Why the Association Term Matters
A joint model links a subject’s longitudinal biomarker trajectory to their event hazard. That link is not uniquely determined by saying that the biomarker is “associated” with risk: the hazard might depend on the marker’s current level, its rate of change, or exposure accumulated over time. Each choice uses a different quantity from the longitudinal model and gives its association coefficient a different meaning.
In software, these choices are usually specified as association terms alongside the longitudinal and event submodels. The important implementation task is to make sure the term requested in the code corresponds to the scientific quantity intended. The terms below refer to the model-implied, latent trajectory, not to a raw observed biomarker value or a finite difference calculated from two visits.
Event-Submodel Terms and Their Meaning
A joint-model hazard can include one or more of these associations. For example, with baseline or other event-submodel covariates \(w_i\), a general specification is
Here \(h_0(t)\) is the baseline hazard, \(\gamma\) contains coefficients for \(w_i\), and \(\alpha_v\), \(\alpha_s\), and \(\alpha_A\) are the respective association parameters. A model need not include all three terms. The subscript on each coefficient is a reminder that the terms have different definitions and units.
- Current value: \(m_i(t)\) is the model-implied marker level at event time \(t\). If the marker is measured in mg/L, \(\alpha_v\) is a log-hazard change per mg/L, conditional on the other model terms.
- Instantaneous slope: \(m_i'(t)\) is the instantaneous rate of marker change. With time in years and marker in mg/L, its units are mg/L per year; \(\alpha_s\) is a log-hazard change per such unit.
- Cumulative exposure: \(\int_0^t m_i(s)\,ds\) is the area under the latent trajectory from the specified origin to \(t\). Its units are marker units multiplied by time units. The coefficient \(\alpha_A\) is defined per unit of that area.
These coefficients are conditional log-hazard associations, not causal effects by themselves. Their numerical values also depend on measurement units and, for cumulative exposure, on the integration convention. A change of time unit or marker scale changes the numerical coefficient needed to describe the same association.
Translating the Terms Into Software
First fit or specify the longitudinal trajectory using the selected time variable, fixed effects, and subject-specific random effects. Then choose the association function for that trajectory in the event submodel. Software packages use different argument names and formula syntax, so the following is a package-neutral R-style specification rather than a claim about one package’s exact API:
In an actual package, replace these placeholders with its documented association-term constructors. Confirm whether the software expects the longitudinal outcome name, a submodel label, or a time variable inside each constructor. Also check whether cumulative association is implemented as an area term and what lower integration limit it uses. Do not assume that similarly named functions in different packages have identical defaults.
The longitudinal submodel must define a trajectory that can be evaluated at event times, and its derivative or integral must be available for any requested term. The usual joint-model interpretation also relies on a correctly specified longitudinal model, an event model with an appropriate baseline hazard and covariates, and a valid joint likelihood for the repeated measurements and event process. The marker association is a model-based link between the latent trajectory and hazard; it does not convert observed biomarker measurements into error-free covariates.
Worked Examples of the Quantities Used
Worked Example: Current Value With a Linear Trajectory
Suppose a hypothetical biomarker is measured in mg/L and time is measured in years since enrollment. The fitted subject-specific trajectory is \(m_i(t)=8+0.6t\). The intended event model uses a current-value association with \(\alpha_v=0.20\), and the subject’s other event covariates are held fixed. Find the marker contribution to the hazard at year 3 relative to a subject with the same other covariates and marker value 8 mg/L.
At year 3, \(m_i(3)=8+0.6(3)=9.8\) mg/L. Relative to 8 mg/L, the difference is 1.8 mg/L. Thus the log-hazard contrast is \(0.20(1.8)=0.36\), and the hazard ratio is \(\exp(0.36)=1.433\). Under this fitted model, the subject with a 9.8 mg/L latent marker value has an estimated hazard 1.433 times as large as the comparison subject at that time, conditional on the other terms. This is an association interpretation, not a causal conclusion.
A software check is to confirm that the current-value constructor uses the fitted marker trajectory at \(t=3\), rather than the last observed measurement. If the model instead used the observed value from an earlier visit, it would not be implementing this latent current-value term.
Worked Example: Instantaneous Slope With a Quadratic Trajectory
For a hypothetical marker measured in mmol/L, suppose the fitted trajectory is \(m_i(t)=5+1.2t-0.10t^2\), with time in years. The event model contains a slope association with \(\alpha_s=0.30\) per mmol/L per year. Calculate the slope and the hazard ratio for a slope difference of 0.5 mmol/L per year at year 2.
