Commensurate Priors for Log-Hazard Effects

Why Link Log-Hazard Effects?

A clinical trial may have historical evidence about a treatment’s effect on the hazard of an event, as well as current-trial evidence. If the historical and current effects are sufficiently similar, the historical information can improve precision. If they differ, borrowing too strongly can pull the current estimate toward an inappropriate value. A commensurate prior makes the assumed degree of similarity explicit: it links the current log-hazard treatment effect to the historical effect through a precision parameter.

The model concerns the relative treatment effect, not equality of the trials’ baseline hazards. In particular, a shared log-hazard effect does not require that historical and current participants have the same baseline event rate. The prior is a modeling assumption about how treatment effects relate across settings; it does not establish that the effects are exchangeable in reality.

Definition: Let $\theta_H$ be the historical log-hazard ratio and $\theta_C$ the current-trial log-hazard ratio, with each defined as the log hazard in the treatment group divided by the hazard in the control group. A commensurate prior is $$ \theta_C\mid\theta_H,\tau \sim N(\theta_H,\tau^{-1}), $$ where $\tau>0$ is the commensurability precision. Larger $\tau$ concentrates the current effect more tightly around the historical effect; smaller $\tau$ allows greater difference.

Under this definition, $\theta<0$ corresponds to a hazard ratio below 1 and therefore a lower hazard in the treatment group, while $\theta>0$ corresponds to a hazard ratio above 1. The log-hazard ratio is dimensionless; the hazard ratio is obtained by exponentiating it.

Likelihood and Historical Information

For a transparent calculation, suppose the historical and current analyses each provide an approximately normal estimate of the log-hazard ratio. Write $y_H$ and $y_C$ for those estimates, with standard errors $s_H$ and $s_C$. The normal approximation may arise from a large-sample analysis such as a Cox proportional-hazards analysis. It is an approximation to the sampling distribution, not a claim that event times themselves are normally distributed.

Working assumptions: The historical and current data are independent conditional on their effects; each log-hazard estimate is adequately approximated by a normal likelihood; the treatment-effect parameter has a consistent orientation in both studies; and the current trial’s analysis supports a proportional-hazards treatment effect over the period being summarized. Any censoring and model conditions needed for the underlying survival analysis must also be reasonable. The baseline hazard may differ between studies. The commensurate prior links $\theta_H$ and $\theta_C$, not their baseline hazards.

Suppose the historical analysis, including its specified prior, yields the posterior $\theta_H\mid D_H\sim N(m_H,V_H)$. Conditional on $\theta_H$ and $\tau$, the current effect has the commensurate prior above. Integrating over uncertainty in the historical effect gives

\theta_C\mid D_H,\tau \sim N\left(m_H,V_H+\tau^{-1}\right).

The historical posterior variance $V_H$ matters: even very large $\tau$ does not make an uncertain historical effect known without error. Combining this normal prior with a current normal likelihood $y_C\mid\theta_C\sim N(\theta_C,s_C^2)$ gives the following posterior.

Posterior result: For fixed $\tau$, define $V_0=V_H+\tau^{-1}$, the variance of the current-effect prior after accounting for historical uncertainty. Then $$ V_C=\left(\frac{1}{V_0}+\frac{1}{s_C^2}\right)^{-1},\qquad m_C=V_C\left(\frac{m_H}{V_0}+\frac{y_C}{s_C^2}\right), $$ so that $\theta_C\mid D_H,D_C,\tau\sim N(m_C,V_C)$.

Why the Posterior Shrinks

The result follows by multiplying the normal prior density for $\theta_C$ by its current-data likelihood. Ignoring factors that do not depend on $\theta_C$, the log of the product is

-\frac{1}{2}\left\{\frac{(\theta_C-m_H)^2}{V_0} +\frac{(y_C-\theta_C)^2}{s_C^2}\right\}.

Collecting the terms involving $\theta_C^2$ gives the total precision $1/V_0+1/s_C^2$. Collecting the linear terms gives the precision-weighted sum $m_H/V_0+y_C/s_C^2$. Completing the square therefore gives variance $V_C=(1/V_0+1/s_C^2)^{-1}$ and mean $m_C=V_C(m_H/V_0+y_C/s_C^2)$, as stated.

Equivalently, the posterior mean is a weighted average:

m_C=w\,m_H+(1-w)y_C,\qquad w=\frac{1/V_0}{1/V_0+1/s_C^2}.

For fixed $\tau$, the weight $w$ depends on the prior and likelihood precisions, not on whether $m_H$ and $y_C$ happen to agree. The posterior still moves toward the current estimate, but its amount of movement is set by those precisions. This distinction is important when interpreting a fixed-precision analysis.

