Translating Odds-Ratio Beliefs Into Log-Odds Priors

From Clinical Beliefs to a Log-Odds Prior

Clinicians often express treatment beliefs as odds ratios: perhaps most plausible values are near 1, or a range of values seems reasonably likely. A trial model, however, commonly represents a treatment effect on the log-odds scale. The conversion is simple once the effect and the elicited quantiles are defined precisely, but small ambiguities about direction or probability can produce a materially different prior.

Let \(p_T\) and \(p_C\) denote the event probabilities in the treatment and control groups for the same endpoint and follow-up period. Define the treatment odds ratio and its log as \(\mathrm{OR}=[p_T/(1-p_T)]/[p_C/(1-p_C)]\) and \(\beta=\log(\mathrm{OR})\). Thus \(\beta\) is the treatment-group log odds minus the control-group log odds. When the event is adverse, \(\mathrm{OR}<1\), equivalently \(\beta<0\), favors treatment; for a favorable event, the direction is reversed.

Definition: A normal prior on the log odds ratio is \(\beta\sim N(\mu_\beta,\sigma_\beta^2)\), where \(\mu_\beta\) is the prior mean of the log odds ratio and \(\sigma_\beta>0\) is its prior standard deviation. Because \(\mathrm{OR}=\exp(\beta)\), this prior induces a lognormal prior on the odds ratio. The odds ratio is positive, while the log odds ratio may take any real value.

This is a prior for a treatment contrast, not a prior for either group’s event probability by itself. Its interpretation depends on the endpoint definition, the direction of the comparison, and the population and follow-up period to which the elicited belief refers.

Matching Elicited Quantiles

Suppose an expert supplies two quantiles of the odds-ratio distribution: \(Q_L\), the \(q_L\) quantile, and \(Q_U\), the \(q_U\) quantile, where \(0<q_L<q_U<1\) and \(Q_L<Q_U\). For a standard normal random variable, write \(z_q\) for its \(q\) quantile. Since the exponential function is increasing, taking logs maps the odds-ratio quantiles to quantiles of \(\beta\). Under the normal-prior assumption,

\log(Q_L)=\mu_\beta+z_{q_L}\sigma_\beta,\qquad \log(Q_U)=\mu_\beta+z_{q_U}\sigma_\beta.

Subtracting the first equation from the second gives the prior standard deviation. Substitution into either equation gives the prior mean:

\sigma_\beta= \frac{\log(Q_U)-\log(Q_L)}{z_{q_U}-z_{q_L}}, \qquad \mu_\beta=\log(Q_L)-z_{q_L}\sigma_\beta.

These equations require that the elicited quantiles be ordered and that the fitted \(\sigma_\beta\) be positive. If \(Q_L=Q_U\) despite \(q_L<q_U\), the elicitation implies no spread and does not yield a non-degenerate normal prior. The calculation also assumes that both values are quantiles of one distribution for the same treatment contrast—not, for example, a mixture of a likely range and an extreme plausible limit.

Worked Example: Using the 10th and 90th Odds-Ratio Quantiles

Suppose a hypothetical elicitation for an adverse endpoint gives a 10th-percentile odds ratio of 0.60 and a 90th-percentile odds ratio of 1.50. Assume these are intended as quantiles of a normal prior on the log odds ratio. Find the prior parameters and check the fitted quantiles.

The standard normal quantiles are \(z_{0.10}=-1.2816\) and \(z_{0.90}=1.2816\), so the difference is 2.5631. The elicited log-quantiles are \(\log(0.60)=-0.5108\) and \(\log(1.50)=0.4055\). Therefore,

\sigma_\beta= \frac{\log(1.50)-\log(0.60)}{2.5631} = \frac{0.4055-(-0.5108)}{2.5631} = 0.3575.

Using the lower quantile to find the mean,

\mu_\beta=\log(0.60)-z_{0.10}\sigma_\beta =-0.5108256-(-1.2815516)(0.3574933) =-0.0526803\approx-0.0527.

The resulting prior is \(\beta\sim N(-0.0527,0.3575^2)\). Substituting the unrounded fitted parameters back into the quantile equations gives log-quantiles \(-0.0526803-1.2815516(0.3574933)\approx-0.5108\) and \(-0.0526803+1.2815516(0.3574933)\approx0.4055\). Exponentiating recovers 0.60 and 1.50, as required. The prior median odds ratio is \(\exp(\mu_\beta)\approx0.9487\); it is not the arithmetic mean odds ratio.

