Joint Level And Slope Associations Across Markers

Why Model Levels and Slopes Together?

Two biomarkers can convey different aspects of a patient’s evolving health. A marker’s current level describes where its trajectory is at a particular time; its slope describes how quickly it is changing. A survival model that includes both can ask whether event hazard is associated with a high current value, a rapid increase or decrease, or both. With two correlated biomarkers, the model also makes each association conditional on the other marker’s level and slope.

This conditional interpretation matters. If the biomarkers tend to move together, a coefficient does not describe the overall association of one marker considered in isolation. It describes the association for a change in that marker while the other modeled marker features and covariates are held fixed. Such a comparison may be scientifically useful, but the observed data may contain relatively little information about it if those features rarely vary independently.

Model Definition and Assumptions

Let \(Y_{ik}\) denote the observed value of marker \(k\) for subject \(i\), recorded at time \(t_{ikj}\), where \(k=1,2\) indexes the two biomarkers and \(j\) indexes visits. The marker-specific latent trajectory is \(m_{ik}(t)\). It represents the subject’s underlying marker value at time \(t\), rather than a single noisy measurement. Its instantaneous rate of change is the derivative \(\dot m_{ik}(t)=d m_{ik}(t)/dt\).

For example, the longitudinal submodel may express each trajectory using fixed effects and subject-specific random effects, with the random effects across the two markers allowed to be correlated. The survival association uses the resulting latent trajectories, not the raw visit measurements. This avoids treating measurement error in an observed value as if it were a genuine instantaneous change in the subject’s underlying marker.

The interpretation relies on several assumptions. The longitudinal models must adequately represent the latent trajectories and their uncertainty; the slopes used in the hazard must be defined by differentiable trajectories. Conditional on those trajectories and \(w_i\), the specified hazard must adequately describe event risk, including the chosen proportional-hazards form: each association coefficient is constant over time unless the model explicitly allows time-varying effects. The usual joint-model formulation also assumes that censoring is non-informative conditional on the modeled information. Inference additionally depends on the assumed distribution and covariance structure for the random effects being adequate.

The model is an association model. Its conditional hazard ratios do not, by themselves, show that changing a biomarker would cause a change in event risk. They also do not describe a marginal association that averages over the distribution of random effects.

Interpreting the Conditional Effects

At a fixed time \(t\), compare two trajectory configurations for the same covariate values. If the first marker’s current value differs by \(\Delta m_1\), its slope by \(\Delta\dot m_1\), and the second marker’s value and slope differ by \(\Delta m_2\) and \(\Delta\dot m_2\), then the conditional hazard ratio is

This result follows by dividing the hazard for one trajectory configuration by the hazard for the other. At the same time \(t\), the baseline hazard \(h_0(t)\) is common to both and cancels. The covariate contribution cancels when \(w_i\) is the same, and subtracting the two linear predictors leaves the four coefficient-times-change terms shown above. Exponentiating this difference gives the hazard ratio.

For a one-unit increase in marker 1’s current value, with its slope and both features of marker 2 held fixed, the conditional hazard ratio is \(\exp(\alpha_1)\). For a one-unit increase in marker 1’s slope, holding the other three features fixed, it is \(\exp(\eta_1)\). The units differ: \(\alpha_1\) is per unit of marker 1, while \(\eta_1\) is per unit of marker 1 per unit of time. A positive coefficient means a higher conditional hazard for the specified increase; it does not automatically mean that the marker is changing in a positive direction.

The comparison must be made at the same time when using this simple ratio. If comparing hazards at different times, the baseline hazards need not cancel. Also, a one-unit increase in a slope means a difference in rate of change, not a one-unit increase in the marker’s current value.

Reading Level and Slope Effects Together

A level coefficient and its corresponding slope coefficient answer distinct conditional questions. They may have different signs without contradiction. For instance, a high current marker value might be associated with higher hazard while a rising trajectory, at the same current value, is associated with lower hazard. Such a pattern is a statement about the model’s conditional association, not a claim that all possible trajectories with those features are biologically plausible or equally well represented in the data.

For a slope interpretation, retain the time unit. Suppose marker 1 has slope coefficient \(\eta_1=0.35\) per (mg/L)/month. A slope difference of \(0.4\) (mg/L)/month, with the two current values and marker 2’s level and slope held fixed, gives the following comparison.

In applications, the two markers’ levels and slopes may be correlated. The coefficient for one feature is consequently conditional on the other three features even when the scientific question began with a single marker. It is important to report the conditioning clearly and to avoid interpreting coefficients as if each came from a separate model.

Model Checks, Alternatives, and Limits

Before interpreting estimates, check that the joint model’s assumptions and fitted components are credible. Plot observed marker measurements alongside fitted trajectories for representative subjects, and examine whether the fitted curves capture the overall patterns and rates of change over the time range relevant to events. Check whether slope estimates are determined by the longitudinal data rather than being driven mainly by the assumed trajectory shape. Review convergence and identifiability evidence, and examine uncertainty in all four association estimates. Very wide intervals or unstable estimates can make a conditional interpretation uninformative.

Units and plausible trajectory ranges are essential diagnostics for interpretation. A slope’s numerical value changes if time is recorded in days rather than months, and its coefficient changes accordingly. A one-unit change may also be much larger than the differences commonly observed. A calculated hazard ratio can be mathematically correct yet describe an implausible comparison if the proposed combination of levels and slopes is not supported by the data.

Common alternatives answer narrower questions. A current-value-only model includes the two \(m_{ik}(t)\) terms but omits slopes. A slope-only model includes the derivatives but omits current levels. A shared-random-effects association instead links hazard to selected latent random effects rather than directly to each marker’s current value and rate of change. These approaches are not interchangeable: they encode different assumptions about which aspects of the longitudinal process carry survival information. The joint level-and-slope specification is useful when both features are scientifically motivated, but it requires estimating and interpreting four conditional associations.

A key pitfall is to read a positive slope coefficient as evidence that a high marker level raises risk. The slope coefficient concerns a faster rate of change while the current level and other modeled features are held fixed. The level coefficient addresses the current value. A second pitfall is to treat a marker’s coefficient as its overall or causal effect despite the conditioning on the other marker features and the model assumptions. State the feature changed, the features held fixed, the units, and the conditional nature of the comparison.