Worked Example: Compatible Historical and Current Estimates

Suppose a hypothetical historical analysis gives $\theta_H\mid D_H\sim N(-0.30,0.04)$. A hypothetical current trial gives $y_C=-0.25$ with standard error $s_C=0.20$. Use a fixed commensurability precision $\tau=100$. These log-hazard effects are close, and the assumed normal approximations and common treatment-effect orientation are taken as adequate for this calculation.

The commensurate variance is $1/\tau=0.01$, so the current-effect prior based on the historical posterior has mean $-0.30$ and variance $V_0=0.04+0.01=0.05$. The current likelihood variance is $s_C^2=0.20^2=0.04$. Thus

V_C=\left(\frac{1}{0.05}+\frac{1}{0.04}\right)^{-1} =(20+25)^{-1}=0.02222,

and

m_C=0.02222\left(\frac{-0.30}{0.05}+\frac{-0.25}{0.04}\right) =0.02222(-6-6.25)=-0.2722.

The posterior standard deviation is $\sqrt{0.02222}=0.1491$. A normal-approximation 95% posterior interval is $-0.2722\pm1.96(0.1491)$, or approximately $[-0.5644,\ 0.0200]$ on the log-hazard-ratio scale. Exponentiating the posterior mean gives a hazard ratio of $\exp(-0.2722)=0.7617$; exponentiating the interval endpoints gives approximately $[0.5687,\ 1.020]$. Conditional on this model and fixed $\tau$, the posterior probability that the log-hazard ratio lies in this interval is approximately 95%. The interval includes a hazard ratio of 1, so it does not exclude no relative hazard effect at this interval level.

Precision, Conflict, and Borrowing

The parameter $\tau$ controls how strongly the prior links the effects. As $\tau$ decreases toward zero, $1/\tau$ grows, $V_0$ grows, and the historical prior precision $1/V_0$ falls. Consequently $w$ approaches zero and the current posterior mean approaches $y_C$. As $\tau$ increases, the commensurate component contributes more precision. It does not eliminate historical uncertainty: as $\tau$ grows without bound, $V_0$ approaches $V_H$, rather than zero.

With a fixed $\tau$, conflicting data do not automatically reduce $w$. This is a limitation of a fixed-precision analysis: it can continue to shrink the current estimate toward the historical effect even when the observed effects disagree. If $\tau$ is uncertain and assigned a prior, the current data can inform it. A conflict may then support smaller values of $\tau$, which in turn weaken borrowing. That adaptation depends on the chosen prior for $\tau$ and the full model; it is not guaranteed merely by writing a commensurate prior.

A useful diagnostic for a proposed fixed $\tau$ is the predictive discrepancy between $y_C$ and the historical mean. Under the historical posterior and commensurate prior, before observing the current estimate, $y_C$ has approximate predictive distribution

y_C\mid D_H,\tau\sim N\left(m_H,\ s_C^2+V_H+\tau^{-1}\right).

A standardized discrepancy is therefore $(y_C-m_H)/\sqrt{s_C^2+V_H+\tau^{-1}}$. A large absolute value signals that the current estimate is surprising under the specified borrowing model. This is a diagnostic, not a universal cutoff: its interpretation depends on the normal approximation, the precision choice, and the surrounding clinical context.

Worked Example: Conflicting Effects at Two Precision Levels

Consider hypothetical data with historical posterior $\theta_H\mid D_H\sim N(-0.50,0.01)$ and current estimate $y_C=0.20$, with $s_C=0.15$. The effects point in opposite directions. Assume the same approximate normal likelihood and treatment-effect definition in both studies. Compare $\tau=100$ with $\tau=4$.

For $\tau=100$, $V_0=0.01+0.01=0.02$, so the prior precision is 50. The current likelihood variance is $0.15^2=0.0225$, with precision $44.444$. The posterior mean and variance are

m_C=\frac{50(-0.50)+44.444(0.20)}{50+44.444} =\frac{-16.111}{94.444}=-0.1706,\qquad V_C=\frac{1}{94.444}=0.01059.

For $\tau=4$, $V_0=0.01+0.25=0.26$, giving prior precision $3.846$. The corresponding calculation is

m_C=\frac{3.846(-0.50)+44.444(0.20)}{3.846+44.444} =\frac{6.966}{48.290}=0.1442,\qquad V_C=\frac{1}{48.290}=0.02071.

The weaker commensurability assumption ($\tau=4$) yields a posterior mean much closer to the current estimate and a larger posterior variance. The stronger setting pulls the estimate toward the historical effect despite the disagreement. With either fixed value of $\tau$, the observed conflict did not itself change the precision; the comparison shows the consequences of choosing different precisions.

