One Biomarker, Two Clinical Stories
Two patients can have nearly the same biomarker concentration today while their trajectories point in opposite directions: one is improving and the other is worsening. A hazard model linked to the current level asks whether being at a higher value is associated with a different instantaneous event rate. A model linked to the current slope instead asks whether the rate of change is associated with that rate. Those are distinct questions, so they can imply contrasting clinical interpretations even when the observed current levels are similar.
The comparison is about the assumed association structure in a joint model, not about proving that a biomarker causes an event. The longitudinal submodel represents each patient’s underlying trajectory, while the event submodel relates the event hazard to a feature of that latent trajectory. For this tutorial, the features are the current value and the instantaneous slope.
Two Conditional Hazard Models
Write \(h_i(t)\) for patient \(i\)’s event hazard at time \(t\), conditional on being event-free immediately before \(t\), the patient’s latent trajectory, and included covariates. Let \(h_0(t)\) be a baseline hazard, \(\mathbf{w}_i\) a vector of covariates, and \(\boldsymbol{\gamma}\) its coefficient vector. A value-linked model is
A slope-linked alternative is
Here \(\alpha_v\) is the value association parameter and \(\alpha_s\) is the slope association parameter. In each model the hazard is positive, and, conditional on the latent trajectory and covariates, a one-unit increase in the linked quantity multiplies the hazard by the exponential of its coefficient. The units differ: \(\alpha_v\) is in inverse biomarker units, while \(\alpha_s\) is in inverse biomarker-units-per-time. Thus the numerical sizes of \(\alpha_v\) and \(\alpha_s\) are not directly comparable.
These alternatives assume that the specified latent feature is the relevant biomarker association for the event hazard, given the included covariates. They also impose the displayed log-linear form: equal changes in the linked quantity have equal effects on the log hazard throughout the modeled range. The slope model additionally requires a trajectory smooth enough for its derivative to be meaningful. These assumptions are not guaranteed by having repeated measurements; they are choices to examine and justify.
Why Similar Levels Can Give Different Comparisons
Suppose two patients have the same covariates and are compared at the same time \(t\). Their baseline hazard and covariate terms are then identical. Under a value link, their hazard ratio depends on the difference in their current values; under a slope link, it depends on their slope difference. This follows directly by taking the ratio of the two hazards: the shared baseline and covariate factors cancel, leaving the exponential of the coefficient multiplied by the relevant difference.
Consequently, if the levels are equal, the value-linked model gives a hazard ratio of 1 for that pair, regardless of their slopes. The slope-linked model can give a ratio different from 1. Conversely, if the slopes are equal, the slope-linked model gives a ratio of 1 even if the current levels differ. This is a contrast between model implications, not evidence that one model must be clinically correct.
Worked Example: Similar Levels, Diverging Slopes
Consider a hypothetical comparison at one year. Patient A’s latent biomarker level is \(5.0\) mg/L and is rising at \(0.4\) mg/L per year. Patient B’s level is \(4.9\) mg/L and is falling at \(0.2\) mg/L per year. Assume their modeled covariates are the same. For illustration, suppose a fitted value-link coefficient is \(\alpha_v=0.30\) per mg/L, and a separate fitted slope-link coefficient is \(\alpha_s=0.80\) per (mg/L per year).
Under the value link, the level difference is \(5.0-4.9=0.1\) mg/L. Therefore,
Under this model, the predicted conditional hazard for A is about 1.03 times B’s, a small contrast associated with their small current-level difference.
Under the slope link, the slope difference is \(0.4-(-0.2)=0.6\) mg/L per year. Thus,
The slope-linked model associates A’s more rapidly rising trajectory with a conditional hazard about 1.62 times B’s at that time. This does not establish that the rise causes the event; it is the model’s conditional association. Nor does 1.62 mean a 62-percentage-point increase in event probability. It is a ratio of instantaneous hazards.
The assumptions for this arithmetic are that the stated coefficients apply at the comparison time, the covariates and baseline conditions are shared, and the latent values and slopes are known for purposes of the illustration. In an analysis, those trajectory quantities are estimated with uncertainty, which must be accounted for in joint-model inference.
Worked Example: Equal Slopes, Different Value-Based Story
Take two other hypothetical patients with the same covariates and the same time-specific slope, \(0.1\) mg/L per year. Their current levels are \(4.0\) mg/L and \(6.0\) mg/L. Use \(\alpha_v=0.30\) per mg/L and \(\alpha_s=0.80\) per (mg/L per year), as illustrative coefficients.
For the value-linked model, comparing the patient at \(6.0\) mg/L with the patient at \(4.0\) mg/L gives
The model therefore associates the higher current level with a conditional hazard about 1.82 times as large. The units cancel because \(0.30\) is per mg/L and the difference is in mg/L.
For the slope-linked model, the slope difference is \(0.1-0.1=0\) mg/L per year, so
That model makes no hazard distinction between these patients on the basis of their current slopes. It does not say their hazards must be identical in every respect; it says the specified slope component contributes no contrast when the slopes match. The arithmetic assumes the same time, covariates, and model coefficient for both patients.
What the Contrasting Interpretations Do—and Do Not—Say
The value-linked interpretation emphasizes where the patient is now; the slope-linked interpretation emphasizes the direction and speed of movement. A rising marker can have a high slope without yet having a high level, while a persistently high marker can have a small slope. Consequently, the models can rank patients differently, especially when current levels are close but trajectories differ.
