Priors for Log Hazard Ratios

Why Specify a Prior on the Log Hazard Ratio?

In a time-to-event trial, a treatment effect is often summarized by a hazard ratio. A clinically stated belief such as “the treatment effect is unlikely to be more extreme than these bounds” is easier to discuss on the hazard-ratio scale, while a normal prior is often easier to specify on the log scale. The conversion is simple once the probability attached to the bounds is clear. But the two scales have different symmetries: equal distances on the log scale correspond to reciprocal, not equally spaced, hazard ratios.

This tutorial uses the convention that the treatment-to-control hazard ratio is \(HR=h_T(t)/h_C(t)\), where \(h_T(t)\) and \(h_C(t)\) are the treatment and control hazards at time \(t\). The model assumes proportional hazards: this ratio is constant over time. Under that assumption, \(HR<1\) means a lower treatment hazard, and \(\theta=\log(HR)<0\) represents a treatment effect in that direction. The prior describes uncertainty about this model parameter before the trial data are incorporated; it is not a statement about the probability that a particular patient will experience an event.

Definition: The log hazard ratio is \(\theta=\log(HR)\), so \(HR=\exp(\theta)\). A normal prior for this parameter is written \(\theta\sim N(m,\sigma^2)\), where \(m\) is the prior mean and \(\sigma\) is the prior standard deviation. Since \(HR\) is positive, taking its logarithm gives a parameter that can range over the real line.

Converting Clinically Interpretable Bounds

Suppose clinicians specify a lower bound \(L\) and upper bound \(U\) for a central prior interval, with \(0<L<U\). If the intention is that the interval contain a stated proportion \(q\) of prior probability, then the probability level is part of the specification. For a normal prior, a central interval has equal tail probabilities. Let \(z_{(1+q)/2}\) be the corresponding standard normal quantile. The bounds then satisfy \( \log(L)=m-z_{(1+q)/2}\sigma \) and \( \log(U)=m+z_{(1+q)/2}\sigma \). Solving these two equations gives the prior parameters.

m=\frac{\log(L)+\log(U)}{2}, \qquad \sigma=\frac{\log(U)-\log(L)}{2z_{(1+q)/2}}.

The centre \(m\) is the midpoint of the log bounds. Equivalently, \(\exp(m)=\sqrt{LU}\), the geometric midpoint of the hazard-ratio bounds. The spread \(\sigma\) depends both on the log distance between the bounds and on the probability level \(q\). A wider interval implies greater prior uncertainty if \(q\) is held fixed; a higher central probability for fixed bounds implies a smaller \(\sigma\).

This conversion requires positive, ordered bounds and a stated probability level. It also assumes the chosen normal distribution is a reasonable representation of the intended prior beliefs. Bounds alone do not uniquely specify a probability distribution: they do not say how much prior probability lies between them, or how probability is distributed within and outside them. The formulas above specifically encode a central interval of a normal prior for \(\theta\), transformed to the hazard-ratio scale.

Worked Example: A 95% Prior Interval That Is Not Centred on No Effect

Suppose a hypothetical trial team believes that a central 95% prior interval for the treatment-to-control hazard ratio runs from \(0.60\) to \(1.40\). The bounds are positive and ordered, and the team has explicitly assigned 95% prior probability to the interval. For a central 95% normal interval, \(z_{0.975}=1.96\).

Transform the bounds and calculate the prior mean:

m=\frac{\log(0.60)+\log(1.40)}{2} =\frac{-0.510826+0.336472}{2} =-0.087177.

The prior standard deviation is:

\sigma=\frac{\log(1.40)-\log(0.60)}{2(1.96)} =\frac{0.847298}{3.92} =0.2161.

Thus the elicited prior is \(\theta\sim N(-0.08718,0.2161^2)\). Its median hazard ratio is \(\exp(m)=\sqrt{0.60\times1.40}=0.9165\), the geometric midpoint of the bounds. The interval is not symmetric around \(HR=1\): its bounds have product \(0.84\), rather than \(1\).

For example, the prior probability of a hazard ratio below 1 is:

P(HR<1)=P(\theta<0) =\Phi\left(\frac{0-(-0.087177)}{0.2161}\right) =\Phi(0.4033) \approx 0.657.

