Posterior Predictive Checks Of Event Histories

Why Check Replicated Event Histories?

A fitted joint model can describe longitudinal trajectories well and still reproduce survival outcomes poorly. It may predict too many events by a clinically relevant time, place events too early, or fail to reproduce the observed amount or timing of censoring. Posterior predictive checks address these questions by comparing the observed event history with histories simulated from the fitted model.

The goal is not to prove that a model is correct. A check asks whether a feature of the observed data would be unusual if the fitted model and its assumptions were adequate. The feature should be chosen to expose a relevant survival-component discrepancy, and the simulation should reproduce the design and conditioning used in the analysis.

Event Histories and Posterior Prediction

For subject \(i\), let \(T_i\) be the event time and \(C_i\) the censoring time. The observed follow-up time and event indicator are \(Y_i=\min(T_i,C_i)\) and \(\Delta_i=I(T_i\le C_i)\), respectively. Thus \(\Delta_i=1\) denotes an observed event and \(\Delta_i=0\) denotes right censoring. Let \(w_i\) denote observed baseline covariates and \(b_i\) the subject-specific random effects in the joint model.

A common survival component specifies an event hazard conditional on model parameters and the longitudinal process. For example, if \(m_i(t)\) is the subject-specific latent marker trajectory, one possible specification is

Here \(\theta\) collects model parameters, including baseline hazard and association parameters. The specific hazard form depends on the fitted joint model; a check must simulate from that fitted form, not from a different survival model. The event-time simulation must also respect the model's risk interval and any competing study-end rules.

A typical workflow draws \(\theta^{(s)}\) from the posterior distribution and generates replicated outcomes \(T_i^{\mathrm{rep},(s)}\). The treatment of \(b_i\) matters. Drawing random effects from their posterior conditional on the subject's data gives a check conditional on information about these particular subjects. Drawing new effects from the estimated population distribution targets new subjects from the modeled population. These answer different questions; conditioning closely on subject-specific information can make a check less sensitive to misspecification.

Specify What Is Held Fixed

Right censoring requires an explicit prediction plan. One option is to hold each observed \(C_i\) fixed, simulate \(T_i^{\mathrm{rep}}\), and form \(Y_i^{\mathrm{rep}}=\min(T_i^{\mathrm{rep}},C_i)\) and \(\Delta_i^{\mathrm{rep}}=I(T_i^{\mathrm{rep}}\le C_i)\). This checks event predictions under the actual observed follow-up limits. It does not check whether a model for censoring is adequate, because the censoring times were not generated.

Alternatively, if the analysis specifies a censoring model, draw \(C_i^{\mathrm{rep}}\) from that model and construct the replicated observed history from both simulated times. This can assess censoring patterns, but requires an adequate censoring model and a defensible assumption about its relationship to event times. In standard survival analyses, censoring is often assumed independent of the event time conditional on modeled covariates and relevant history. If censoring is informative and the model does not account for it, ordinary posterior predictive comparisons may be misleading.

The posterior predictive distribution of a summary \(D\) is the distribution of \(D^{\mathrm{rep}}\) across replicated histories, with uncertainty in fitted quantities included. A one-sided posterior predictive tail area can be estimated as

For a lower-tail question, reverse the inequality. The added one prevents reporting a zero estimate when no simulated value is as extreme as the observed value. This tail area is a model-checking measure, not a frequentist p-value with a uniform null distribution. Because the same data inform the posterior and the discrepancy, posterior predictive tail areas can be conservative.

Choose Survival Summaries That Match the Question

Useful discrepancies include the total number of events by a fixed time, the distribution of event-free time, and the number or timing of censorings. These summaries should be defined before examining the resulting predictive comparisons, and time points should be meaningful and within the observed follow-up range.

