Shared Random Effects For Correlated Endpoints

Why Share a Subject-Level Effect?

A biomarker and a symptom can be related because subjects differ in a persistent underlying tendency that affects both. For example, some subjects may tend to have higher biomarker values and greater symptom propensity across visits. A joint model can represent that source of association with a shared subject-level random effect, while also allowing each endpoint to have its own subject-specific variation.

A shared effect does not mean that the two endpoints have the same measurement scale or the same response model. A biomarker can be modeled with a continuous-outcome model, while a binary symptom can be modeled with a logit link. Their shared effect links the models through subject-level variation. This develops the distinction between separate endpoint fits and a joint model introduced in the earlier material on separate endpoint fits.

A Joint Model for a Repeated Biomarker and Symptom

Let \(i\) index subjects and \(j\) index occasions. Let \(B_{ij}\) be a continuous biomarker and \(S_{ij}\) be an indicator that a symptom is present, with values 0 or 1. Let \(m_{Bij}\) and \(m_{Sij}\) denote the fixed-effect parts of the two models; these may depend on time, treatment, exposure, and other covariates.

Here \(U_i\) is the shared subject effect, \(A_{Bi}\) is the biomarker-specific subject effect, and \(A_{Si}\) is the symptom-specific subject effect. The logit function is \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The coefficients \(\lambda_B\) and \(\lambda_S\) are loadings: they determine how strongly and in what direction the shared effect enters each endpoint model.

For the basic model, assume \(U_i\sim N(0,\sigma_U^2)\), \(A_{Bi}\sim N(0,\tau_B^2)\), and \(A_{Si}\sim N(0,\tau_S^2)\). Assume these three effects are mutually independent. The biomarker residuals \(\varepsilon_{Bij}\) have mean zero and variance \(\sigma_B^2\), are independent across occasions, and are independent of all subject effects. Conditional on the subject effects, the biomarker measurements and symptom outcomes are independent across endpoints and occasions. The symptom outcome is Bernoulli with the probability specified by its logit model.

These assumptions are part of the model, not consequences of having repeated observations. In particular, the model says that the shared effect is the only source of cross-endpoint association after conditioning on the fixed effects and the subject effects. If a residual association remains, this model does not represent it.

Deriving the Induced Cross-Endpoint Covariance

The biomarker conditional mean is \(m_{Bij}+\lambda_B U_i+A_{Bi}\). The symptom linear predictor is \(m_{Sik}+\lambda_S U_i+A_{Si}\), where \(k\) can be any occasion for the symptom. Their fixed parts do not contribute to covariance. By independence, all cross-products involving an endpoint-specific effect and an effect from the other endpoint have zero covariance. The shared term is therefore the only remaining contribution.

Here \(\eta_{Sik}=m_{Sik}+\lambda_S U_i+A_{Si}\) is the symptom linear predictor. The same covariance applies between the biomarker linear predictor and the symptom linear predictor. Its sign is determined by the product \(\lambda_B\lambda_S\), and its magnitude also depends on the variance of the shared effect. If either loading is zero, or \(\sigma_U^2=0\), the shared effect induces no cross-endpoint covariance.

The covariance formula concerns linear predictors, not necessarily observed outcomes. For a binary symptom, the logit-scale predictor is not the observed 0-or-1 response. The probability transformation is nonlinear, so one must not report the covariance of linear predictors as the covariance of the biomarker and the binary symptom.

Repeated Measurements and Interpretation

The shared effect also contributes to within-endpoint dependence over time. For distinct biomarker occasions \(j\ne k\), the model gives

because the shared and biomarker-specific subject effects persist across occasions, while the occasion-specific residuals are independent. The symptom linear predictors at two occasions have covariance

The cross-endpoint covariance and the within-endpoint covariances are related but not interchangeable. Cross-endpoint association is due to the shared effect under this model; endpoint-specific effects add within-endpoint dependence but, by assumption, do not create cross-endpoint dependence.

For binary outcomes, the relationship with the observed response can be expressed through conditional means. Define \(g(u)=E_{A_{Si}}[\operatorname{expit}(m_{Sik}+\lambda_Su+A_{Si})]\), where \(\operatorname{expit}(x)=1/(1+e^{-x})\). Under the stated independence assumptions,

The endpoint-specific effects do not contribute to this covariance because they are independent across endpoints. If \(\lambda_S>0\), then \(g(u)\) is increasing in \(u\); if \(\lambda_S<0\), it is decreasing. For a nondegenerate \(U_i\), the covariance of \(U_i\) with an increasing function of \(U_i\) is nonnegative, and with a decreasing function is nonpositive. Consequently, the observed biomarker–symptom covariance has the direction implied by \(\lambda_B\lambda_S\), under this model. Its numerical magnitude is not generally \(\lambda_B\lambda_S\sigma_U^2\), because the symptom probability is obtained through the nonlinear expit transformation.

Assumptions, Diagnostics, and Alternatives

The covariance result is useful only if its assumptions are plausible. In practice, assess whether the shared-effect structure is compatible with the observed repeated data and whether the endpoint-specific terms are adequate. Relevant warning signs include systematic time patterns left in residuals, poor prediction of endpoint combinations, or substantial differences between observed and model-predicted within-subject association. For a binary endpoint, compare predictions and observations across clinically meaningful groups and visits, rather than treating a logit-scale covariance as an observed-response correlation.

The estimated loadings and variances should also be considered together. A weakly estimated loading, a shared-effect variance near zero, or unstable estimates under reasonable model changes can make the implied cross-endpoint association uncertain. Repeated measurements of both endpoints and variation in the fixed-effect predictors help distinguish persistent subject-level dependence from other patterns. These diagnostics do not prove that the shared effect is the true biological mechanism; the random effect is a statistical representation of residual subject heterogeneity.

The principal alternatives answer different modeling needs. Separate endpoint fits can describe each endpoint, but do not estimate a shared random effect or provide a model-based cross-endpoint covariance. A model with endpoint-specific random effects only allows persistent heterogeneity within each endpoint but, under independence, implies no cross-endpoint covariance. A shared effect adds a structured, low-dimensional source of dependence. Other dependence structures may be needed if association remains after conditioning on the shared effect; residual covariance across repeated endpoints is a distinct modeling approach. Each choice requires assumptions suited to the endpoint scales and sampling design.

A shared random effect is parsimonious and interpretable, but it imposes a particular dependence pattern: the endpoints are associated through a common subject-level quantity with endpoint-specific loadings. It may be too restrictive if the association changes with time or covariates, differs across parts of the response distribution, or persists in occasion-specific residual variation. The model is therefore a useful starting structure, not a universal description of endpoint dependence.