\mu_\delta)/\tau_\delta)\). When the elicited values imply a positive standard deviation, this equation can be solved for \(\tau_\delta\):
Here, \(\Phi\) is the standard normal cumulative distribution function. The numerator and denominator must have the same sign for \(\tau_\delta\) to be positive. If \(q=0.5\), a normal distribution assigns probability 0.5 above the threshold only when \(\delta_M=\mu_\delta\); this probability alone cannot determine the standard deviation. More generally, one probability and one fixed mean determine a standard deviation only when they provide a non-degenerate, coherent constraint.
Worked Example: Solving from a Meaningful-Benefit Probability
Suppose the outcome is a function score measured in points, with larger values better. Clinicians set the prior mean treatment effect at 2 points and define a meaningful benefit as an improvement greater than 5 points. They judge that the probability of exceeding 5 points should be 20%. Assume a normal prior is considered reasonable.
Here \(\mu_\delta=2\), \(\delta_M=5\), and \(q=0.20\). Since \(\Phi^{-1}(0.80)=0.8416\), solving the probability equation gives:
The prior is \(\delta\sim N(2,3.564^2)\), with variance about \(12.70\) points squared. Substitution verifies the elicited probability:
For additional interpretation, the implied central 90% interval is \(2\pm1.645(3.564)\), or approximately \([-3.86,7.86]\) points. The prior therefore includes some probability of effects favoring control as well as effects that exceed the meaningful-benefit threshold.
Checking Whether Elicited Judgments Agree
When both a range and a probability statement are available, use one to calculate the prior and the other to check it. Agreement supports the chosen normal representation; disagreement is a reason to clarify the judgments, not to average incompatible numbers without explanation. The following calculation provides a direct check.
Worked Example: Checking a Range Against a Benefit Probability
For a higher-is-better symptom score measured in points, a group gives a central 90% range from -4 to 8 points and says that meaningful benefit is an effect greater than 4 points. They estimate a 30% chance of exceeding that threshold. Assume a symmetric normal prior.
The range gives a midpoint of \((-4+8)/2=2\) points and a half-width of 6 points. Thus:
The range-derived prior implies:
The implied probability is about 29.2%, close to the stated 30%. Given that clinical judgments are often approximate, these inputs are coherent enough to retain as a useful elicitation. The range sets the parameters; the probability question provides an interpretable cross-check.
Diagnostics, Alternatives, and Limitations
Several checks help detect a specification that does not represent the elicited judgments:
- Check the direction and units. Confirm that a positive effect has the agreed clinical meaning, and express both the range and threshold in the same units as \(\delta\).
- Check the interval definition. Verify whether the range is central and whether its stated coverage is 90%, 95%, or another proportion. Do not use a quantile formula for an interval with unspecified coverage.
- Check probability implications. Calculate the prior probability of exceeding the meaningful threshold. If it is far from the elicited probability, revisit the mean, range, threshold, or normality assumption.
- Inspect tails and symmetry. The normal distribution allows effects of any size. If its implied tail probabilities are clinically implausible, a normal prior may not reflect the judgment even if its center and range match.
- Check for incompatible constraints. A threshold and probability can imply no positive \(\tau_\delta\) for a fixed mean. That signals a conflict in the elicited statements or an unsuitable symmetric normal model.
A central range is a direct way to specify spread when experts can judge plausible endpoints and coverage. A probability question focuses attention on a decision-relevant threshold, but usually needs an independently elicited center to determine the normal prior. Direct judgments about a mean and standard deviation are another option, but many experts find a standard deviation harder to interpret than a range in clinical units. These approaches can be used together as checks; they should not be treated as interchangeable inputs unless their implied distributions agree.
The key limitation is the normal model's symmetry and unbounded support. A normal prior can assign probability to effects that are technically possible on a mathematical scale but impossible or implausible clinically. It can also misrepresent uncertainty that is strongly skewed or concentrated near a boundary. If a normal approximation remains inadequate after the effect scale and elicitation are reviewed, choose a distribution that better matches the clinical constraints and document the reason. The parameters should not be adjusted solely to produce a preferred trial conclusion.
A Short Derivation of the Range Formula
The range conversion follows directly from the standardization property of the normal distribution. If \(\delta\sim N(\mu_\delta,\tau_\delta^2)\), then \(Z=(\delta-\mu_\delta)/\tau_\delta\sim N(0,1)\). For a central \(1-\alpha\) interval, the standardized lower and upper bounds are \(-z_{1-\alpha/2}\) and \(z_{1-\alpha/2}\). Therefore:
Adding these equations gives \(L+U=2\mu_\delta\), so \(\mu_\delta=(L+U)/2\). Subtracting the first equation from the second gives \(U-L=2z_{1-\alpha/2}\tau_\delta\), so \(\tau_\delta=(U-L)/(2z_{1-\alpha/2})\). This derivation also shows why the midpoint assumption matters: the formula encodes a central, symmetric interval, not an arbitrary pair of endpoints.
Common Mistake
Check Your Understanding
Work each problem, then compare your result with the solutions below.
- A higher-is-better endpoint has treatment effect in points. A central 90% plausible range is 1 to 9 points. Find the normal prior mean and standard deviation.
- For a continuous outcome measured in days, suppose \(\mu_\delta=1\) day, meaningful benefit is greater than 4 days, and the elicited probability of meaningful benefit is 10%. Find \(\tau_\delta\).
- A range-based normal prior has mean 2 points and standard deviation 3 points. What probability does it assign to an effect greater than 5 points?
- Explain why a range of -2 to 6 points cannot be converted to a central symmetric normal interval with mean 3 points.
Solutions. (1) The midpoint is \((1+9)/2=5\) points. The half-width is 4 points, so \(\tau_\delta=4/1.645\approx2.432\) points and the prior is \(N(5,2.432^2)\). (2) With \(q=0.10\), \(\Phi^{-1}(0.90)=1.282\). Therefore \(\tau_\delta=(4-1)/1.282\approx2.340\) days. (3) Standardize: \(z=(5-2)/3=1\). Thus \(P(\delta>5)=1-\Phi(1)\approx0.1587\), or 15.9%. (4) The interval midpoint is \((-2+6)/2=2\), not 3. A central symmetric normal interval must be centered on its mean, so the stated range and mean are incompatible; clarify the elicitation rather than applying the central-range formula unchanged.