Shared Random Effects as the Marker-to-Hazard Link

Why Link Multiple Marker Trajectories Through Random Effects?

When several biomarkers are measured over time, each trajectory can vary from its population mean because of subject-specific characteristics. Those characteristics may also be related to event risk. A shared-random-effects model expresses this idea by letting the same latent subject-specific effects help generate the marker trajectories and enter the event hazard. The link is therefore between an event and selected features of the subject’s latent trajectories, not automatically between the event and each observed marker value at every time.

This distinction matters for interpretation and prediction. A marker observation can inform the event-time prediction only insofar as it updates the posterior distribution of the random-effects combination that appears in the hazard. Updating some other random effect is not enough.

Model and Assumptions

Let \(Y_{ik}(t_{ikj})\) denote measurement \(j\) of marker \(k\) for subject \(i\), taken at time \(t_{ikj}\). Let \(b_{ik}\) be the vector of subject-specific random effects for marker \(k\), and stack them as \(b_i=(b_{i1}^{T},\ldots,b_{iK}^{T})^{T}\). A usual multimarker longitudinal model is

Here \(X_{ik}\) and \(Z_{ik}\) are known design vectors, \(\beta_k\) is a marker-specific fixed-effects vector, and \(\epsilon_{ik}\) is measurement error. The stacked random effects are commonly modeled as \(b_i\sim N(0,D)\), where \(D\) allows random effects for different markers to be correlated. The longitudinal model may also include an appropriate within-visit residual covariance across markers.

Let \(T_i^*\) be the event time, \(C_i\) the censoring time, \(T_i=\min(T_i^*,C_i)\), and \(\delta_i=I(T_i^*\le C_i)\), where \(\delta_i=1\) indicates an observed event. Let \(w_i\) be a vector of survival covariates. A shared-random-effects hazard is

The hazard is conditional on \(b_i\) and \(w_i\). Holding these quantities fixed, a one-unit increase in component \(b_{ir}\) changes the hazard by a factor of \(\exp(\alpha_r)\), with the other random-effect components held fixed. The unit is the unit of that random effect, which may not be the unit of an observed marker measurement. Thus \(\alpha_r\) is not automatically a hazard ratio per one-unit increase in a measured marker.

The model relies on several substantive assumptions. The random-effects distribution and the marker mean and covariance models must describe the longitudinal data adequately. Conditional on the random effects and modeled covariates, the event process is specified by the stated hazard; marker errors and the event process are taken to be conditionally independent in the joint model. Censoring must be noninformative conditional on the modeled quantities, or its mechanism must be modeled. Finally, the chosen hazard form assumes a proportional-hazards relationship conditional on \(b_i\), even though the marginal hazard after averaging over random effects need not be proportional.

From Conditional Hazard to Joint Contribution

Write \(H_0(t)=\int_0^t h_0(u)\,du\). Conditional on \(b_i\) and \(w_i\), the cumulative hazard and survival function are

For an observed event, the conditional time-to-event likelihood contribution is the hazard at \(T_i\) multiplied by survival through \(T_i\). For right censoring, it is survival through \(T_i\). Both cases are represented by

If \(Y_i\) denotes all of subject \(i\)’s marker observations and \(f_Y(Y_i\mid b_i)\) is their conditional likelihood under the multimarker model, the subject-level contribution after integrating out the unobserved random effects is

Here \(\phi(b_i;0,D)\) is the multivariate normal density. The integration is essential: the random effects are latent, not observed covariates to be substituted with known values. The marker data and event information both contribute information about \(b_i\), while the population parameters are estimated across subjects.

How Shared Effects Create Dependence

The dependence can be seen directly from the conditional survival function. Given \(b_i\), the longitudinal measurements and event time are modeled through their respective conditional distributions. Marginally, however, both depend on the distribution of \(b_i\). Information from a marker observation can therefore change the event-time prediction when it changes the distribution of \(\alpha^{T}b_i\).

A short Gaussian calculation makes the qualification precise. Suppose \(b_{i1}\) and \(b_{i2}\) are independent \(N(0,1)\), a marker observation is \(Y=b_{i1}+\epsilon\), and \(\epsilon\sim N(0,1)\) is independent of the random effects. Suppose the hazard depends on \(\eta=b_{i1}+b_{i2}\). Normal conditioning gives

Because \(b_{i2}\) is independent of \(Y\), its conditional mean and variance remain 0 and 1. Hence

The marker observation changes the distribution of the hazard-relevant combination \(\eta\), so it changes the event-time prediction. In contrast, if the hazard depended only on \(b_{i2}\), the same marker observation would leave that hazard-relevant effect’s distribution unchanged. A nonzero association vector by itself is not enough: the marker data must update the particular combination selected by that vector.

A Conditional Likelihood Calculation

Interpretation, Diagnostics, and Alternatives

The shared-effects link is useful when latent subject characteristics plausibly influence both marker trajectories and event risk. It provides a coherent way to combine multiple noisy, unequally observed trajectories with event or censoring information, while accounting for uncertainty about the latent effects. It also clarifies what the association parameters mean: they describe how the conditional hazard varies with the specified random-effects combination.

The model does not establish that a marker causes an event. A random effect can capture unmeasured subject heterogeneity, and its association with the hazard is conditional on the model and covariates. Nor does this formulation automatically imply that every marker is informative for survival. A marker whose random effects are unrelated to \(\alpha^{T}b_i\) may contribute little or no information to event prediction.

Useful checks include assessing whether the fitted longitudinal model reproduces marker means, variability, and cross-marker covariance; checking whether the hazard-relevant random-effects combination is estimable with the observed follow-up; and examining sensitivity to plausible random-effects and baseline-hazard specifications. Simulation-based parameter recovery and stability checks are particularly relevant when the association parameters are weakly identified. Poor convergence, large uncertainty, or strong sensitivity to the covariance structure can signal that the data do not distinguish the survival association from random-effects variation well.

Compared with fitting separate longitudinal and survival models, the shared-effects approach explicitly represents dependence through latent subject effects and propagates their uncertainty into the joint likelihood. Separate models do not generally make this connection. A different class of joint models links the hazard to a time-varying marker value rather than directly to random effects. That is a different scientific specification: the shared-effects model described here associates event risk with latent trajectory features selected by \(\alpha\), not necessarily with the marker’s current value.

A common mistake is to argue that any nonzero \(\alpha\) guarantees that marker observations change event-time predictions. That conclusion ignores which random effects the measurements update. For example, if \(b_{i1}\) and \(b_{i2}\) are independent, marker data inform only \(b_{i1}\), and \(\alpha=(0,1)^T\), then the hazard depends on \(b_{i2}\) alone. The marker data do not update the distribution of \(\alpha^{T}b_i=b_{i2}\), and therefore do not change the event-time prediction through this link. The correct question is whether the marker data update the hazard-relevant combination.