Sampling Baseline Hazard Parameters

Why Baseline Hazard Updates Need the Whole Follow-Up

In a shared random-effects joint model, the survival likelihood depends on the baseline hazard as well as on each subject’s current risk, which is determined by covariates and latent effects. Updating baseline hazard parameters therefore requires accounting for both the hazard at observed event times and the cumulative hazard over each subject’s follow-up. The latter contribution applies to censored subjects too.

Here the baseline hazard is \(h_0(t;\theta)\), with cumulative baseline hazard \(H_0(t;\theta)=\int_0^t h_0(u;\theta)\,du\). Let \(Y_i=\min(T_i,C_i)\) be subject \(i\)’s observed follow-up time and let \(\delta_i=1\) if the event is observed, and \(0\) if the observation is right-censored. Conditional on the current random effects and other model parameters, write the subject-specific hazard as \(h_i(t)=h_0(t;\theta)r_i(t)\), where \(r_i(t)>0\) is the subject-specific risk multiplier. For example, it may be the exponential of a linear predictor involving covariates and the subject’s latent longitudinal trajectory.

This is a conditional posterior, not a posterior that treats the latent effects or association parameters as known forever. In an MCMC algorithm they have current sampled values during this update and are updated in other steps. The longitudinal likelihood need not be repeated in the baseline-hazard update if it contains no baseline-hazard parameter: it is constant with respect to \(\theta\) conditional on the rest.

How Events and Censoring Enter

For an observed event, the likelihood includes the hazard at the event time as well as survival up to that time. A censored subject contributes survival up to the censoring time, but no event-hazard factor there. This distinction follows directly from the right-censored likelihood: the observed-data contribution is a density for an event and a survival probability for a censored time.

The censoring mechanism is assumed independent of the event process conditional on the modeled information, so its distribution can be omitted from this parameter update. If censoring is informative even after conditioning on that information, this likelihood is not adequate without modeling the censoring process. In all the calculations below, exposure means the integrated risk multiplier over the observed time, not simply the number of subjects at risk.

Conjugate Updates for Constant and Piecewise Hazards

The constant-hazard calculation is a special case of a useful conjugate pattern. If the baseline hazard is constant within a time interval, its likelihood contains a power of that interval’s hazard parameter, from events in the interval, and an exponential penalty involving accumulated exposure in the interval. A Gamma prior then yields a Gamma conditional posterior.

For a piecewise-constant baseline, partition follow-up time into intervals \((a_{k-1},a_k]\), and let the baseline hazard on interval \(k\) be \(\lambda_k\). Define \(d_k\) as the number of observed events in that interval and define its total risk-weighted exposure as \(A_k=\sum_i\int_{(a_{k-1},a_k]\cap[0,Y_i]}r_i(u)\,du\). The integral stops at each subject’s observed follow-up time. With independent priors \(\lambda_k\sim\operatorname{Gamma}(a_k^*,b_k^*)\), the conditional posterior for each interval is \(\operatorname{Gamma}(a_k^*+d_k,b_k^*+A_k)\). Here \(a_k^*,b_k^*\) are prior shape and rate parameters; they are distinct from the interval boundary notation \(a_k\).

This update remains conjugate when \(r_i(u)\) varies over time: its integral contributes a quantity that is fixed with respect to \(\lambda_k\) in this conditional update. The risk multiplier must be evaluated using the current values of the other parameters. If the model instead imposes a prior linking the \(\lambda_k\)’s, the conditional distribution for a single interval generally no longer has this independent Gamma form.

Parametric Hazards Beyond a Constant Rate

For a parametric baseline hazard, the same event-and-exposure likelihood applies, but the integral may couple several parameters and prevent a closed-form update. A useful strategy is to identify any parameters that retain conjugate conditional distributions, then sample the remaining parameters from their conditional posterior kernels using an appropriate transition method. The posterior kernel must include the prior and every likelihood term that depends on the parameter being updated.

For example, consider a Gompertz baseline \(h_0(t;a,b)=a\exp(bt)\), where \(a>0\) is a scale parameter and \(b\) is a shape parameter. For fixed \(b\), define \(q_b(t)=\int_0^t\exp(bu)\,du\), which equals \((\exp(bt)-1)/b\) when \(b\ne0\), and \(t\) when \(b=0\). With risk multipliers equal to 1, the event likelihood contributes \(a^D\exp(b\sum_{\delta_i=1}Y_i)\), while survival contributes \(\exp\{-a\sum_i q_b(Y_i)\}\). Thus a Gamma prior on \(a\) gives a Gamma conditional update. If \(b\) is unknown, its conditional posterior generally has no Gamma form and must retain its own prior and the likelihood terms that depend on \(b\).

Derivation, Sampling, and Checks

The conjugate update can be derived directly rather than memorized. For a constant hazard \(\lambda\) on one interval, an observed event contributes \(\lambda\) from its hazard term and every subject contributes \(\exp(-\lambda \times \text{risk-weighted time})\) from survival. Multiplying over subjects gives the likelihood kernel \(\lambda^d\exp(-A\lambda)\). Multiplying by a Gamma prior kernel yields

The expression on the right is the kernel of a Gamma distribution with shape \(a+d\) and rate \(b+A\). This also explains why a rate parameter, not a scale parameter, is added to exposure. If software uses a Gamma scale parameter, it must instead receive the reciprocal of the rate.

A direct Gamma draw is an exact conditional update when the stated conjugacy conditions hold. For other parametric parameters, first form the log posterior kernel from the prior plus the event log-hazards minus the cumulative hazards. A proposed update must respect the parameter space, and its acceptance calculation must use the full conditional target, including any transformation Jacobian if sampling on a transformed scale. Do not treat an unavailable closed-form draw as evidence that a likelihood term can be dropped.

Useful checks include verifying that total exposure includes all observed follow-up, confirming that each event is assigned to exactly one interval, and checking that the calculated integrated hazard is nonnegative. For a constant hazard, increasing a subject’s follow-up while holding the event count fixed should increase the posterior rate and lower the conditional posterior mean. If a supposedly conjugate update produces a rate smaller than its prior rate despite positive exposure, inspect the exposure calculation and Gamma parameterization.

Comparison and Limitations

These are alternative ways to represent the baseline hazard, not interchangeable assumptions. A piecewise-constant model can adapt to different hazard levels across intervals but may be sensitive to sparse events or arbitrary cut points. A smooth parametric form is more compact, but imposes its chosen time pattern. Conditional updates also depend on the current latent effects and association parameters; uncertainty about those quantities is represented only through the full sequence of joint-model updates, not by a single baseline update.

Common mistake: “A censored subject had no event, so that subject contributes nothing to the hazard update.” This is wrong because censoring still establishes survival through the censoring time. The correct contribution is the survival factor, which adds integrated exposure to the likelihood; only the event-hazard factor is absent. Another frequent error is to count exposure only up to the beginning of an interval rather than integrating over the portion of the interval the subject actually remained under observation.