Worked Right-Censored Survival Posterior Analysis

From Trial Records to Posterior Comparisons

A survival analysis can look abstract when it begins with likelihoods and distributions. In a trial, however, the information entering the model is concrete: for each participant, the observed follow-up time and whether an event occurred or follow-up ended first. This tutorial works through how those records produce posterior inferences about each arm’s hazard, survival at a specified time, and the hazard ratio.

The model is deliberately simple: within each arm, event times have a constant hazard. This makes the calculations transparent, but also makes the assumption consequential. The results describe a hypothetical trial; they are not evidence about any actual treatment.

Model, Data, and Assumptions

The right-censored likelihood contribution from an event is the event-time density, \(\lambda_j\exp(-\lambda_jY_{ij})\). A censored observation contributes the probability of surviving through its observed time, \(\exp(-\lambda_jY_{ij})\). Multiplying the contributions gives a likelihood proportional to \(\lambda_j^{D_j}\exp(-\lambda_jE_j)\). Thus the data enter through two summaries: event count and total exposure.

The calculation requires several substantive assumptions. Event times within an arm follow a constant hazard over the modeled period; participants’ event processes are conditionally independent given the arm hazard; and censoring is noninformative for the event time, so censoring does not selectively hide events in a way left unaccounted for by the model. Follow-up times and event indicators must be recorded correctly. For a causal interpretation of an arm contrast, randomization and the trial’s conduct also matter; a hazard comparison alone does not establish that every modeling assumption is appropriate.

Use independent Gamma priors for the two arm hazards, with the shape-rate convention:

Here \(a_j\) is the shape and \(b_j\) is the rate parameter in the Gamma density. Because \(\lambda_j\) has units of inverse time, \(b_j\) has units of follow-up time, so \(b_j\lambda_j\) is dimensionless. Combining this prior with the likelihood gives:

Worked Example: Summarizing Participant Records

Posterior Hazards and Fixed-Time Survival

For a complete worked analysis, assign each arm the prior \(\operatorname{Gamma}(1,12\text{ months})\). Its prior mean hazard is \(1/12\) per month. The prior rate of 12 months is a time quantity, consistent with a hazard measured per month. Applying the update to the records above gives:

The posterior mean of a Gamma(shape, rate) variable is shape divided by rate. Therefore the control-arm posterior mean hazard is \(4/42=0.09524\) events per participant-month, and the treatment-arm posterior mean hazard is \(3/48=0.06250\) events per participant-month. These are posterior summaries under the specified model and priors, not observed event rates that account for no uncertainty.

Uncertainty about fixed-time survival can also be represented by transforming a credible interval for the hazard. Since \(S_j(t)=\exp(-t\lambda_j)\) decreases as \(\lambda_j\) increases, the hazard interval’s endpoints must be reversed when transformed. For example, approximate equal-tail 95% posterior intervals for the hazards are \(0.02595\) to \(0.2087\) per month in control and \(0.01289\) to \(0.1505\) per month in treatment. Transforming the upper hazard endpoint to the lower survival endpoint gives approximate 95% posterior intervals for 12-month survival of \(0.0817\) to \(0.7324\) in control and \(0.1643\) to \(0.8567\) in treatment. These wide intervals show why point summaries alone do not capture the uncertainty.

Comparing the Arms With a Hazard Ratio

Given independent priors and arm-specific data, the two posterior hazards are independent. In the example, the posterior mean hazard ratio can be computed using independence and the Gamma inverse-moment formula. If \(X\sim\operatorname{Gamma}(a,b)\) with \(a>1\), then \(E(1/X)=b/(a-1)\). Hence:

This posterior mean is dimensionless. It is distinct from the ratio of posterior mean hazards, \((3/48)/(4/42)=0.6563\); in general, the mean of a ratio is not the ratio of the means. To describe posterior uncertainty or the probability of a beneficial direction, use the joint posterior distribution rather than treating either point summary as definitive.

Derivation, Diagnostics, and Alternatives

The conjugate update follows directly from multiplying the likelihood and prior. Suppressing constants that do not depend on \(\lambda_j\), the posterior density is proportional to:

This is the kernel of a Gamma density with shape \(a_j+D_j\) and rate \(b_j+E_j\), which justifies the update. Notice that an event increases the shape, while all observed follow-up—including censored follow-up—increases the rate. Both terms affect the posterior.

Diagnostics begin with the records and the model’s implied event-time behavior. Check that each participant contributes only the observed time up to event or censoring, that event indicators are coded correctly, and that exposure units are consistent with the prior rate and target time. The constant-hazard assumption implies an exponential survival curve; substantial time variation in the observed hazard is a warning that this model may be too restrictive. Informative censoring, such as loss to follow-up related to unobserved prognosis, can also undermine the analysis. A Gamma prior with little data may materially influence the result, so prior sensitivity is important.

The exponential model is concise and conjugate, but it is not the only approach. Kaplan–Meier estimation is a nonparametric frequentist alternative for describing survival under right censoring, while Weibull or piecewise-exponential models allow hazards to vary over time. Those alternatives answer related questions with different assumptions and inferential frameworks. The present calculation’s advantage is a direct posterior distribution for hazards and their functions; its limitation is that apparent precision cannot compensate for a misspecified constant hazard, problematic censoring, or a poorly chosen prior.