Identifying Marker Effects Separately From Covariance

Why Separate Covariance From Hazard Association?

When two longitudinal markers are correlated, that correlation does not by itself tell us whether either marker is associated with the event time. Correlation describes how measurements vary together; a hazard-association parameter describes how a marker’s latent value relates to the event hazard, conditional on the model’s other terms. A joint model contains both kinds of quantities, and it is important to know which data components inform each one.

We use a simplified two-marker random-intercept model. It makes the distinction visible without requiring a complicated trajectory specification. The same likelihood logic applies when the longitudinal mean trajectories include time trends or when the hazard uses marker slopes or other functions of the latent trajectories.

The entries of \(D\) describe between-person variation in the random intercepts, including how the two markers’ person-specific deviations co-vary. The entries of \(R\) describe residual variation and covariance between the two measurements taken at a visit, after accounting for the modeled means and random intercepts. The parameters \(\alpha_1,\alpha_2\) describe conditional associations between the latent markers and the hazard. For example, \(\alpha_1\) is the change in log hazard per one-unit increase in marker 1 when marker 2 and the other model terms are held fixed.

Where the Parameters Enter the Likelihood

Let \(T_i\) be the observed follow-up time and \(\delta_i\) indicate whether the event was observed (\(\delta_i=1\)) or the observation was right-censored (\(\delta_i=0\)). Conditional on \(b_i\), the longitudinal measurements and event outcome contribute separate factors. Under independent censoring, the person’s likelihood contribution, after integrating over the unobserved random effects, is

Here \(\phi_2(\,\cdot\,;a,V)\) denotes a bivariate normal density with mean \(a\) and covariance matrix \(V\). The integral is necessary because the person-specific random effects are not observed. The event factor includes both an event-time density contribution when \(\delta_i=1\) and a survival contribution through \(T_i\) for every person.

Before integration, the roles of the parameters are especially clear. The conditional longitudinal measurement density contains \(R\), but not \(D\); the random-effects density contains \(D\); and the conditional event factor contains \(\alpha_1,\alpha_2\) through the hazard. After integration, these components are linked: the longitudinal measurements help determine plausible random effects, and those random effects enter the conditional hazard. Thus, although \(R\) has no direct term in the conditional event factor, it can affect estimates of the association parameters by affecting how the latent marker values are inferred from the measurements. The marginal longitudinal likelihood, which integrates the measurement density over \(b_i\), can inform both \(R\) and \(D\).

Why Repeated Measurements Help Separate Covariances

The random-intercept model gives a direct covariance argument. For different visits \(j\ne k\), the shared random intercept contributes covariance across visits, while the independent residuals do not. At a single visit, both sources contribute.

For example, the covariance between the two markers at the same visit contains the between-person covariance \(D_{12}\) plus the within-visit residual covariance \(R_{12}\). The cross-visit covariance between marker 1 at one visit and marker 2 at another is \(D_{12}\) under the stated assumptions. Comparing same-visit and cross-visit patterns can therefore separate the two sources. This distinction relies on the model assumptions: correlated residuals across visits, time-varying random effects, or misspecified mean trajectories would change the covariance relationships.

A related separation concerns \(D\) and the hazard associations. The longitudinal data inform the distribution of latent marker values through the random-effects model; the event data inform how those latent values relate to event risk. For the association parameters to be distinguished, there must be usable variation in the latent marker values and event follow-up that provides information about the hazard. A flexible baseline hazard, limited follow-up, or little variation in a marker can make the data less informative about an association. Repeated measurements help estimate latent values, but they do not guarantee that every hazard association is precisely determined.

Worked Hazard Contributions

For a fixed latent marker profile, the conditional hazard association is explicit. If the marker values are measured in their original units, each \(\alpha_k\) has units of inverse units of marker \(k\). The hazard ratio for a change in marker values \((\Delta m_1,\Delta m_2)\), holding other terms fixed at the same time, is \(\exp(\alpha_1\Delta m_1+\alpha_2\Delta m_2)\). This is a conditional comparison; it is not generally the hazard ratio after averaging over the random-effects distribution.

Conditions, Diagnostics, and Alternatives

The separation described above depends on assumptions that should be checked against the study design and data. The model assumes that the two-marker residual vector is adequately represented by a bivariate normal distribution with covariance \(R\); that the random-intercept vector follows the specified bivariate normal distribution with covariance \(D\); and that residuals are independent across visits conditional on the random effects and modeled means. It also assumes that the hazard form and its marker association are appropriate, and that censoring is independent of the event process conditional on the modeled information. Visit timing and missing measurements must not introduce unmodeled dependence that invalidates the likelihood.

Useful diagnostics include comparing observed within-person and cross-visit marker covariance patterns with those implied by the fitted model, checking residual distributions and patterns over time, and examining whether the fitted hazard and survival predictions are compatible with observed event and censoring patterns. A mismatch between same-visit and cross-visit covariance patterns can indicate that the proposed \(R\) and \(D\) structure is inadequate. Implausible estimates, large uncertainty, or strong sensitivity to reasonable model changes are warnings that the data or specification may not support a clean separation.

A separate longitudinal analysis can describe marker covariance without using event information, but it cannot estimate marker-to-hazard associations. A survival analysis using observed marker values as if they were error-free predictors does not make the same allowance for latent trajectories and measurement error. The joint likelihood uses both measurement and event information while accounting for latent random effects. Its advantage does not remove its limitations: conclusions depend on the longitudinal and hazard specifications, distributional assumptions, and available follow-up. A fitted association is conditional on the specified model and is not, by itself, a causal effect.

Common Mistake

Wrong reasoning: “The markers have a positive covariance, so marker 1 must have a positive hazard association.” Covariance between markers only describes their joint variation. It gives neither the sign nor the size of either marker’s conditional association with the hazard.

Correction: Identify the source of each quantity in the likelihood. The longitudinal measurement density informs \(R\), the random-effects distribution informs \(D\), and the conditional event factor contains the \(\alpha_k\). The integrated likelihood links the information, but marker covariance alone does not determine a hazard association. Also, even if no events are observed, censored follow-up still contributes survival terms involving the integrated hazard; there is no event-occurrence or event-time evidence, but the follow-up is not devoid of information about \(\alpha_k\).