What the Normal Prior Implies on the Odds-Ratio Scale

A normal prior on \(\beta\) is symmetric around \(\mu_\beta\), but its induced odds-ratio distribution is generally right-skewed. Three summaries should not be confused:

The median odds ratio is \(\exp(\mu_\beta)\).
The \(q\) quantile of the odds ratio is \(\exp(\mu_\beta+z_q\sigma_\beta)\).
The prior mean odds ratio is \(\exp(\mu_\beta+\sigma_\beta^2/2)\), which is generally not \(\exp(\mu_\beta)\).

The mean formula follows from the moment-generating function of a normal random variable: if \(\beta\sim N(\mu_\beta,\sigma_\beta^2)\), then \(E[\exp(\beta)]=\exp(\mu_\beta+\sigma_\beta^2/2)\). The distinction matters when an elicitation describes a “typical” odds ratio. The median is the natural center on the log scale; the arithmetic mean can be pulled upward by the right tail.

Quantile matching also gives a direct check on the elicitation. Once two quantiles determine \(\mu_\beta\) and \(\sigma_\beta\), every other quantile is implied by the same normal model. A third elicited quantile that is far from its implied value signals either elicitation inconsistency or a poor fit to the lognormal shape; it should not be silently treated as if all three values were matched.

Worked Example: Checking a Median and an Upper Quantile

Suppose, hypothetically, that experts specify a median odds ratio of 1.20 and a 90th-percentile odds ratio of 2.00 for a favorable event. Under a normal prior for \(\beta\), find the parameters and the implied 10th percentile.

The median corresponds to the 50th percentile, for which \(z_{0.50}=0\). Thus \(\mu_\beta=\log(1.20)=0.1823\). With \(z_{0.90}=1.2816\),

\sigma_\beta= \frac{\log(2.00)-\log(1.20)}{1.2816} = \frac{0.6931-0.1823}{1.2816} = 0.3986.

The implied 10th-percentile log odds ratio is \(0.1823-1.2816(0.3986)=-0.3285\), so the implied odds ratio is \(\exp(-0.3285)=0.7200\). The fitted prior is therefore \(\beta\sim N(0.1823,0.3986^2)\), with a 10th-to-90th percentile odds-ratio interval of approximately 0.7200 to 2.00. If the experts had separately stated that the 10th percentile was 0.90, those three elicited values would not be compatible with this normal-log-odds model. The discrepancy would require discussion, not a change in arithmetic.

Using the Prior for Directional Beliefs

A quantile-based prior also makes it possible to state how much prior probability lies on either side of the no-effect value. Since \(\mathrm{OR}=1\) exactly when \(\beta=0\), and the exponential transformation is increasing, for an adverse event the prior probability of a treatment-favorable odds ratio is \(P(\mathrm{OR}<1)=P(\beta<0)\). Under the normal prior, this equals the standard normal cumulative probability at \((0-\mu_\beta)/\sigma_\beta\). This is a probability implied by the prior distribution, not a probability that the treatment truly works after observing data.

Worked Example: Prior Probability of Benefit for an Adverse Event

Use the prior from the first example, \(\beta\sim N(-0.0527,0.3575^2)\). The event is adverse, so odds ratios below 1 favor treatment. Calculate the prior probability of a favorable direction.

Standardizing the no-effect value gives

P(\mathrm{OR}<1)=P(\beta<0) = P\left(Z<\frac{0-(-0.0527)}{0.3575}\right) = P(Z<0.1474) \approx 0.5586.

Thus this prior assigns about 55.86% probability to an odds ratio below 1 and about 44.14% to an odds ratio above 1. The probability is only slightly above one half, consistent with the elicited quantiles spanning 1 and the fitted log-scale mean being close to zero. It does not describe the posterior probability after trial results are observed.

Assumptions, Diagnostics, and Alternatives

Before using quantile matching, verify the assumptions that make its output meaningful. The elicited odds ratios must refer to the same endpoint, event coding, treatment contrast, analysis population, and follow-up period. The treatment-to-control orientation must be fixed throughout. The quantile probabilities must be understood—for example, a 10th and 90th percentile pair describes a central 80% interval, not a 90% interval. Finally, the normality assumption applies to \(\beta=\log(\mathrm{OR})\), not to the odds ratio itself.

