Weibull Baseline Hazards in Bayesian Joint Models

Why Model the Baseline Hazard Parametrically?

In a joint model, the event process is linked to a subject’s longitudinal trajectory, often through shared random effects or a time-varying measure of the biomarker. The baseline hazard supplies the event-time pattern before that association term is applied. A Weibull baseline is useful when a smooth, two-parameter hazard pattern is a reasonable working model and a compact posterior representation is desirable.

This tutorial uses \(T_i\) for subject \(i\)'s observed event or censoring time, \(\delta_i\) for the event indicator (\(\delta_i=1\) for an observed event and \(0\) for right censoring), and \(b_i\) for that subject's random effects. Write the hazard multiplier associated with the longitudinal and covariate information as \(\exp\{\eta_i(t)\}\). Conditional on \(b_i\) and model parameters, the subject-specific hazard is \(h_0(t)\exp\{\eta_i(t)\}\). Thus \(h_0(t)\) is the baseline hazard for \(\eta_i(t)=0\), not generally the hazard of a typical subject.

The shape controls how the baseline hazard changes over time: it decreases when \(\kappa<1\), is constant when \(\kappa=1\), and increases when \(\kappa>1\). These statements concern the baseline hazard, holding the other model terms fixed. The Weibull form is not flexible enough to represent an arbitrary sequence of increases and decreases.

Right-Censored Likelihood Contribution

Assume a continuous event time, conditionally independent subjects, and right censoring that is noninformative conditional on the information included in the model. The event-time likelihood contribution is the event density when \(\delta_i=1\), and the survival probability through \(T_i\) when \(\delta_i=0\). Since a density can be written as hazard times survival, both cases can be written in one expression: $$ L_i^{(T)}=h_i(T_i)^{\delta_i}S_i(T_i) =h_i(T_i)^{\delta_i}\exp\{-H_i(T_i)\}. $$ Here \(h_i(t)=h_0(t)\exp\{\eta_i(t)\}\) and $$ H_i(T_i)=\int_0^{T_i}h_0(u)\exp\{\eta_i(u)\}\,du. $$ Taking logarithms and substituting the Weibull baseline gives $$ \ell_i^{(T)} =\delta_i\left[\log\kappa-\kappa\log\lambda +(\kappa-1)\log T_i+\eta_i(T_i)\right] -\frac{1}{\lambda^\kappa}\int_0^{T_i}\kappa u^{\kappa-1}\exp\{\eta_i(u)\}\,du. $$ The event term is present only for an observed event. The integrated hazard term is present for every subject, including those censored, because censoring still tells us the subject remained event-free up to \(T_i\).

If \(\eta_i(t)\) is constant over time, the integrated hazard has a closed form: $$ H_i(T_i)=\exp\{\eta_i\}\left(\frac{T_i}{\lambda}\right)^\kappa. $$ When the association term depends on the evolving biomarker trajectory, the integral will generally depend on that trajectory and may require numerical integration. The conditional likelihood remains the same; only its evaluation changes.

Priors and the Joint Posterior

A common choice is to put normal priors on log scale and log shape: $$ \log\lambda\sim N(m_\lambda,s_\lambda^2),\qquad \log\kappa\sim N(m_\kappa,s_\kappa^2). $$ These priors ensure positive parameters. Their medians on the original scale are \(\exp(m_\lambda)\) and \(\exp(m_\kappa)\). Their spreads should be chosen in light of the follow-up time units and plausible hazard shapes; a prior that seems diffuse in one time unit can imply very different beliefs after rescaling time. Prior predictive checks can reveal whether the induced event times and hazard trajectories are plausible.

For clarity, let \(\theta\) denote all parameters other than \(\lambda,\kappa\), including longitudinal-model parameters, association parameters, and random effects. Conditional on the random effects and model parameters, and assuming the usual conditional factorization of longitudinal and event processes, the posterior kernel is $$ p(\theta,\lambda,\kappa,\mathbf b\mid \mathbf y,\mathbf T,\boldsymbol\delta) \propto \left[\prod_i p(y_i\mid b_i,\theta)\right] \left[\prod_i h_i(T_i)^{\delta_i}\exp\{-H_i(T_i)\}\right] p(\mathbf b\mid\theta)\,p(\theta)\,p(\lambda)\,p(\kappa). $$ The Weibull parameters enter both the event-hazard terms for observed events and every subject's integrated hazard. They do not replace or remove the longitudinal likelihood or the random-effects distribution.

With normal priors on \(\log\lambda\) and \(\log\kappa\), the prior density with respect to \((\lambda,\kappa)\) includes the change-of-variables factor \(1/(\lambda\kappa)\). Equivalently, if computation is performed in coordinates \((\log\lambda,\log\kappa)\), the prior is directly normal in those coordinates. A sampler's target density must match the coordinates and density convention being used; omitting or duplicating the Jacobian changes the posterior.

Interpreting Posterior Draws

A posterior draw \((\lambda^{(m)},\kappa^{(m)})\) defines a complete baseline survival curve and hazard curve. Transform each draw before summarizing quantities such as survival at a clinically relevant time. For example, compute \(S_0(t)^{(m)}=\exp[-(t/\lambda^{(m)})^{\kappa^{(m)}}]\) for every draw, then summarize those values. A posterior interval for \(S_0(t)\) is generally not obtained by inserting separate marginal interval endpoints for \(\lambda\) and \(\kappa\), because their posterior dependence matters.

Assumptions, Diagnostics, and Alternatives

The parametric baseline assumes a Weibull-shaped hazard over the modeled follow-up interval. The shared-effects structure also requires that the chosen random effects and association term adequately represent how the longitudinal process relates to the event process. Conditional independence assumptions, the censoring mechanism, and the event-time scale must be defensible. In particular, if censoring is informative after conditioning on modeled variables, the likelihood above does not account for that additional dependence.

Check whether the model implies plausible survival and hazard curves over the observed and scientifically relevant time ranges, not just near the center of follow-up. Examine prior predictive curves and posterior draws; extreme values of \(\lambda\) or \(\kappa\) can produce nearly immediate events, very long survival, or sharply changing hazards. Also check that numerical integration of \(H_i(T_i)\) is accurate when \(\eta_i(t)\) varies over time. Integration error can distort the posterior target and its gradients.

A frequent mistake is to treat \(\lambda\) as a median survival time or to interpret the baseline hazard as the hazard for every subject. The correction is that \(S_0(\lambda)=e^{-1}\), not \(0.5\), and the subject-specific hazard includes the multiplier \(\exp\{\eta_i(t)\}\). Another mistake is to omit the integrated hazard for censored subjects; censoring removes the event-density term, not survival information through the censoring time.

A Weibull baseline is more restrictive than a piecewise exponential baseline, which allows a stepwise hazard across intervals, but it offers a smooth curve with only scale and shape parameters. A spline baseline can represent more varied hazard shapes, at the cost of additional parameters and smoothing choices. The appropriate alternative depends on event information, substantive expectations, and the need for flexibility; neither flexibility nor simplicity alone guarantees a well-fitting model.