Shared Random Effects for Recurrence and Terminal Risk

Why Share a Random Effect?

A subject who experiences many recurrent events may also have a different risk of a terminal event, such as death. One way to represent this dependence is to introduce a latent subject characteristic that influences both processes. For example, an unobserved degree of frailty might raise recurrence intensity and terminal risk; another latent characteristic could raise recurrence intensity but be associated with lower terminal risk. The model captures these patterns as statistical dependence, not as evidence that recurrences cause the terminal event or vice versa.

The key idea is to condition on a subject-level random effect. Given that effect and measured covariates, the recurrence and terminal-event processes are assumed independent. When the random effect is unobserved and averaged over, however, the processes are generally dependent. This construction is useful when a common source of subject heterogeneity is scientifically plausible and the data contain information about both event processes.

Here \(\lambda_i(t\mid b_i)\) is the recurrent-event intensity, and \(h_i(t\mid b_i)\) is the terminal-event hazard. The at-risk indicator \(Y_i(t)\) is one while subject \(i\) is under observation and has not yet had the terminal event, and zero otherwise. The functions \(\lambda_{R0}(t)\) and \(h_{D0}(t)\) are baseline intensity and hazard functions; \(x_i\) and \(z_i\) are measured covariate vectors, with coefficients \(\beta\) and \(\gamma\). The parameters \(\alpha\) and \(\delta\) are the loadings of the shared random effect. For scale and interpretation, take \(b_i\sim N(0,\sigma_b^2)\), independently across subjects, and assume it is independent of the measured covariates.

Assumptions and Interpretation

The model requires more than writing \(b_i\) into two equations. A common assumption is that, conditional on \(b_i\) and measured covariates, the recurrent-event process and terminal-event time are independent. This is a modeling assumption about what remains associated after conditioning; it is not guaranteed by the data. The chosen distribution for \(b_i\), often normal on the log-rate scale, is also an assumption. The recurrent process must stop contributing events at the terminal event, and subjects’ random effects are usually taken to be independent.

The log-linear form gives a direct interpretation of each loading. Holding covariates fixed, a one-unit increase in \(b_i\) multiplies recurrent intensity by \(\exp(\alpha)\) and terminal hazard by \(\exp(\delta)\). Since the scale of \(b_i\) depends on \(\sigma_b\), a more comparable change is one standard deviation, \(\sigma_b\): the respective multipliers are \(\exp(\alpha\sigma_b)\) and \(\exp(\delta\sigma_b)\). A positive \(\alpha\) means larger \(b_i\) corresponds to greater recurrence intensity; a negative \(\delta\) means it corresponds to lower terminal hazard.

The signs of \(\alpha\) and \(\delta\) determine the direction of the association induced by this shared effect. If their signs agree, subjects with larger \(b_i\) tend to have both greater recurrence intensity and greater terminal hazard, or both lower. If their signs differ, higher recurrence intensity tends to accompany lower terminal hazard across values of \(b_i\). This describes latent heterogeneity, not necessarily a causal relationship or every possible form of dependence.

How Dependence Appears After Averaging

Although the processes are conditionally independent given \(b_i\), they need not be independent in the population. The same latent value enters both conditional rates. A simple calculation shows the mechanism. At a fixed time with \(Y_i(t)=1\), write the covariate-adjusted baseline rates as \(c_R=\lambda_{R0}(t)\exp(x_i^\top\beta)\) and \(c_D=h_{D0}(t)\exp(z_i^\top\gamma)\). Then the conditional rates are \(c_R\exp(\alpha b_i)\) and \(c_D\exp(\delta b_i)\).

For a normal random effect, the moment-generating function gives \(E[\exp(qb_i)]=\exp(q^2\sigma_b^2/2)\). Consequently, the covariance across subjects of these two conditional rates is

To see the sign, note that \((\alpha+\delta)^2-(\alpha^2+\delta^2)=2\alpha\delta\). Since the exponential function is increasing, the covariance is positive when \(\alpha\delta>0\), negative when \(\alpha\delta<0\), and zero when either loading is zero. This is a covariance of conditional rates across the random-effect distribution; it is not by itself the covariance of a recurrent-event count and a terminal time. The latter involves event-process variation and the fact that follow-up ends at the terminal event, but the calculation isolates how the shared effect creates dependence.

Subject Likelihood and Latent-Effect Updating

For each subject, the random effect is unobserved. The likelihood contribution therefore averages the conditional contribution over its distribution. Under a conditionally specified model, the general form is \(L_i=\int L_i(b)f(b)\,db\), where \(L_i(b)\) is the subject’s contribution conditional on \(b\) and \(f\) is the random-effect density. This integration is essential: substituting \(b=0\) for everyone would ignore the heterogeneity and the dependence it induces.

For a transparent calculation, assume that conditional on \(b_i\), recurrent events follow a Poisson process over the subject’s observed exposure, and the terminal time follows the stated hazard. If the cumulative baseline recurrent intensity through time \(t\) is \(\Lambda_{R0}(t)\), and the cumulative baseline terminal hazard is \(H_{D0}(t)\), a subject with \(n_i\) recurrences and terminal-event indicator \(d_i\) has conditional likelihood, up to factors not involving \(b_i\),

The factor involving \(d_i\) supplies the terminal-event density contribution when the terminal event is observed; when it is censored, \(d_i=0\) and the survival contribution remains. This compact expression assumes the simple Poisson recurrent-event likelihood just stated. More general recurrent-event specifications require the likelihood appropriate to their conditional intensity and observation process.

Diagnostics, Alternatives, and Limitations

A shared random effect is a parsimonious dependence structure, not a guarantee of adequate fit. Warning signs include systematic mismatch between observed and model-predicted recurrence counts across follow-up, poor calibration of terminal survival, extreme or unstable estimates of \(\alpha\), \(\delta\), or \(\sigma_b^2\), and strong sensitivity to the assumed random-effect distribution. Sparse recurrences, few terminal events, or limited overlap in follow-up can make the two loadings difficult to separate: recurrence data inform \(\alpha\), terminal data inform \(\delta\), and the estimated distribution of \(b_i\) connects them. Inspect uncertainty and sensitivity rather than treating a fitted latent effect as directly observed.

The main alternatives answer different modeling needs. A model with no shared random effect can be reasonable if residual dependence is not needed, but it may fail to represent common latent heterogeneity. A model with separate, correlated random effects allows more flexible links between the two processes, at the cost of additional parameters and stronger data demands. A shared scalar effect is simpler and gives an interpretable dependence direction, but forces both processes to depend on the same latent dimension. The choice should reflect the scientific question, available information, and fit checks.

The shared-effect model has important limitations. Conditional independence may be implausible if dependence remains after conditioning on \(b_i\) and covariates. A normal distribution on the log scale may not represent the actual heterogeneity, and a single effect cannot capture every pattern of dependence. Moreover, the sign of a loading does not establish a causal mechanism: \(b_i\) is latent, and associations may reflect omitted subject characteristics or model assumptions. Report the model’s conditional and marginal interpretations separately.