Several practical checks help detect errors or a poor fit:

Confirm \(Q_L<Q_U\) and \(q_L<q_U\), and check that the calculated \(\sigma_\beta\) is positive.
Substitute \(\mu_\beta\) and \(\sigma_\beta\) back into both quantile equations and exponentiate; the results should recover the elicited odds ratios up to rounding.
Check at least one additional elicited quantile when available. Large disagreement suggests that a single normal distribution on the log scale may not represent the stated beliefs well.
Ask whether an elicited “range” means probabilistic quantiles or strict bounds. A normal prior has unbounded support on \(\beta\), and its induced odds-ratio prior has support over all positive values. Quantile matching does not make the elicited endpoints hard limits.

A normal prior on \(\beta\) is equivalent, under the exponential transformation, to a lognormal prior on the odds ratio. It is often convenient because it matches a common treatment-effect parameterization directly and guarantees positive odds ratios. A prior specified directly on the odds-ratio scale is another way to communicate beliefs, but its implications must still be checked against the trial model’s parameterization. A uniform prior on the odds-ratio scale is not the same prior as a uniform prior on the log-odds-ratio scale; changing parameterization changes the implied distribution. If elicited quantiles strongly conflict with lognormal shape, forcing them into one normal prior conceals that conflict. The limitation of this method is not the algebra, but the distributional assumption used to summarize beliefs.

Common Pitfalls

Using the wrong interval probability. Calling the 10th-to-90th percentile range a central 90% interval leads to incorrect normal quantiles. It contains 80% of the probability because \(0.90-0.10=0.80\).
Reversing the treatment contrast. If an elicitation is control-to-treatment but the model defines treatment-to-control, the log odds ratio changes sign and the odds ratio is reciprocated.
Confusing median and mean. \(\exp(\mu_\beta)\) is the median odds ratio under this model; the mean is \(\exp(\mu_\beta+\sigma_\beta^2/2)\).
Treating an elicited range as an absolute bound. Quantiles describe probabilities, not impossible values outside the range.
Ignoring a mismatch at a third quantile. Two quantiles always determine a fitted normal distribution when valid, but that does not guarantee the distribution captures every stated belief.

Key takeaway: Define the odds-ratio direction first. Then take logs of the elicited quantiles, match them to the corresponding standard-normal quantiles, and check the resulting prior against the original beliefs. The calculation is exact for the selected quantiles; the adequacy of the normal log-odds model remains an assumption to assess.

Check Your Understanding

Try each problem, then compare your work with the solutions below.

For an adverse event, the treatment-to-control odds-ratio quantiles are 0.80 at the 25th percentile and 1.25 at the 75th percentile. What is the corresponding log-odds prior, using \(z_{0.25}=-0.6745\) and \(z_{0.75}=0.6745\)?
If \(\beta\sim N(0.10,0.20^2)\), what is the median odds ratio and the mean odds ratio?
For an adverse event, if \(\beta\sim N(-0.20,0.30^2)\), express the prior probability of a treatment-favorable direction as a standard normal probability.
Why can a normal prior on the log odds ratio not make an elicited 10th-to-90th percentile range into strict lower and upper bounds?

Solutions. (1) The standard-normal quantile difference is \(0.6745-(-0.6745)=1.3490\), and \(\log(1.25)-\log(0.80)=0.4463\). Hence \(\sigma_\beta=0.4463/1.3490=0.3308\), and using the displayed rounded values, \(\mu_\beta=\log(0.80)-(-0.6745)(0.3308)\approx-0.000019\). Since 0.80 and 1.25 are reciprocals and the quantile probabilities are symmetric, their exact log-scale midpoint is 0, so \(\mu_\beta=0\) (approximately \(0.0000\)). Thus \(\beta\sim N(0,0.3308^2)\). (2) The median odds ratio is \(\exp(0.10)=1.105\); the mean is \(\exp(0.10+0.20^2/2)=\exp(0.12)=1.127\). (3) The favorable direction is \(\mathrm{OR}<1\), so its probability is \(P(\beta<0)=P(Z<(0-(-0.20))/0.30)=P(Z<0.6667)\). (4) A normal distribution has positive probability in both tails beyond any finite quantiles, and exponentiating it gives positive odds ratios with no finite upper bound. Elicited quantiles specify probability locations, not impossible values outside them.