Current Biomarker Value Versus Current Slope

Two Ways to Link a Biomarker Trajectory to Event Risk

A biomarker trajectory contains information about both a patient’s current status and the direction and rate of change in that status. A joint model can associate event risk with either feature. A current-value link makes the hazard depend on the patient’s latent biomarker level at time \(t\). A current-slope link instead makes it depend on the instantaneous rate at which that latent level is changing at time \(t\). These are distinct scientific specifications, not two names for the same association.

The distinction matters because a value describes a state, whereas a slope describes a rate of change. For example, two patients can have the same biomarker value at a given time while one is improving and the other is deteriorating. A current-value link assigns them the same biomarker-related hazard contribution at that instant; a slope link need not. Conversely, patients with different levels but the same slope receive the same contribution under a slope link.

Model Specifications and Assumptions

A typical longitudinal submodel writes the observed biomarker measurement \(Y_{ij}\), taken for patient \(i\) at time \(t_{ij}\), as

Here \(\mathbf{x}_i(t)\) and \(\mathbf{z}_i(t)\) are specified covariate vectors, \(\boldsymbol{\beta}\) is a vector of fixed effects, and \(\mathbf{b}_i\) is a patient-specific random-effects vector. The error \(\epsilon_{ij}\) represents measurement variation around the latent trajectory. This distinction is essential: the event link refers to \(m_i(t)\) or \(m_i'(t)\), not directly to a single noisy measurement \(Y_{ij}\) or to a finite difference between two observed measurements.

Let \(T_i\) be the event time, \(\mathbf{w}_i\) a vector of event-model covariates, and \(h_0(t)\) a baseline hazard. The two alternative specifications are

for the current-value link, and

for the current-slope link. These are generally fitted and assessed as alternative association structures. The models assume that, conditional on the included covariates and patient-specific random effects, the specified latent trajectory feature gives the modeled association with the event hazard. They also rely on an adequately specified longitudinal trajectory, a suitable baseline hazard, and appropriate treatment of censoring. In the usual joint-model formulation, longitudinal measurement errors are modeled separately from the event process conditional on the latent trajectory and random effects; censoring is assumed noninformative conditional on the modeled information.

The hazard is an instantaneous event rate among patients still event-free at time \(t\), conditional on the model’s information. It is not itself a probability. The hazard-ratio interpretations below hold at the same time \(t\), with the other covariates and model terms held fixed.

Interpreting the Association Parameters

Under a current-value link, compare two otherwise identical patients whose latent values at time \(t\) differ by \(\Delta m\). Their conditional hazard ratio is

Thus, \(\alpha\) is a log hazard ratio per one-unit increase in the latent biomarker value. If the biomarker is measured in mg/L, \(\alpha\) has units of inverse mg/L, so that \(\alpha m_i(t)\) is dimensionless. The ratio concerns the modeled association at a given time and does not establish that changing the biomarker would cause a change in risk.

For a current-slope link, compare two otherwise identical patients whose instantaneous slopes differ by \(\Delta s\). Their conditional hazard ratio is

Here \(\eta\) is a log hazard ratio per one-unit increase in slope. If time is measured in months and the biomarker in mg/L, slope is measured in mg/L per month, and \(\eta\) has the reciprocal units. The reference contrast must be stated: a positive change in slope means a faster increase, or a less rapid decline, in the biomarker. Whether that change corresponds to a higher hazard depends on the sign of \(\eta\).

A Derivation and a Direct Comparison

The hazard-ratio formulas follow directly from the log-linear form of each model. For the current-value specification, hold time, covariates, random effects other than the compared trajectory feature, and baseline hazard fixed. Replacing \(m_i(t)\) by \(m_i(t)+\Delta m\) changes the exponent by \(\alpha\Delta m\). The baseline hazard and all unchanged terms cancel in the ratio:

For the slope specification, the same cancellation leaves \(\exp(\eta\Delta s)\). This derivation also shows why the numerical values of \(\alpha\) and \(\eta\) cannot be compared directly: they multiply different quantities, measured on different scales. Meaningful comparisons require specifying and interpreting contrasts in the units of each link.

Assumptions, Diagnostics, and Failure Modes

The current-value link presumes that, on the log-hazard scale, the modeled association is linear in the latent value and has the specified coefficient structure. The current-slope link makes the analogous assumption for the instantaneous derivative. Both specifications ordinarily impose a constant association parameter over time unless the model explicitly allows otherwise. Neither specification automatically captures every aspect of a trajectory.

Useful checks focus on whether the trajectory and event components support the chosen link. Inspect fitted longitudinal trajectories against observed measurements, including their behavior near event times and at the edges of the observed time range. Check whether the functional form used for \(m_i(t)\) is plausible. A slope is a derivative of that fitted function, so misspecified curvature can distort estimated slopes even when fitted values appear reasonable. In particular, sparse or irregular measurements can leave the derivative weakly determined; a smooth-looking estimate should not be mistaken for precise slope information.

Assess event-model adequacy as well: examine convergence, uncertainty in the association estimate, sensitivity to reasonable trajectory specifications, and whether the modeled hazard behavior is compatible with the data. Compare predictive performance or model fit using criteria appropriate to the analysis, rather than selecting a link solely because its coefficient has a small p-value. A failure to converge, very large uncertainty, or results that change sharply under modest changes to the longitudinal trajectory are warnings that the data may not identify the proposed association reliably.

Other association structures, including cumulative and lagged links, encode different summaries of the latent trajectory. They are alternatives when the scientific question concerns accumulated or delayed exposure rather than the instantaneous level or rate. The current-value and current-slope specifications also have limitations: each reduces a potentially complex trajectory to one feature at a time, and either may miss a relevant association if the other feature matters. A slope link can be especially sensitive to how the longitudinal trajectory is modeled and to the timing and density of measurements.

Common Mistake

A common mistake is to compare the numerical estimates of \(\alpha\) and \(\eta\) and conclude that the larger one indicates a stronger association. This is not justified: \(\alpha\) is per biomarker unit, whereas \(\eta\) is per biomarker unit per time. The correction is to choose interpretable contrasts, compute \(\exp(\alpha\Delta m)\) or \(\exp(\eta\Delta s)\), and state the time, units, reference, and conditional nature of the comparison. Even then, a contrast-based interpretation does not by itself establish that the link fits better or that the association is causal.