Worked Comparison of Current-Value and Cumulative Links

Why Compare These Two Links?

A joint model can connect a patient’s longitudinal biomarker trajectory to their event hazard in more than one way. A current-value link relates the hazard at time \(t\) to the patient’s latent biomarker value at that time. A cumulative-exposure link relates it to the biomarker history accumulated up to \(t\). These models ask different scientific questions: does risk track the current state, or does it reflect exposure built up over time?

Comparing the links requires more than placing their fitted coefficients side by side. The coefficients have different units, and the two models summarize a trajectory differently. A useful comparison calculates hazard contrasts for specified patients or exposure histories, then asks what features of those histories account for any difference. The calculations below use an entirely hypothetical biomarker cohort.

Define the Links and Their Scales

Let \(m_i(t)\) denote the latent mean biomarker trajectory for subject \(i\) at time \(t\), as specified by the longitudinal submodel. The latent trajectory, rather than a single noisy measurement, is used in the hazard link. Let \(w_i\) denote baseline covariates, \(\gamma\) their coefficient vector, and \(h_0(t)\) the baseline hazard. For simplicity, the equations show no additional time-varying covariates.

Definition: A current-value link specifies \(h_i(t)=h_0(t)\exp\{\gamma^\mathsf{T}w_i+\alpha_{\mathrm{CV}}m_i(t)\}\). A cumulative-exposure link specifies \(h_i(t)=h_0(t)\exp\{\gamma^\mathsf{T}w_i+\alpha_{\mathrm{CE}}A_i(t)\}\), where \(A_i(t)=\int_0^t m_i(s)\,ds\). Thus \(A_i(t)\) is the area under the latent biomarker trajectory from time 0 to \(t\).

These forms presume that the stated link is an adequate description of how the longitudinal process relates to the hazard, conditional on the included covariates and model components. They also presume that the longitudinal model provides a sufficiently appropriate trajectory over the time interval used in the link. In practice, the trajectory is estimated from repeated measurements, so sparse visits or trajectory misspecification can affect the link comparison.

If the biomarker is measured in units \(U\) and time in years, \(m_i(t)\) has units \(U\), while \(A_i(t)\) has units \(U\)-years. Accordingly, \(\alpha_{\mathrm{CV}}\) is on a per-\(U\) log-hazard scale and \(\alpha_{\mathrm{CE}}\) is on a per-\(U\)-year log-hazard scale. Neither coefficient is, by itself, a scale-free measure of association.

A Contrast Formula for Each Link

Consider two subjects whose baseline covariates are the same, and compare their hazards at the same time \(t\). Under the current-value link, the common baseline hazard and covariate contribution cancel in the ratio. Under the cumulative link, they cancel in the same way, but the trajectory histories enter through their accumulated areas.

\frac{h_1(t)}{h_0^{\ast}(t)} =\exp\{\alpha_{\mathrm{CV}}[m_1(t)-m_0(t)]\}, \qquad \frac{h_1(t)}{h_0^{\ast}(t)} =\exp\{\alpha_{\mathrm{CE}}[A_1(t)-A_0(t)]\}.

Here \(h_0^{\ast}(t)\) denotes the hazard for the comparison subject, not the baseline hazard \(h_0(t)\). The first ratio is for the current-value model; the second is for the cumulative-exposure model. Writing the comparison subject’s hazard as \(h_{\mathrm{ref}}(t)\) avoids any possible ambiguity:

\log\frac{h_1(t)}{h_{\mathrm{ref}}(t)} =\alpha_{\mathrm{CV}}\{m_1(t)-m_{\mathrm{ref}}(t)\}

\log\frac{h_1(t)}{h_{\mathrm{ref}}(t)} =\alpha_{\mathrm{CE}}\int_0^t \{m_1(s)-m_{\mathrm{ref}}(s)\}\,ds.

The second expression follows by substituting the definition of \(A_i(t)\): \(A_1(t)-A_{\mathrm{ref}}(t)=\int_0^t\{m_1(s)-m_{\mathrm{ref}}(s)\}\,ds\). This is the central reason the links can differ. The current-value contrast depends only on the separation between trajectories at the comparison time. The cumulative contrast depends on the separation over the entire preceding interval.

