Interpreting Posterior Hazard Ratios and Survival Differences

Two Summaries, Two Questions

A posterior hazard ratio and a posterior difference in survival probability can both summarize a treatment comparison, but they answer different questions. The hazard ratio is relative: it compares instantaneous event rates between arms. A survival-probability difference is absolute: it compares the probabilities of being event-free at one specified time. A favorable relative comparison does not, by itself, tell a clinician how many more patients are expected to be event-free at that time.

Let \(T\) denote the treatment arm and \(C\) the control arm, and let \(\lambda_T\) and \(\lambda_C\) denote their event hazards. For this tutorial’s exponential examples, each hazard is constant over time, so \(S_a(t)=\exp(-\lambda_a t)\) for arm \(a\). Hazards have units of inverse time; survival probabilities are unitless. Define the hazard ratio as treatment divided by control, and define the survival difference as treatment survival minus control survival. Under these conventions, values below one for the hazard ratio and values above zero for the survival difference favor treatment.

In Bayesian analysis, both quantities are functions of the arm-specific parameters. For each paired draw from the joint posterior of \((\lambda_T,\lambda_C)\), calculate \(HR=\lambda_T/\lambda_C\) and \(\Delta_S(t)=\exp(-\lambda_Tt)-\exp(-\lambda_Ct)\). The resulting draws approximate the posterior distributions of the two summaries. Their posterior means, medians, credible intervals, and probabilities of clinically relevant events can then be reported. The draws must remain paired: breaking their dependence can alter uncertainty in either comparison.

Assumptions Behind the Exponential Comparison

The exponential model assumes a constant event hazard within each arm during the modeled follow-up. It therefore implies \(S_a(t)=\exp(-\lambda_a t)\) and a constant hazard ratio over time. The event definition and time origin must be the same in both arms, and the horizon must be meaningful for the trial’s follow-up. If observations are right-censored, inference also relies on censoring being non-informative conditional on the modeled information; otherwise the event-time likelihood may not adequately describe the observed data.

An arm-specific hazard ratio is a treatment comparison, but a causal interpretation additionally requires a design and analysis that justify it. In a randomized trial, randomization supports treatment-group comparability at assignment; the interpretation still depends on appropriate handling of follow-up, missingness, censoring, and the defined treatment strategies. In particular, an exponential model is not made appropriate simply by randomization.

Independent priors and a likelihood that separates by arm are sufficient for posterior independence of the arm-specific hazards. More generally, posterior independence is justified when the joint posterior factors by arm. If the posterior does not factor, paired posterior draws must retain the dependence when forming ratios or survival differences.

Worked Examples

Why the Survival Difference Depends on Time

The fixed-horizon difference is a function of both hazards and \(t\). In the exponential model, if treatment has the lower hazard, write \(\lambda_T=r\lambda_C\), where \(0<r<1\). Then

At time zero the difference is zero, since both arms start with survival probability one. As \(t\) grows very large, both exponential survival probabilities approach zero, so their difference also approaches zero. The difference can therefore rise and later fall, even when the hazard ratio remains constant. To locate its maximum, differentiate with respect to \(t\):

Setting this derivative to zero and dividing by the positive quantity \(\lambda_C\exp(-r\lambda_Ct)\) gives \[ 1=r\exp[(1-r)\lambda_Ct]. \] Taking logarithms yields \[ t^*=\frac{\log(1/r)}{(1-r)\lambda_C}. \] For \(0<r<1\), this is positive. The derivative is positive before \(t^*\) and negative after it, so the difference reaches its maximum there. This result emphasizes why a survival difference must always be tied to a stated time horizon.

For the first example, \(r=0.8\) and \(\lambda_C=0.050\) per month, so \[ t^*=\frac{\log(1/0.8)}{(1-0.8)(0.050)} =\frac{0.2231}{0.010}=22.31\text{ months}. \] Thus the 12-month difference is before the model’s peak difference. This mathematical feature should not be mistaken for a general clinical pattern: real hazards need not be constant, and the exponential model may not describe long-term survival adequately.

Posterior Reporting, Diagnostics, and Alternatives

For posterior reporting, state the direction of each comparison, the horizon for \(\Delta_S(t)\), and the units for the hazards. A credible interval for a transformed quantity should be obtained by transforming each paired posterior draw and then taking the desired posterior quantiles. Do not assume that a ratio of separate posterior summaries equals the posterior summary of the ratio. Likewise, do not substitute posterior mean hazards into the survival formula and call the result the posterior mean survival difference: because the exponential transformation is nonlinear, these generally differ.

Diagnostics should assess whether the model and data support the interpretation. Check whether observed survival patterns and posterior predictive survival curves are compatible with a constant hazard in each arm. Marked changes in event rates over time, systematic disagreement between observed and predicted survival, or changing treatment effects across follow-up can signal that the exponential model is inadequate. Also check whether the horizon is supported by follow-up in both arms; extrapolation beyond observed follow-up can make a precise-looking posterior difference strongly model-dependent.

A hazard ratio is useful when a relative instantaneous-rate comparison is scientifically relevant and a proportional-hazards description is credible. A survival difference at a prespecified time is often easier to interpret as an absolute benefit or harm, but it is horizon-specific and depends on baseline risk. Neither summary replaces the other. When hazards are not constant, an exponential-model hazard ratio may obscure time variation; alternatives include a time-varying hazard comparison or reporting survival probabilities at clinically relevant times. The latter remains an absolute comparison, but its interpretation still depends on the chosen horizon and the survival model or estimator.

A frequent mistake is to interpret \(HR=0.80\) as a 20-percentage-point increase in survival, or as a 20% increase in the probability of survival. The correction is to say that the treatment hazard is 20% lower under the stated hazard-ratio model. Calculate \(\Delta_S(t)\) separately to describe the absolute survival difference at a particular time. Another pitfall is to report a survival difference without naming its horizon: 12-month and 24-month differences are not interchangeable.