Posterior Uncertainty in Subject-Specific Hazards

Why quantify uncertainty in an individual’s hazard?

A shared-random-effects joint model connects a subject’s longitudinal trajectory with their event hazard through latent subject-specific effects. For a particular subject, the hazard at time \(t\) is therefore uncertain: the subject’s random effects are estimated rather than known, and the hazard parameters are uncertain as well. Reporting only a hazard calculated at estimated parameter values can conceal that uncertainty.

The target here is the posterior distribution of one subject’s hazard at one specified time, conditional on the observed joint-model data. It is not the distribution of hazards across a new population, and a hazard is not itself a probability of experiencing an event.

Define the subject-specific hazard at the chosen time

Let \(D\) denote the observed data used to fit the joint model, including the longitudinal measurements and event or censoring information. Write \(\theta\) for the model parameters, including longitudinal fixed effects, association parameters, and baseline-hazard parameters. Let \(b_i\) be the random effects for subject \(i\). One common current-value shared-random-effects specification is

Here \(m_i(t;\beta,b_i)\) is the modelled longitudinal value at time \(t\); \(h_0(t;\gamma)\) is the baseline hazard; and \(\eta\) and \(\alpha\) are regression and association parameters. The exponent is dimensionless, so the hazard has the same units as the baseline hazard, such as events per year. The formula is one standard shared-effects form; a model with another specified association structure must use that structure in the calculation.

This is a subject-specific, conditional hazard: at time \(t\), it describes the instantaneous event rate for subject \(i\), conditional on being event-free just before \(t\), given the subject’s latent effects and model parameters. Its posterior distribution reflects uncertainty about those quantities. It does not describe variability between hypothetical new subjects unless the target is explicitly changed to include new random effects drawn from their population distribution.

From posterior draws to a hazard distribution

Suppose there are \(M\) draws from the joint posterior, written \((\theta^{(s)},b_i^{(s)})\) for \(s=1,\ldots,M\). Evaluate the hazard for each draw using the same subject and the same specified time:

These values are posterior draws of the hazard. Their empirical mean estimates the posterior mean, their empirical quantiles estimate posterior quantiles, and the fraction above a threshold estimates a posterior exceedance probability. For example,

The computation is justified because the posterior mean is the integral of the hazard function over the joint posterior of the random effects and parameters. A posterior sample approximates that integral by averaging the function evaluated at its draws. Likewise, the posterior probability above \(c\) is the integral of the indicator \(I\{h_i(t)>c\}\), approximated by the proportion of draws for which the indicator is one. For dependent simulation draws, these summaries remain defined the same way; adequate effective sample size is needed for them to be estimated precisely.

A central 95% equal-tailed credible interval is formed from the 2.5th and 97.5th posterior quantiles of the hazard draws. Under the fitted model and conditional on \(D\), this interval contains 95% of the posterior probability for the hazard at that time. It is not a statement that a fixed parameter has a 95% frequentist probability of lying in the realized interval.

Why paired draws and the chosen time matter

Each hazard draw should use a coherent joint posterior draw: its baseline-hazard parameters, association parameter, fixed effects, and subject-specific random effects must come from the same draw. These quantities can be posterior dependent. Combining a random effect from one draw with an association parameter from another generally creates combinations that do not represent the fitted joint posterior, and can distort the uncertainty in the hazard.

The time \(t\) is part of the target, not just a label. In a random-intercept-and-slope model, uncertainty about the slope changes the estimated longitudinal value as time changes. Consequently, the hazard distribution at one time does not automatically describe the distribution at another time. The calculation should also use consistent time units: evaluating a baseline hazard expressed per year at a time recorded in months without conversion can produce a numerically plausible but incorrectly scaled result.

Interpretation, alternatives, and limitations

The earlier material on posterior summaries for biomarker-survival association concerns uncertainty in association parameters. Subject-specific hazard uncertainty uses the same posterior information but transforms the joint draws into a different target: a hazard value for a particular subject at a particular time. Because the hazard is a nonlinear function of the parameters and random effects, calculating it at posterior mean parameter values need not equal its posterior mean. For example, in general \(E[\exp(X)\mid D]\ne\exp(E[X\mid D])\).

A posterior hazard interval can be wide or skewed, particularly when uncertainty in a random effect is amplified by the exponential link. It is conditional on the fitted model and available data; it does not account automatically for model misspecification or unrepresented sources of uncertainty. Posterior uncertainty also differs from Monte Carlo error: too few effectively independent draws can make the computed summaries imprecise even when the underlying posterior is well-defined.

Useful checks include confirming that every computed hazard is positive, checking that the specified time is in the model’s supported range, and verifying that the baseline-hazard units agree with the time scale. Inspect whether a few extreme draws dominate the mean, and compare summaries across well-mixed portions of the posterior simulation. If results are implausibly extreme, check exponent calculations, covariate units, and whether paired draws were retained. These checks diagnose computation or interpretation problems; they do not establish that the model itself is adequate.

A common mistake: treating hazard as risk

A hazard of \(0.040\) events per year is an instantaneous rate, not a 4% probability of an event in the next year. To obtain an event probability over an interval, one needs the hazard over that interval and the corresponding cumulative hazard. For a constant hazard \(h=0.040\) per year over one year, the event probability conditional on being event-free at the interval’s start is \(1-\exp(-0.040)\approx0.0392\), not exactly \(0.040\). If the hazard varies with time, the interval calculation must account for that variation. Therefore, do not label a posterior hazard interval as an interval for event probability.