For numerical illustrations, suppose both links are fitted to the same hypothetical cohort, using the same longitudinal model, covariates, and time origin. Suppose their fitted association coefficients are \(\hat{\alpha}_{\mathrm{CV}}=0.12\) per biomarker unit and \(\hat{\alpha}_{\mathrm{CE}}=0.040\) per biomarker-unit-year. These are stipulated hypothetical fit results, not estimates from a real study. Each example calculates a model-based hazard contrast conditional on the specified trajectories and equal covariates.

Worked Example: Same Trajectory Separation Throughout Follow-Up

Suppose the comparison subject’s fitted trajectory is \(m_{\mathrm{ref}}(s)=20+2s\), while the other subject’s is \(m_1(s)=22+2s\), with the biomarker measured in units \(U\) and time in years. Compare their hazards at year 4.

At year 4, the current-value difference is \((22+2(4))-(20+2(4))=2\ U\). The fitted current-value model therefore gives

\widehat{\mathrm{HR}}_{\mathrm{CV}}(4) =\exp\{0.12(2)\} =\exp(0.24) =1.271.

For the cumulative link, the trajectory difference is 2 units at every time from 0 to 4 years. Its accumulated difference is \(\int_0^4 2\,ds=8\ U\text{-years}\). Thus

\widehat{\mathrm{HR}}_{\mathrm{CE}}(4) =\exp\{0.040(8)\} =\exp(0.32) =1.377.

Under these fitted models and this contrast, the current-value link estimates a hazard 1.271 times the comparison hazard, while the cumulative link estimates a hazard 1.377 times the comparison hazard at year 4. Neither ratio is a probability of an event. They are instantaneous hazard ratios, conditional on the model and the stated trajectories.

Why Equal Current Values Need Not Mean Equal Risks

A cumulative link can distinguish subjects who have the same biomarker value now but arrived there along different paths. The current-value link, by contrast, gives the same linked hazard contribution to those subjects at that time, provided their covariates match. This is a structural difference between the models, not a discrepancy in arithmetic.

Worked Example: Equal Current Values, Different Histories

At year 4, suppose subject A has trajectory \(m_A(s)=22+1.5s\), and subject B has trajectory \(m_B(s)=16+3s\), for \(0\leq s\leq4\). Both have a biomarker value of \(28\ U\) at year 4. Assume equal covariates.

The current-value difference is \(m_A(4)-m_B(4)=28-28=0\ U\). The fitted current-value link therefore gives

\widehat{\mathrm{HR}}_{\mathrm{CV}}(4)=\exp\{0.12(0)\}=1.

For the cumulative link, integrate each trajectory. Subject A’s accumulated exposure is \(\int_0^4(22+1.5s)\,ds=22(4)+0.75(4^2)=88+12=100\ U\text{-years}\). Subject B’s is \(\int_0^4(16+3s)\,ds=16(4)+1.5(4^2)=64+24=88\ U\text{-years}\). The cumulative difference is \(12\ U\)-years, so

\widehat{\mathrm{HR}}_{\mathrm{CE}}(4) =\exp\{0.040(100-88)\} =\exp(0.48) =1.616.

The current-value link assigns equal linked hazards at year 4 because the current biomarker values are equal. The cumulative link estimates a 1.616 hazard ratio for A versus B because A has the greater accumulated exposure. This example assumes the trajectory shapes between visits are adequately represented by the specified linear functions; in real data, that assumption needs scrutiny.

Compare Models on a Shared Scientific Contrast

A fair comparison specifies a trajectory contrast that has a clear interpretation under both models. For example, ask about a one-unit biomarker elevation sustained over a three-year interval. At the end of the interval, that contrast is one unit under the current-value link and three unit-years under the cumulative link. Comparing the resulting hazard ratios is more meaningful than comparing \(0.12\) directly with \(0.040\), because the coefficients multiply quantities with different units.

Worked Example: A Sustained Elevation Over Three Years

Suppose one trajectory is exactly \(1\ U\) higher than the reference trajectory at every time from year 0 through year 3. At year 3, the current-value difference is \(1\ U\); the integrated difference is \(\int_0^3 1\,ds=3\ U\)-years. Using the stipulated fitted coefficients,

\widehat{\mathrm{HR}}_{\mathrm{CV}}(3) =\exp\{0.12(1)\} =\exp(0.12) =1.128, \qquad \widehat{\mathrm{HR}}_{\mathrm{CE}}(3) =\exp\{0.040(3)\} =\exp(0.12) =1.128.