• Event counts: compare the observed number of events by a time \(t\) with replicated counts. Counts are easy to interpret but may conceal when events occurred or which subgroups drive a discrepancy.
• Event-time distribution: compare a survival curve or selected survival probabilities. When follow-up is censored, a Kaplan–Meier estimate uses both event and censoring information; a histogram of observed event times alone discards censored observations and can distort the comparison.
• Censoring pattern: compare counts censored by selected times or the distribution of censoring times when censoring is simulated. If observed censoring times are held fixed, the replicated censoring pattern is fixed by construction and is not a test of the censoring mechanism.

Compare Event-Time Distributions With Censoring Present

A Kaplan–Meier summary is often more appropriate than comparing only observed event times. At each event time, it estimates the probability of remaining event-free by multiplying the conditional survival proportions among those still at risk. Censoring removes a subject from subsequent risk sets but is not counted as an event. The method relies on censoring being non-informative for event time under the analysis conditions.

For a predictive check, calculate the same summary from observed data and every replicated data set. This gives a predictive distribution for the selected survival probability or curve feature. The Kaplan–Meier estimate is a summary for comparison, not an additional data-generation step. A small sample can make its uncertainty wide, so a visual or numerical discrepancy should be interpreted alongside the number at risk and the predictive spread.

Check Censoring Patterns When They Are Simulated

A censoring check requires replicated censoring times, not just replicated event times. For example, a censoring model might generate \(C_i^{\mathrm{rep}}\) conditional on administrative study end and observed predictors. A useful discrepancy could be the number censored before 12 months. The check is meaningful only if the censoring model and its assumptions are part of the analysis and the same censoring summary is computed for observed and replicated histories.

Justification and Practical Workflow

The central calculation follows directly from conditional probability. If \(D^{\mathrm{rep}}\) is a chosen discrepancy, then its posterior predictive distribution averages its sampling distribution over uncertainty about the fitted quantities:

In simulation, the integral is approximated by drawing \(\theta^{(s)}\) from the posterior, simulating a history under that draw and the chosen design, and computing \(D^{\mathrm{rep},(s)}\). Averaging the indicators \(I\{D^{\mathrm{rep},(s)}\ge D^{\mathrm{obs}}\}\) estimates the corresponding tail probability. Thus posterior uncertainty and outcome variability both contribute to the predictive distribution; plugging in a single parameter estimate would omit the former.

Diagnostics, Alternatives, and Limitations

A predictive distribution that is too narrow, shifted, or systematically unlike the observed summaries signals that the model may not capture the relevant feature. Check that simulated histories obey the same entry, follow-up, and administrative censoring rules as the observed data. Also confirm that event indicators and time scales were encoded consistently. Monte Carlo uncertainty matters: with few replications, tail-area estimates are noisy, so increase \(S\) when a decision depends on a small tail area.

Posterior predictive checks complement, rather than replace, survival residual diagnostics such as martingale, Cox–Snell, and event-time probability integral transform residuals. Residual methods summarize subject-level departures in particular ways; posterior predictive checks directly ask whether selected features of whole replicated histories resemble the observed data. Neither approach alone guarantees adequate fit. A global event count can look reasonable while event timing is wrong, and an acceptable event-time summary can conceal poor fit in a subgroup.

These checks are limited by the summaries selected, the fitted model, and the censoring assumptions. Repeatedly searching many summaries and reporting only the most extreme comparison exaggerates apparent evidence of misfit. Also, posterior predictive tail areas are not calibrated frequentist significance levels, especially when the discrepancy is strongly informed by the same data used to estimate the posterior. Treat an unusual comparison as a prompt for focused model assessment, not an automatic instruction to change one parameter.

Common Pitfall

Wrong reasoning: “The replicated data use the observed censoring times, and their censoring pattern resembles the data, so the model reproduces censoring well.” Correction: with censoring times fixed, the replicated pattern is imposed, not predicted. This design can assess event outcomes under the observed follow-up limits, but checking the censoring process requires generating censoring times from an explicit censoring model. A second pitfall is comparing only the times of observed events as if they were a complete sample of event times; censoring makes that subset selective. Use a censoring-aware summary, such as a Kaplan–Meier estimate, and state the assumptions needed for its interpretation.