Posterior Inference for Median Survival

Why Infer Median Survival?

A hazard is expressed in inverse-time units, while median survival is expressed in time units. The two quantities describe the same exponential survival model from different perspectives: the hazard is the instantaneous event rate, and the median is the time by which the model assigns a survival probability of one half. In a clinical trial, median survival can be easier to communicate clinically, but its uncertainty must be calculated from the posterior distribution rather than by simply transforming a point estimate and keeping the same uncertainty measure.

The introductory material on posterior survival probability at a fixed time considers uncertainty in \(S(t)\), the probability of surviving beyond a specified time \(t\). Here the target is different: the time \(m\) at which the survival curve reaches one half. We derive its full posterior distribution from the posterior for the hazard.

Exponential Median and Model Conditions

The exponential model assumes that, conditional on \(\lambda\), the event-time hazard is constant over follow-up. It also assumes the event times for different participants are conditionally independent under the model. With right-censored observations, the censoring mechanism must be non-informative for the event-time analysis: after accounting for the model, censoring must not convey additional information about a participant’s event time. These assumptions may be unsuitable when the hazard changes substantially over time or when censoring is informative.

Suppose the posterior distribution for the hazard, after combining the prior and trial data, is \(\lambda\mid\text{data}\sim\operatorname{Gamma}(a,b)\), using the shape-rate parameterization. Thus \(a>0\), \(b>0\), and the density is proportional to \(\lambda^{a-1}\exp(-b\lambda)\). The hazard has units of inverse time, so \(b\) has units of time. We treat \(a\) and \(b\) as fixed posterior parameters in the transformation below.

Transforming the Hazard Posterior

Because \(M=\log(2)/\lambda\) decreases as \(\lambda\) increases, a large hazard corresponds to a short median survival time, and vice versa. This reversal matters for both probabilities and credible-interval endpoints. Write \(c=\log(2)\). For any \(m>0\),

where \(F_{\lambda\mid\text{data}}\) is the posterior cumulative distribution function of the hazard. The inequality reverses because \(\lambda\) is positive and \(c/\lambda\) is decreasing in \(\lambda\). This formula determines the entire posterior distribution of median survival.

The density follows by the change-of-variables rule. Since \(\lambda=c/m\), the absolute derivative is \(|d\lambda/dm|=c/m^2\). Multiplying the Gamma density evaluated at \(c/m\) by this Jacobian gives

This is an inverse-Gamma distribution with shape \(a\) and scale \(bc\). The derivation uses only the posterior for \(\lambda\) and the deterministic relationship between \(\lambda\) and \(M\); no additional sampling approximation is needed.

Posterior Summaries and Credible Intervals

The posterior mean of median survival exists only if \(a>1\). When it exists, it is

For \(a>2\), the posterior variance also exists and is

These existence conditions arise because the inverse-Gamma density has a long right tail: when the hazard is very close to zero, its corresponding median survival is very large. If \(a\leq1\), the posterior mean of \(M\) is infinite; if \(1<a\leq2\), the mean exists but the variance is infinite. A finite posterior mean is therefore not guaranteed simply because the hazard posterior is proper.

To obtain an equal-tailed \(100(1-\alpha)\%\) credible interval for \(M\), first find the \(\alpha/2\) and \(1-\alpha/2\) quantiles of the hazard posterior. The lower endpoint for \(M\) comes from the upper hazard quantile:

Conditional on the model and data, this interval contains the unknown model median with posterior probability \(1-\alpha\). Since the transformation is nonlinear, the interval will generally be asymmetric around a posterior point summary. Transforming the endpoints correctly is essential; retaining their original order would produce an invalid interval.

Worked Trial-Data Update

For independent exponential event times with right censoring, let \(d\) be the number of observed events and let \(T\) be the total observed follow-up time across participants. Under the constant-hazard model and non-informative censoring, the likelihood for \(\lambda\) is proportional to \(\lambda^d\exp(-\lambda T)\). With a Gamma prior \(\operatorname{Gamma}(a_0,b_0)\) in the shape-rate convention, the posterior is \(\operatorname{Gamma}(a_0+d,b_0+T)\). The example applies this update and then transforms the resulting posterior to median survival.

Probabilities and Interpretation on the Time Scale

The transformed distribution also answers questions about clinically meaningful thresholds. For a threshold \(t_0>0\), the posterior probability that the model median exceeds \(t_0\) is

This is a posterior probability about the parameter \(M\), not the probability that a particular participant survives beyond \(t_0\). The latter is \(S(t_0\mid\lambda)\) for a specified hazard, or a posterior distribution for \(S(t_0\mid\lambda)\) when the hazard is uncertain. Keeping the parameter target distinct from an individual event-time probability avoids a common interpretive error.

Diagnostics, Alternatives, and Limitations

The transformation is exact for the exponential model, but it does not protect the analysis from model failure. Before interpreting the median, examine whether a constant hazard is plausible over the trial’s follow-up. Strong time variation, such as an early high hazard followed by a lower hazard, makes a single exponential median an inadequate summary of the survival pattern. Also investigate whether censoring could depend on prognosis after accounting for the model; informative censoring can undermine the likelihood and posterior update.

A practical diagnostic is to compare the exponential survival curve implied by the fitted hazard with the observed event and censoring pattern over time. Marked systematic departures, especially across clinically meaningful follow-up periods, are a warning that the constant-hazard assumption may not describe the data well. Sparse events can also leave the posterior sensitive to the prior, particularly near zero hazard, where very long median times occur. Report the prior and assess whether plausible alternative priors materially change the transformed interval or threshold probability.

The main alternative summaries answer different questions. A posterior interval for the hazard is natural for rate-based decisions, but it is not an interval on the time scale. A posterior survival probability at a fixed time describes survival beyond that time, rather than the time at which survival reaches one half. A median is clinically direct under an exponential model, but it compresses the survival curve to one number and may conceal changes in hazard over time. The transformation also produces a long-tailed posterior: a small amount of posterior probability near zero hazard can substantially increase the mean or make it infinite.