For this particular sustained-elevation contrast, the two fitted models yield the same hazard ratio. That agreement does not make the links equivalent: the equality follows from the chosen coefficients and exposure history. A different trajectory, or different fitted coefficients, can produce different contrasts.

How to Fit and Inspect the Comparison

Fit the two association structures as distinct candidate joint models using the same cohort, longitudinal submodel, covariates, time origin, and comparable baseline-hazard specification. The longitudinal submodel supplies \(m_i(t)\); the current-value model uses its value at \(t\), and the cumulative model uses its integral up to \(t\). Then report the fitted association coefficient with its units and calculate hazard contrasts for clinically interpretable trajectory differences. This comparison focuses on the link-specific implications; the coefficient estimates are not interchangeable across models.

Check that the longitudinal trajectory is credible over the integration interval, that the time origin and integration window are explicitly defined, and that visits provide enough information to distinguish plausible histories. Examine whether unusual trajectories or sparse measurements have undue influence. Also assess whether the assumed baseline hazard and covariate structure are sufficiently comparable across the fitted models. A difference between link estimates is difficult to attribute to the association structure if other major model components differ.

Feature	Current-value link	Cumulative-exposure link
Trajectory feature used	Latent biomarker value at time \(t\)	Area under the trajectory from the chosen origin to \(t\)
Coefficient units	Per biomarker unit	Per biomarker-unit-year
Same current value, different histories	Same linked contribution if covariates match	May give different contributions
Key sensitivity	Trajectory estimate near the event time	Trajectory estimates across the full integration period
Scientific interpretation	Association with current biomarker level	Association with accumulated biomarker exposure

Common Mistakes and Limitations

Comparing coefficient magnitudes directly. A coefficient per biomarker unit and one per biomarker-unit-year have different units. Compare predicted hazard ratios for a stated trajectory contrast instead.
Treating cumulative exposure as the current value. The integral incorporates past values. Two people with identical current measurements can have different cumulative exposures, as the second example shows.
Interpreting a hazard ratio as a risk ratio. A hazard is an instantaneous event rate among those still at risk at time \(t\); a hazard ratio is not automatically a ratio of event probabilities over a follow-up period.
Assuming the calculated contrast proves causation. These are model-based associations conditional on the included terms. The link comparison alone does not establish that changing the biomarker would change event hazard.
Ignoring accumulated uncertainty. A cumulative exposure depends on the estimated trajectory throughout an interval. Sparse visits or a misspecified trajectory can distort its integral even when the fitted value at the event time seems plausible.

Neither link is universally preferable. A current-value link may be a natural representation when the present biomarker state is scientifically central; a cumulative link may be relevant when duration or accumulated burden matters. A cumulative link also depends on the chosen time origin and integration interval, and may not represent mechanisms involving recent exposure, delayed effects, or changing exposure weights. Those alternatives require their own explicit model structures. Link sensitivity is informative when it reveals which conclusions depend on the exposure summary, but it does not by itself identify the scientifically correct structure.

Key takeaway: State each link with its exposure scale, then compare fitted models through hazard ratios for the same clearly defined trajectory contrast. Differences between those ratios reflect both the fitted association and the distinct information each link extracts from the biomarker history.

Check Your Understanding

Answer each question using the link definitions above. Brief solutions follow the questions.

A subject’s trajectory is \(1.5\ U\) above the reference trajectory throughout two years. What are the current-value and cumulative-exposure contrasts at year 2?
Two subjects have equal biomarker values at year 5, but one has a larger accumulated exposure. Which link can distinguish them at year 5, assuming equal covariates, and why?
Why is it not meaningful to conclude that a cumulative-link coefficient is smaller merely because its numerical value is less than a current-value coefficient?
Using the hypothetical fitted coefficients in this tutorial, calculate the cumulative-link hazard ratio for a cumulative exposure difference of \(5\ U\)-years.

Solutions. (1) The current-value difference is \(1.5\ U\); the cumulative difference is \(1.5(2)=3\ U\)-years. (2) The cumulative link can distinguish them because it uses the integrated history; the current-value link uses only the equal values at year 5. (3) Their units differ: one is per biomarker unit and the other per biomarker-unit-year, so their raw magnitudes do not measure association on a common scale. (4) \(\exp\{0.040(5)\}=\exp(0.20)=1.221\). The fitted cumulative model estimates a hazard 1.221 times as high for the trajectory with the greater accumulated exposure, conditional on the stated model and covariates.