A coefficient’s sign also has a specific, conditional meaning. If \(\alpha_v>0\), higher current values are associated with higher hazard under the value-linked model, holding the stated covariates and time conditions fixed. If \(\alpha_s>0\), more positive slopes are associated with higher hazard under the slope-linked model. A negative slope is not automatically “protective”: the hazard comparison depends on the difference between slopes and the sign of \(\alpha_s\). These are associations conditional on model specification, not causal effects.
A link-sensitivity analysis compares the substantive conclusions under plausible alternative specifications. It should use a common comparison that is meaningful in context, such as the two patient trajectories above, rather than comparing coefficient magnitudes with different units. If the clinical conclusion changes, that change is important information about dependence on the assumed association structure. It is not, by itself, a formal test that selects the correct link.
| Question | Value-linked model | Slope-linked model |
|---|---|---|
| Which trajectory feature enters? | Current latent biomarker level | Instantaneous latent rate of change |
| Units of the association coefficient | Inverse biomarker units | Inverse biomarker units per time unit |
| Equal feature for two patients implies | No hazard contrast from the value term | No hazard contrast from the slope term |
| Clinical emphasis | Current burden or status | Current direction and pace of change |
| Central vulnerability | May understate trajectory dynamics | Derivative can be uncertain or unstable |
Diagnostics, Failure Modes, and Limitations
A slope link is particularly demanding of the longitudinal trajectory model. A derivative is not directly observed at a visit: it is inferred from the fitted trajectory. With sparse or noisy measurements, small changes in the trajectory fit can produce large changes in the estimated slope, and the resulting hazard association may be imprecise or sensitive to model choices. A smooth mathematical trajectory can still be a poor description of a biomarker that changes abruptly.
Useful checks include examining whether the fitted longitudinal trajectories describe the observed repeated measurements, whether inferred slopes are plausible over the observed time range, and whether the event-model fit and calibration are adequate under each candidate link. Uncertainty in the latent trajectory must be carried through the joint analysis rather than treating estimated levels or slopes as error-free observed covariates. Instability, wide uncertainty, poor fit, or conclusions that change substantially under reasonable trajectory specifications are warnings against a confident clinical interpretation.
The value-only and slope-only models are alternatives with different assumptions. The value model can miss risk information carried by a changing trajectory. The slope model may be difficult to estimate and can assign similar levels different hazards because their directions of change differ. Neither model is universally preferable. Their interpretation is limited to the modeled population, time scale, covariates, trajectory assumptions, and observed range; extrapolating beyond those conditions is not justified. Choosing between them requires scientific reasoning and model assessment, not just selecting the one with a larger association coefficient.
Worked Example: Direction Matters Under a Slope Link
Suppose two hypothetical patients have the same current level, \(5.2\) mg/L, and the same covariates. One patient’s biomarker is rising at \(0.3\) mg/L per month; the other’s is falling at \(0.3\) mg/L per month. Let the slope-link coefficient be \(\alpha_s=0.50\) per (mg/L per month). Comparing the rising patient with the falling patient gives a slope difference of \(0.3-(-0.3)=0.6\) mg/L per month, so
The slope-linked model associates the rising trajectory with a conditional hazard 1.35 times that of the falling trajectory. The same-level value-linked comparison gives
for any finite \(\alpha_v\). Thus the models tell different stories for this pair: one distinguishes direction of change, while the other does not distinguish them by current value. This comparison assumes a common evaluation time, common covariates, and slopes expressed in the same time unit. Changing the time unit would change the numerical slope and coefficient, so the unit must be stated and used consistently.
Common Mistake
A common error is to say, “The slope model’s coefficient is larger, so the slope is more strongly associated with risk.” That comparison is invalid when coefficients have different units, and it ignores the scale of clinically relevant differences. Another error is to translate a hazard ratio directly into a probability difference. A hazard is an instantaneous event rate conditional on remaining event-free up to that time; it is not an event probability.
Check Your Understanding
Answer each question, then compare with the concise solutions below.
- Under a value-linked model, two patients have the same current biomarker level and the same covariates. What is their hazard ratio contribution from the value term?
- With \(\alpha_s=0.40\) per (mg/L per year), compare slopes of \(0.5\) and \(0.0\) mg/L per year. Calculate the slope-linked hazard ratio.
- Why is it not meaningful to declare a value coefficient of \(0.30\) stronger than a slope coefficient of \(0.80\) solely because of their numerical sizes?
- A slope-linked model gives a hazard ratio of 1.5 for one slope contrast. Does this mean the event probability is 50 percentage points higher? Explain.
Solutions. (1) The level difference is zero, so the ratio is \(\exp\{\alpha_v(0)\}=1\). (2) The slope difference is \(0.5-0.0=0.5\) mg/L per year, giving \(\exp\{0.40(0.5)\}=\exp(0.20)=1.221\); the first slope has an associated conditional hazard about 1.22 times the second’s. (3) The coefficients have different units and act on different quantities. Compare hazard ratios for stated, clinically interpretable differences instead. (4) No. A hazard ratio compares instantaneous hazards, not event probabilities or percentage-point differences in probability.