This prior assigns about 65.7% probability to a lower treatment hazard, not 50%. That is consistent with the interval’s geometric midpoint being below 1. The probability is a prior probability about the model parameter, not a probability that the null hypothesis is true.

What Symmetry on the Log Scale Means

A normal distribution for \(\theta\) is symmetric around \(m\): points equally far above and below \(m\) have equal prior density. On the hazard-ratio scale, exponentiation turns those points into ratios equally distant on the log scale, or equivalently into reciprocal ratios around \(\exp(m)\). For a prior centred at \(m=0\), the reciprocal pairs are \(r\) and \(1/r\), and the median hazard ratio is 1. The hazard-ratio distribution is not generally symmetric in ordinary additive distance around 1.

m=0 \quad\Longrightarrow\quad \exp(m-a)=\exp(-a)=\frac{1}{\exp(a)}.

For instance, if one bound is \(0.75\), its reciprocal is \(1/0.75=1.333\ldots\), not \(1.25\). The log-scale distances from 1 are equal because \(\log(0.75)=-\log(1.333\ldots)\). A normal prior for \(\theta\) therefore yields a lognormal prior for \(HR\). Its median is \(\exp(m)\); the median is not generally the mean hazard ratio.

Worked Example: A Prior Symmetric Around No Effect

Suppose a hypothetical clinical team gives a central 90% prior interval of \(0.75\) to \(1.3333\) for the treatment-to-control hazard ratio. These bounds are reciprocal to the displayed precision, so their log midpoint is zero. The central 90% standard normal quantile is \(z_{0.95}=1.644854\).

The prior parameters are:

m=\frac{\log(0.75)+\log(1.3333)}{2}\approx 0, \qquad \sigma=\frac{\log(1/0.75)-\log(0.75)}{2(1.644854)} =\frac{0.575364}{3.289708} =0.1749.

Thus \(\theta\sim N(0,0.1749^2)\), and the median hazard ratio is 1. By symmetry, \(P(HR<1)=0.500\). A stronger benefit, \(HR<0.80\), has prior probability:

P(HR<0.80) =\Phi\left(\frac{\log(0.80)-0}{0.1749}\right) =\Phi(-1.276) \approx 0.101.

The 90% prior interval assigns 5% probability below 0.75 and 5% above 1.3333. Its bounds are equally far from zero on the log scale, even though their ordinary distances from 1 are \(0.25\) and about \(0.3333\). The asymmetry in additive distances is expected, not a calculation error.

Working from Quantiles and Checking Implied Beliefs

The same logic works when clinical judgments are expressed as quantiles rather than as a central interval. If the median hazard ratio and an upper quantile are elicited, take their logarithms and use the corresponding normal quantile to determine \(m\) and \(\sigma\). The resulting prior implies all other quantiles and probabilities; checking those implications is an important part of deciding whether the prior represents the intended beliefs.

Worked Example: Using a Median and an Upper Quantile

Suppose a hypothetical team states that the median plausible hazard ratio is \(0.85\), and that only 5% of prior values should exceed \(1.10\). These are the 50th and 95th percentiles, respectively. Under a normal prior for \(\theta\), the median determines \(m\), and the 95th percentile is \(m+1.644854\sigma\).

First, the median gives \(m=\log(0.85)=-0.162519\). The upper quantile gives:

\sigma=\frac{\log(1.10)-m}{1.644854} =\frac{0.095310-(-0.162519)}{1.644854} =\frac{0.257829}{1.644854} =0.1568.

The prior is therefore approximately \(\theta\sim N(-0.1625,0.1568^2)\). As a consistency check, its 5th percentile on the log scale is \(m-1.644854\sigma=-0.420348\), which transforms to \(HR=\exp(-0.420348)\approx0.6568\). Thus the implied central 90% hazard-ratio interval is approximately \(0.657\) to \(1.10\).

The implied prior probability of a lower treatment hazard is:

P(HR<1)=\Phi\left(\frac{0-(-0.162519)}{0.1568}\right) =\Phi(1.037) \approx 0.850.