Worked Example: Checking Predictive Compatibility

Use the compatible example’s historical posterior $N(-0.30,0.04)$, current standard error $s_C=0.20$, and $\tau=100$. The predictive standard deviation for the current estimate is

\sqrt{0.20^2+0.04+0.01}=\sqrt{0.09}=0.30.

If the observed current estimate is $y_C=-0.25$, its standardized discrepancy is $(-0.25-(-0.30))/0.30=0.167$, a small discrepancy under this predictive model. If instead a hypothetical current estimate were $y_C=0.35$, the discrepancy would be $(0.35-(-0.30))/0.30=2.17$. The latter is substantially farther from the historical mean relative to the predictive uncertainty, so it would prompt scrutiny of the commensurability assumption and study comparability. It does not, by itself, prove that the studies are incompatible or dictate a particular value of $\tau$.

Choosing Among Borrowing Approaches

A commensurate prior is one way to express a continuum of similarity between historical and current effects. With no borrowing, the current analysis uses its own likelihood and a current-only prior. With direct pooling, the effects are treated as equal, effectively imposing a much stronger relationship than a finite-precision commensurate prior. A fixed-$\tau$ commensurate model lies between these choices, though the amount of borrowing also depends on $V_H$ and the current standard error. Robust mixture approaches, covered in the earlier material on contamination priors, represent borrowing through alternative prior components rather than a single normal link. These approaches answer related but not identical modeling questions.

The assumptions should be checked in context. For example, if treatment effects are not proportional over time, a single log-hazard ratio may be an inadequate summary. If treatment definitions, eligibility criteria, outcome ascertainment, or follow-up differ materially, the historical effect may not be commensurate with the current one even if their numerical estimates are close. Conversely, a discrepancy may arise from sampling variability. Numerical similarity alone is not evidence that all other design differences are irrelevant.

Common Mistake and Practical Checks

Mistake: “The historical and current point estimates are close, so a large $\tau$ is automatically justified; if they disagree, the commensurate prior will automatically stop borrowing.” Neither conclusion follows. A fixed $\tau$ expresses a specified degree of linkage, and the fixed-$\tau$ posterior weight is determined by the two precisions, not by the observed gap. Choosing $\tau$ based only on observed agreement can also make the analysis data-dependent in ways not represented by the stated model.

Correction: Prespecify or carefully justify the precision model, report sensitivity to plausible values, and examine predictive compatibility. When $\tau$ is estimated or given a prior, report how its uncertainty and posterior distribution affect the treatment-effect inference. Check that the log-hazard ratios use the same treatment/control orientation and that the survival-model assumptions are credible in each setting. A borrowing result is conditional on these choices; it is not a design-independent guarantee of improved inference.

Key takeaway: The commensurate prior $\theta_C\mid\theta_H,\tau\sim N(\theta_H,\tau^{-1})$ translates historical information into a current-effect prior whose variance includes both historical uncertainty and lack of commensurability. Larger $\tau$ increases potential borrowing, but with fixed $\tau$ the borrowing weight does not react to conflict in the observed estimates.

Check Your Understanding

Answer each question using the model and assumptions stated above. Brief solutions follow the questions.

If $\tau$ decreases while $V_H$ and $s_C$ stay fixed, what happens to $V_0$ and the historical weight $w$?
Suppose $m_H=-0.20$, $V_H=0.01$, $y_C=-0.10$, $s_C=0.10$, and $\tau=99$. Calculate the posterior mean and variance for $\theta_C$.
Why does $\tau\to\infty$ not necessarily make the historical effect known exactly?
For fixed $\tau$, does a large difference between $m_H$ and $y_C$ automatically lower the historical weight? Explain.
What does a posterior log-hazard ratio of $-0.25$ imply about the hazard ratio, and what is its approximate value?

Solutions. (1) $1/\tau$ increases, so $V_0=V_H+1/\tau$ increases, its precision falls, and $w=(1/V_0)/(1/V_0+1/s_C^2)$ decreases. (2) Here $V_0=0.01+1/99=0.02010$, so its precision is about 49.75; the current variance is $0.01$, with precision 100. Thus $V_C=1/(149.75)=0.006678$ and $m_C=0.006678[49.75(-0.20)+100(-0.10)]\approx-0.1332$. (3) As $\tau\to\infty$, $V_0\to V_H$; uncertainty in the historical effect remains unless $V_H$ is itself zero. (4) No. With fixed $\tau$, the weight depends on $V_0$ and $s_C^2$, not the observed difference; conflict changes the weighted mean’s implication, not the weight. (5) The hazard ratio is $\exp(-0.25)\approx0.7788$, indicating an estimated hazard about 0.779 times the control hazard under the stated treatment/control orientation.