The Association Term in a Shared Random-Effects Hazard

Why Separate the Association Term?

A joint model can use the same subject-specific random effects to describe repeated marker measurements and the event hazard. This creates an association between the longitudinal and event processes even after accounting for measured baseline covariates. To interpret the model correctly, it is essential to distinguish the term that carries this shared-random-effects association from the terms for baseline covariates.

The association term is not automatically a direct effect of a measured marker on event risk. It describes how latent subject-specific features of the longitudinal trajectory, represented by random effects, enter the hazard conditional on those effects. The distinction matters because the random effects describe departures from the population-average trajectory, whereas baseline covariates describe recorded characteristics such as age or treatment group.

A Simple Shared-Random-Effects Model

For subject \(i\), let \(Y_{ij}\) be a marker measured at time \(t_{ij}\). A simple linear mixed model is

Here, \(\beta_0+\beta_1t\) is the population-average trajectory, while \(b_{0i}+b_{1i}t\) is subject \(i\)'s deviation from that trajectory. The random intercept \(b_{0i}\) shifts the subject's trajectory, and the random slope \(b_{1i}\) changes its rate of change. Write \(\mathbf b_i=(b_{0i},b_{1i})^\mathsf T\), with \(\mathbf b_i\sim N(\mathbf 0,\mathbf D)\). The measurement errors are commonly modeled as independent \(N(0,\sigma^2)\) variables, independent of \(\mathbf b_i\).

A corresponding conditional event hazard can be written

In this expression, \(h_0(t)\) is the baseline hazard, \(\mathbf w_i\) is a vector of baseline covariates, and \(\boldsymbol\gamma\) contains their hazard coefficients. The term \(\alpha_0 b_{0i}+\alpha_1 b_{1i}\) is the association term for this model. It links the event hazard to the random effects shared with the longitudinal model. The parameters \(\alpha_0\) and \(\alpha_1\) quantify the hazard association with the random intercept and random slope, respectively.

The model assumes a specified longitudinal distribution and random-effects distribution, together with a correctly specified conditional hazard. It also assumes that, conditional on the shared random effects and included covariates, the longitudinal measurement errors and event process have the conditional independence structure used in the joint likelihood. Censoring must be handled under an appropriate conditional independent-censoring assumption. These are modeling assumptions, not consequences of sharing random effects.

The random effects have units inherited from the longitudinal model. For example, \(b_{0i}\) may be in marker units and \(b_{1i}\) in marker units per year. Since the exponent must be dimensionless, \(\alpha_0\) and \(\alpha_1\) have the reciprocal units of their respective random effects. The baseline covariate coefficients likewise depend on how the covariates are measured or coded.

What the Association Term Does—and Does Not—Mean

The association term operates on the conditional log hazard. If two subjects have the same baseline covariates and random effects \(\mathbf b_i\) and \(\mathbf b_k\), their conditional hazard ratio is

This follows by substituting each subject's hazard and canceling the common baseline hazard and common baseline-covariate factor. The ratio is conditional on the random effects. It does not generally equal a hazard ratio obtained by integrating over the random-effects distribution, because averaging conditional hazards can change the hazard comparison.

By contrast, when the random effects are held fixed and baseline covariates differ, their contribution to the conditional hazard ratio is \(\exp\{\boldsymbol\gamma^\mathsf T(\mathbf w_i-\mathbf w_k)\}\). If both the baseline covariates and random effects differ, both contributions appear in the exponent. Keeping these pieces separate clarifies which model component produces a comparison.

Baseline Covariate Effects Are a Separate Component

A baseline covariate can also be associated with the event hazard, but it enters through its own coefficient in \(\boldsymbol\gamma\). It is not part of the shared-random-effects association term merely because it predicts the event or is also used in the longitudinal model. For example, a baseline covariate may explain differences in average marker trajectories through the fixed effects while having a distinct coefficient in the hazard.

Derivation and Interpretation

The conditional hazard ratio formula follows directly from the model, but writing out the cancellation is useful for avoiding sign errors. Let \(\mathbf w_i=\mathbf w_k=\mathbf w\). Then

The common baseline hazard and baseline-covariate factor cancel. If the baseline covariates differ, the same algebra leaves the additional term \(\boldsymbol\gamma^\mathsf T(\mathbf w_i-\mathbf w_k)\) in the exponent. This derivation concerns conditional hazards at the same time; it does not establish a corresponding proportional-hazards relationship after averaging over random effects.

The signs of the association coefficients require care. If \(\alpha_0>0\), a larger random intercept raises the conditional hazard with other quantities fixed. A negative \(\alpha_1\), for example, means a larger random slope lowers the conditional hazard, conditional on the other model terms. The interpretation is in terms of the random-effect coordinate and its scale, not necessarily a simple change in one observed marker measurement.

Assumptions, Diagnostics, and Limitations

The association term is meaningful only within the model that defines the random effects. Important checks include whether the longitudinal mean and covariance structure describe the repeated measurements adequately, whether the assumed random-effects distribution is plausible, and whether the event model and baseline hazard are adequately specified. In practice, convergence warnings, unstable association estimates, large standard errors, or strong sensitivity to initial values can signal weak identification or model misspecification.

The data must contain enough information to distinguish the random-effect contribution from baseline covariate effects. Strong correlation between covariates and random effects, few repeated measurements, or limited event information can make that separation difficult. Estimates should be examined on their stated scales, with uncertainty, and under reasonable alternative specifications. Residual and fit diagnostics for the longitudinal and event components can reveal problems, although no single diagnostic verifies the full joint model.

A shared-random-effects model also imposes a particular structure: dependence between the marker and event processes is represented through the selected shared random effects. If that structure is wrong, the estimated association can be misleading. The model does not prove that marker changes cause events, and its conditional coefficients should not be described as causal effects without additional design and causal assumptions. Nor should the conditional hazard ratio be presented as a population-average hazard ratio.

Compared with a survival model containing baseline covariates only, the shared-random-effects model can represent event-risk differences tied to latent subject-specific longitudinal features, rather than leaving those differences unexplained. Compared with a model that substitutes observed marker measurements directly into the hazard, the random-effects formulation explicitly represents the latent trajectory and measurement error. These approaches answer different model-based questions; choosing the shared-random-effects formulation requires support for its assumptions and sufficient information in the data.

Common Mistake

Wrong reasoning: “Age is associated with both the marker and the event, so its hazard coefficient belongs to the association term.” This confuses the source of a covariate's association with the model component where it appears. Correction: a baseline covariate's hazard contribution is \(\boldsymbol\gamma^\mathsf T\mathbf w_i\); the shared-random-effects association is the specified function of \(\mathbf b_i\), such as \(\alpha_0b_{0i}+\alpha_1b_{1i}\). A covariate may appear in more than one model component, but each occurrence has a distinct parameter and interpretation.