Shared-Random-Effects Joint Models in Multi-Omics Integration

Why Share Random Effects?

A longitudinal omics assay is a noisy measurement of a subject’s evolving biological state. That state may also be related to the subject’s event risk. If a subject’s underlying omics trajectory is associated with event timing, the observed assay history is not simply an external covariate: it is a noisy and incomplete indication of a latent subject-specific process. Treating each observed value as error-free can therefore give a misleading picture of how the trajectory and event risk are connected.

A shared-random-effects joint model represents the trajectory and event process together by giving them common subject-specific random effects. The longitudinal measurements inform those latent effects, and the event hazard also depends on them. This provides a principled way to connect a measured trajectory to event risk without substituting a noisy assay value directly into the hazard.

Model Specification

For subject \(i\), let \(Y_{ij}\) be an omics measurement at time \(t_{ij}\), such as a normalized log-expression value. A linear mixed model represents the measurement as a latent trajectory plus assay error:

Here \(X_i(t)\) and \(Z_i(t)\) are specified covariate vectors, \(\beta\) is a vector of population-level trajectory coefficients, and \(b_i\) is the subject-specific random-effects vector. For example, \(b_i\) may contain a random intercept and random slope. A common starting assumption is \(b_i\sim N(0,D)\) and independent measurement errors \(\epsilon_{ij}\sim N(0,\sigma^2)\), independent of \(b_i\). The model also assumes the chosen mean trajectory and error structure adequately describe the repeated measurements.

Let \(T_i\) be the event time, \(C_i\) the censoring time, \(\tilde T_i=\min(T_i,C_i)\) the observed time, and \(\delta_i=I(T_i\le C_i)\) the event indicator. A shared-effects hazard model can be written as

In this expression, \(h_0(t)\) is the baseline hazard, \(W_i\) contains specified covariates, and \(\gamma\) is their coefficient vector. The vector \(\alpha\) connects the random effects to the conditional hazard. Given \(b_i\) and \(W_i\), the hazard is the instantaneous event rate at time \(t\), among subjects still event-free just before \(t\). It is not a probability of an event at exactly \(t\).

The two models share \(b_i\), not the assay error \(\epsilon_{ij}\). Thus the event model uses the latent subject-specific trajectory features represented by the random effects, rather than treating a single noisy observation as the subject’s true biological state. A current-value version can instead put \(m_i(t)\) in the hazard exponent. Since \(m_i(t)\) is a function of the same \(b_i\), this also links the outcomes through shared random effects, but it imposes a particular time-varying form of that link. The model here focuses on the more general \(\alpha^\top b_i\) formulation.

Why the Shared Structure Addresses Endogeneity

The term endogenous biomarker history refers here to an observed longitudinal predictor whose values are related to the event process through subject-specific latent characteristics. Subjects with different underlying trajectories may have different event risks. In addition, measurements are noisy, and later values are observed only while a subject remains under observation and, in many studies, event-free. Directly using the observed history as if it were a complete, error-free external covariate ignores these features.

A simple derivation shows how shared effects create dependence. Fix a measurement time \(t\) and condition on the covariates. Write \(Y_i(t)=X_i(t)^\top\beta+Z_i(t)^\top b_i+\epsilon_i(t)\), and let the random-effects part of the log hazard be \(L_i=\alpha^\top b_i\). Under the stated model, the fixed terms do not vary across subjects after conditioning, and the measurement error is independent of \(b_i\). Therefore,

The final equality follows because \(\operatorname{Cov}(b_i)=D\) and \(\operatorname{Cov}(\epsilon_i(t),b_i)=0\). Unless this covariance is zero, a measurement and the log-hazard contribution are associated through the shared random effects. This is a model-based dependence result; it does not by itself imply a causal effect of the biomarker.

Assumptions, Estimation Context, and Diagnostics

The model’s interpretation depends on assumptions about both outcome processes and their connection. In particular, the model assumes a suitable trajectory form, a suitable distribution for \(b_i\), and a suitable hazard structure. It also assumes that, conditional on the shared effects and included covariates, the longitudinal measurement errors do not carry additional information about event risk. Censoring must be noninformative under the analysis: conditional on modeled information, the censoring mechanism must not selectively conceal event times in a way that changes the event-process inference. The observation schedule is typically treated as fixed by design or otherwise ignorable for the modeled outcomes.

These assumptions should be considered before interpreting fitted results. Useful checks include comparing observed longitudinal patterns with fitted trajectories, examining residuals for systematic time trends or changing spread, and checking whether the random-effects distribution is a plausible representation of subject-level variation. For the event component, assess whether the chosen hazard structure is adequate; for example, a proportional-hazards specification requires the modeled covariate effects to operate as constant multipliers over time. Sensitivity analyses can compare plausible random-effects structures and trajectory forms.

Warning signs include residual patterns that vary with time, poor fit for subjects with extreme trajectories, implausible random-effects estimates, or event-risk patterns that the specified hazard cannot represent. A weakly supported random-effects structure can also make estimates unstable. Such findings motivate revisiting the model specification; they do not justify interpreting the shared-effects link as causal.

How This Model Differs From Alternatives

The shared-effects approach is most useful when the scientific model plausibly has subject-level trajectory features connected to risk, and repeated measurements are noisy. It is not automatically preferable in every setting. It relies on distributional and structural assumptions, may be sensitive to misspecification, and can be computationally demanding. If event risk depends on aspects of the trajectory not represented by the selected random effects, the model may not capture the relevant connection. Likewise, it does not resolve confounding or turn an association into a causal effect.