This is a consequence of the specified median and upper quantile under the normal-on-log-scale assumption. If the team finds the implied lower quantile or benefit probability implausible, it should revisit the judgments or the distributional assumption rather than silently changing the calculated parameters.

Assumptions, Diagnostics, and Alternatives

The conversion is appropriate only when the hazard ratio is the parameter of interest under a proportional-hazards model, and the elicited bounds refer to uncertainty about that parameter. If treatment effects vary substantially over time, a single constant hazard ratio may not describe the clinical effect; a prior on that single parameter cannot fix a mismatch between the model and the scientific question. The prior also does not specify the baseline hazard or other model parameters.

Before using a converted prior, check that the bounds have the intended treatment-versus-control orientation, that they are positive and correctly ordered, and that the probability level and quantile convention are stated. Then verify that the calculated prior reproduces the elicited quantiles and inspect clinically important probabilities such as \(P(HR<1)\). These checks often reveal an accidental reversal of the ratio or an unintended shift away from no effect.

A particularly important orientation check is that replacing treatment-to-control \(HR\) with control-to-treatment \(1/HR\) changes \(\theta\) to \(-\theta\). The mean changes sign while the standard deviation stays the same. The prior must use the same orientation as the trial model; otherwise the interpretation of benefit and harm is reversed.

A normal prior on the log hazard ratio is a convenient choice, not a consequence of knowing two bounds. A prior specified directly on \(HR\) is another possibility, but it must respect the positive range and its symmetry and tail behaviour may differ. A different distribution for \(\theta\) could also match selected quantiles while assigning different probabilities elsewhere. The earlier material on skeptical priors for treatment effects provides one way to express beliefs concentrated near no effect; here the essential issue is converting hazard-ratio-scale judgments while preserving their intended quantile meaning.

Key takeaway: To convert a central hazard-ratio interval into a normal prior for the log hazard ratio, take the logarithms of both bounds, use their midpoint for \(m\), and divide their log distance by twice the appropriate normal quantile for \(\sigma\). Log-scale symmetry means reciprocal hazard ratios around the median, not equal additive distances around 1.

Common Mistakes

Omitting the probability level. Bounds such as 0.70 and 1.30 do not determine a prior standard deviation by themselves. State the central probability or quantiles those bounds represent.
Using the arithmetic midpoint. The prior median is the geometric midpoint \(\sqrt{LU}\), not generally \((L+U)/2\). Calculate the centre after taking logarithms.
Assuming symmetry on the wrong scale. Equal log distances from 1 give reciprocal bounds, not bounds equally spaced from 1 in ordinary units.
Reversing the ratio. A control-to-treatment ratio is the reciprocal of a treatment-to-control ratio. Reversing it changes the sign of the log hazard ratio and the interpretation of benefit.
Calling a prior interval a confidence interval. A prior interval summarizes the specified distribution before seeing the trial data; it is not a frequentist confidence interval.

Check Your Understanding

Answer each question, then compare with the solutions below.

A central 95% prior interval for \(HR\) is \(0.80\) to \(1.25\). What are \(m\) and \(\sigma\) for a normal prior on \(\theta\)?
For \(\theta\sim N(\log(0.90),0.20^2)\), what is the median hazard ratio and the approximate central 95% interval for \(HR\)?
If the ratio is redefined as control-to-treatment instead of treatment-to-control, how do the normal prior parameters change?
Why are the bounds \(0.80\) and \(1.20\) not symmetric around no effect on the log scale?

Solutions. (1) Since \(0.80\times1.25=1\), \(m=0\). The standard deviation is \(\log(1.25)/1.96=0.223144/1.96=0.1138\). (2) The median is \(\exp(\log(0.90))=0.90\). The central 95% limits are \(\exp(\log(0.90)\pm1.96(0.20))\), or approximately \(0.608\) to \(1.332\). (3) The log ratio changes sign, so \(N(m,\sigma^2)\) becomes \(N(-m,\sigma^2)\). (4) The bounds are not reciprocal: \(1/0.80=1.25\), not \(1.20\). Their log distances from zero therefore differ.

What You'll Learn