Differentiate the trajectory: \(m_i'(t)=1.2-0.20t\) mmol/L per year. At year 2, \(m_i'(2)=1.2-0.20(2)=0.8\) mmol/L per year. For a comparison subject whose slope at that time is 0.3 mmol/L per year, the slope difference is \(0.8-0.3=0.5\) mmol/L per year. The log-hazard contrast is \(0.30(0.5)=0.15\), so the hazard ratio is \(\exp(0.15)=1.162\).
This calculation uses the derivative of the fitted trajectory. It does not use the difference between the last two recorded measurements divided by the time between visits. Such a finite difference is affected by measurement error and visit timing, and is not the model-implied instantaneous slope.
Worked Example: Cumulative Exposure and the Time Origin
Suppose a hypothetical marker has trajectory \(m_i(t)=4+0.5t\) mg/L, with time in years since enrollment. The cumulative association is defined from enrollment, \(A_i(t)=\int_0^t m_i(s)\,ds\), and \(\alpha_A=0.04\) per mg/L-year. Calculate cumulative exposure at year 4 and the hazard ratio relative to a subject with cumulative exposure 2 mg/L-year lower, holding other terms fixed.
Integrate the trajectory:
The specified exposure difference is 2 mg/L-year, so the log-hazard contrast is \(0.04(2)=0.08\). The hazard ratio is \(\exp(0.08)=1.083\). Thus the model associates a 2 mg/L-year greater accumulated area with an estimated hazard 1.083 times as large, conditional on the remaining terms.
If the software integrates from a different lower limit, or uses a rescaled time variable, the cumulative quantity will differ. The integration interval and time units must therefore be checked against the scientific definition before interpreting \(\alpha_A\).
A Short Derivation: Why a Linear Slope Term Is Constant
A simple derivation helps identify a software mistake that can otherwise go unnoticed. Suppose the longitudinal trajectory is linear in time, \(m_i(t)=\beta_0+b_{0i}+(\beta_1+b_{1i})t\), where \(b_{0i}\) and \(b_{1i}\) are subject-specific random effects. Differentiating with respect to \(t\) gives
The derivative has no \(t\) term: each subject’s fitted linear trajectory has a constant slope. Therefore a slope association in this model distinguishes subjects by their fitted slopes, but does not vary within a subject over time. If the scientific intent is to let the instantaneous slope change over follow-up, the longitudinal trajectory must permit curvature, for example by including a quadratic time term or suitable spline terms. A software option requesting a slope cannot create curvature that the longitudinal model does not contain.
Implementation Checks, Alternatives, and Limitations
Before fitting, write down the intended mathematical quantity and compare it with the package’s syntax. The following checks are especially useful:
- Verify that the longitudinal model response and time variable are the intended ones, and that the event and longitudinal records use compatible time origins.
- Check whether the association uses the latent trajectory, including its random effects, rather than the raw observed biomarker.
- For a slope term, confirm the derivative is with respect to the intended time scale. A slope per month and a slope per year differ by a factor of 12.
- For a cumulative term, check the lower limit, upper limit, and units. Determine whether the definition includes time before delayed entry or begins at another specified origin.
- Check whether the package centers or scales the marker, time, or association covariate. Such transformations affect coefficient interpretation.
- Confirm which association terms are actually included in the event model; a longitudinal formula alone does not establish the event association.
Current value is often the most direct choice when risk is hypothesized to depend on present marker level. Slope is useful when the rate of increase or decline is scientifically relevant, while cumulative exposure represents accumulated burden. These are competing scientific specifications, not interchangeable software options. Including several terms simultaneously is possible in principle, but it can make interpretation more conditional and may create strong dependence among the terms. The choice should be motivated by the question and supported by the trajectory model.
Common implementation failures include requesting a slope from a model whose trajectory is linear when a changing slope was intended, using a cumulative term with an unnoticed time origin, or interpreting the association coefficient as a change per observed unit when the software internally rescales the marker. A further pitfall is to conclude that a larger current value and a larger cumulative value are the same predictor: two subjects can have the same current level but different prior areas under their trajectories.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A marker is measured in mg/L and time in years. What are the units of its instantaneous slope and its cumulative area under the trajectory?
- For \(m_i(t)=3+2t\), calculate \(m_i'(t)\) and explain whether it changes over time.
- Why is the last observed biomarker value generally not the same quantity as a latent current-value association at event time?
- Two subjects have the same marker value at year 3 but different trajectories before year 3. Can their cumulative associations differ? Explain.
Solutions. (1) The slope is mg/L per year; the cumulative area is mg/L-year. (2) Differentiation gives \(m_i'(t)=2\) mg/L per year, constant in time for this linear trajectory. (3) The observed value may be from an earlier visit and includes measurement error, whereas the latent current value is the fitted trajectory evaluated at the event time. (4) Yes. The cumulative term integrates each trajectory over the interval, so different earlier levels can produce different areas even when the trajectories agree at year 3.