Posterior Hazard Ratios for Two Exponential Trial Arms

From Arm-Specific Hazards to a Hazard Ratio

A trial comparison is often summarized by a hazard ratio, but that ratio is a derived quantity: uncertainty in both arm-specific hazards contributes to uncertainty in their ratio. With exponential event times and independent gamma posteriors for the two hazards, the ratio has a tractable posterior distribution. This makes it possible to calculate probabilities such as the posterior probability that the treatment hazard is lower than the control hazard, rather than comparing only two point estimates.

Let \(\lambda_T\) and \(\lambda_C\) denote the treatment-arm and control-arm event hazards, respectively. In this tutorial the hazard ratio is defined as treatment divided by control, \(HR=\lambda_T/\lambda_C\). Thus \(HR<1\) corresponds to a lower instantaneous event hazard in the treatment arm under the model. Hazards have units of inverse time, while the ratio is dimensionless.

In this parameterization, a \(\operatorname{Gamma}(a,b)\) distribution has density proportional to \(\lambda^{a-1}e^{-b\lambda}\), mean \(a/b\), and rate \(b\). Combining this prior with the likelihood gives \( \lambda_j\mid D_j\sim\operatorname{Gamma}(a_j,b_j)\), where \(a_j=a_{j0}+d_j\) and \(b_j=b_{j0}+Y_j\). The posterior rate has units of time, so \(a_j/b_j\) has units of inverse time.

Deriving the Posterior Distribution of the Ratio

A ratio of independent gamma variables is not generally another gamma variable. A useful transformation instead converts it to a scaled beta-prime ratio. Define \(U=b_T\lambda_T\) and \(V=b_C\lambda_C\). Under the independent posteriors, \(U\sim\operatorname{Gamma}(a_T,1)\) and \(V\sim\operatorname{Gamma}(a_C,1)\), independently. The sum-and-proportion property of independent unit-rate gamma variables gives \(Q=U/(U+V)\sim\operatorname{Beta}(a_T,a_C)\).

This transformation supplies the complete posterior distribution of \(HR\). For any positive value \(h\), set \(q_h=h b_T/(b_C+h b_T)\). Since the transformation from \(Q\) to \(HR\) is increasing, \(P(HR\le h\mid D)=P(Q\le q_h\mid D)\). In particular, the posterior probability that the treatment hazard is lower than the control hazard is the beta cumulative probability at \(q_1=b_T/(b_T+b_C)\).

If \(q_L\) and \(q_U\) are the lower and upper quantiles of the \(\operatorname{Beta}(a_T,a_C)\) posterior, then a central equal-tailed \(100(1-\gamma)\%\) credible interval for the hazard ratio is \(\bigl((b_C/b_T)q_L/(1-q_L),\ (b_C/b_T)q_U/(1-q_U)\bigr)\), where \(q_L\) and \(q_U\) are the \(\gamma/2\) and \(1-\gamma/2\) beta quantiles. This is a posterior interval: conditional on the model and data, it contains the hazard ratio with posterior probability \(1-\gamma\).

Credible Intervals and Arm-Specific Survival

For the same hypothetical data, the equal-tailed 95% quantiles of \(\operatorname{Beta}(2,2)\) are approximately \(q_L=0.0943\) and \(q_U=0.9057\). Since \(b_C/b_T=10/12\), transforming both endpoints gives:

The 95% equal-tailed posterior credible interval is approximately 0.0868 to 8.00. It is wide and includes 1, reflecting substantial uncertainty about the relative hazards. Its asymmetry is natural for a positive ratio; an interval for \(HR\) need not be centered on a point estimate.

The ratio does not replace arm-specific survival summaries. Under an exponential model, survival at time \(t\) is \(S_j(t)=\exp(-\lambda_jt)\). With a gamma posterior, its posterior mean is the Laplace transform of that gamma distribution:

This posterior mean is not obtained by simply substituting the posterior mean hazard into the survival function. In general, \(E\{\exp(-\lambda_jt)\mid D_j\}\ne\exp[-E(\lambda_j\mid D_j)t]\). The following example shows both arm-specific survival summaries and how they complement a hazard-ratio summary.

Posterior Summaries, Diagnostics, and Limitations

The posterior distribution can be summarized in several ways. A posterior probability such as \(P(HR<1\mid D)\) directly answers a directional question. A credible interval describes a range of plausible ratio values and their posterior probability. A posterior mean or median gives a single summary, but should not obscure skewness or a long upper tail.

The posterior mean of \(HR\) exists only if \(a_C>1\), because the inverse of a \(\operatorname{Gamma}(a_C,b_C)\) variable has a finite expectation only under that condition. When it exists, independence gives:

For example, in the second hypothetical trial this mean is \((9/120.1)(100.1/10)\approx0.7501\). This number is not the ratio of posterior mean hazards, which is approximately \(0.07494/0.1099\approx0.682\). A ratio posterior can be highly skewed, and the two summaries answer different questions. If \(a_C\le1\), the posterior mean ratio is infinite or undefined, though posterior probabilities and finite quantiles remain well-defined.

A practical check is to compare the implied hazards and survival summaries with the clinical time scale and observed event/exposure information. An implausibly high hazard, a survival curve inconsistent with the event pattern, or a ratio interval spanning many orders of magnitude should prompt checks of units, event counts, exposure totals, prior parameters, and computation. Sparse events can produce wide or prior-sensitive posteriors. The exponential model also imposes a constant hazard over follow-up; systematic time patterns in observed event timing may make that assumption unsuitable. Independent-arm posteriors further rely on independent priors and arm-separated likelihoods, so shared parameters or informative dependencies require a different model.

For numerical work, the beta transformation provides a direct route to probabilities and quantiles. Another valid approach is to draw posterior samples independently from the two gamma distributions, divide each treatment draw by its paired control draw, and summarize the resulting ratios. Simulation is flexible, but introduces Monte Carlo error; the beta transformation is an exact distributional result for these independent gamma posteriors. Comparing arm-specific hazards alone is not a substitute for either method, because it does not quantify posterior uncertainty in their ratio.

Justification of the Beta Transformation

Common Pitfalls

• Reversing the ratio. If the stated estimand is treatment divided by control, the scale factor is \(b_C/b_T\), not \(b_T/b_C\). Write the ratio explicitly before transforming.
• Using only posterior mean hazards. The ratio of two posterior means is not generally the posterior mean or median of their ratio. Use the ratio posterior distribution for uncertainty statements.
• Calling a credible interval a confidence interval. The interval here has a posterior probability interpretation conditional on the model, prior, and data; it is not a repeated-sampling coverage statement.
• Reading \(HR<1\) as a survival probability statement. It indicates a lower constant event hazard in treatment under the exponential model. It does not by itself give the magnitude or uncertainty of a survival difference at a chosen time.
• Ignoring model fit and censoring. Gamma conjugacy does not make the exponential or non-informative censoring assumptions true. Check the time scale and censoring mechanism before relying on the posterior comparison.