One trajectory, three association questions
A time-to-event model can relate a subject’s event hazard to different features of that subject’s longitudinal marker. A current-value link asks whether the marker’s level at time \(t\) is associated with the hazard at \(t\). A slope link asks whether its instantaneous rate of change is associated with that hazard. A cumulative link asks whether accumulated marker exposure up to \(t\) is associated with it. These are different scientific questions, and even when they use the same latent trajectory they can imply different patterns over time.
The calculations below isolate those patterns along one hypothetical trajectory. They do not establish which link is appropriate for a particular application: that requires scientific reasoning and model assessment. In particular, the association-multiplier comparisons below are not automatically comparisons of full hazards, because the baseline hazard can also change over time.
This comparison assumes a correctly specified latent trajectory for the purpose of calculating its value, derivative, or integral; a proportional association form in which the relevant coefficient \(\alpha\) is constant over time; and a defined time origin for the cumulative term. A cumulative link also requires a finite integral over the interval considered. The illustrative comparisons hold the subject’s covariates and latent trajectory fixed. When models are fitted to data, their baseline hazards and association coefficients may differ, so the pattern calculations alone do not identify the best-fitting model.
Deriving the relative association pattern
For a given link, define its association multiplier as \(\exp\{\eta_i(t)\}\). To describe how that component changes relative to its value at time zero, define \(R_k(t)=\exp\{\eta_{ik}(t)-\eta_{ik}(0)\}\), where \(k\) denotes the link. This is a ratio of association multipliers. It is not, in general, the ratio of the full conditional hazards.
The fixed-covariate term cancels because the same subject and covariates are being compared. The baseline-hazard ratio does not cancel unless it equals one. Thus \(R_k(t)\) isolates the change attributable to the association term, while the full conditional hazard ratio also reflects the baseline hazard.
The relative patterns follow directly from substituting each link into \(R_k(t)\). Differentiating their log ratios, when the required derivatives exist, shows which part of the trajectory controls the pattern:
The last equality follows from the fundamental theorem of calculus for a continuous trajectory. For the slope link, differentiating the log ratio requires a second derivative; if the trajectory is only differentiable, the slope link itself is defined, but this additional rate-of-change statement may not be. These expressions show why the links need not tell the same time-course story: current-value change follows the slope, slope-link change follows curvature, and cumulative-link change follows the current marker level.
Worked comparison on a linear trajectory
Use the same hypothetical trajectory for all three links: \(m_i(t)=2+0.5t\), where the marker is measured in marker units and \(t\) is in years. Thus \(m_i'(t)=0.5\) marker units per year, and \(A_i(t)=\int_0^t(2+0.5s)\,ds=2t+0.25t^2\) marker-unit-years. To make the arithmetic concrete, take \(\alpha_V=0.4\) per marker unit, \(\alpha_S=0.6\) per marker-unit per year, and \(\alpha_C=0.1\) per marker-unit-year. These coefficients are hypothetical and are chosen for illustration, not for comparison of clinical importance across links.
Worked Example: Calculating All Three Association Patterns
Calculate the association terms, multipliers, and relative multipliers at \(t=0,2,4\) years. The trajectory is smooth and linear, so its value, slope, and integral are defined throughout this interval.
For the current-value link, \(\eta_V(t)=0.4(2+0.5t)=0.8+0.2t\). At \(t=2\), for example, \(\eta_V(2)=1.2\), so its multiplier is \(e^{1.2}=3.320\). Relative to time zero, \(R_V(2)=e^{1.2-0.8}=e^{0.4}=1.492\).
For the slope link, \(\eta_S(t)=0.6(0.5)=0.3\) at every time. Its association multiplier is \(e^{0.3}=1.350\) throughout, and \(R_S(t)=e^{0.3-0.3}=1\).
For the cumulative link, \(\eta_C(t)=0.1(2t+0.25t^2)\). At \(t=2\), \(A_i(2)=4+1=5\) marker-unit-years, so \(\eta_C(2)=0.1(5)=0.5\), and \(R_C(2)=e^{0.5-0}=1.649\).
| Time (years) | Current \(m_i(t)\) | \(R_V(t)\) | Slope \(m_i'(t)\) | \(R_S(t)\) | Cumulative \(A_i(t)\) | \(R_C(t)\) |
|---|---|---|---|---|---|---|
| 0 | 2 | 1.000 | 0.5 | 1.000 | 0 | 1.000 |
| 2 | 3 | 1.492 | 0.5 | 1.000 | 5 | 1.649 |
| 4 | 4 | 2.226 | 0.5 | 1.000 | 12 | 3.320 |
For example, \(A_i(4)=2(4)+0.25(4^2)=8+4=12\), giving \(R_C(4)=e^{0.1(12)}=e^{1.2}=3.320\). The current-value relative multiplier increases exponentially with time on this linear trajectory. The slope relative multiplier stays at one because the slope is constant. The cumulative relative multiplier increases more rapidly over this interval because the marker level is positive and the accumulated exposure grows. These are association-component patterns, not statements that the full hazard has those same ratios.
Worked Example: Including the Baseline-Hazard Ratio
Suppose, hypothetically, that under the current-value model the baseline hazard is \(0.020\) per year at time zero and \(0.030\) per year at year 2. Use the current-link calculation above to find the full conditional hazard ratio at year 2 versus time zero, holding the subject’s covariates and latent trajectory fixed.
The association-multiplier ratio is \(R_V(2)=1.492\), and the baseline-hazard ratio is \(0.030/0.020=1.5\). Therefore, \[ \frac{h_i(2\mid b_i)}{h_i(0\mid b_i)} = 1.5(1.492)=2.238. \]
The full conditional hazard at year 2 is about 2.238 times the full conditional hazard at time zero under these hypothetical values. The association component alone contributes a ratio of 1.492; the additional factor of 1.5 comes from the baseline hazard. Both hazards are rates in inverse years, and their ratio is dimensionless. This calculation presumes a positive baseline hazard at time zero, as required to form the ratio.
Worked Example: Comparing Changes From Year 2 to Year 4
Compare the association multipliers at years 4 and 2, rather than comparing each with time zero. This makes clear that a relative pattern depends on the reference time.
For the current-value link, the difference in association terms is \(1.6-1.2=0.4\), so the multiplier ratio is \(e^{0.4}=1.492\). For the slope link, the terms are both \(0.3\), so the ratio is \(e^0=1\). For the cumulative link, \(A_i(4)-A_i(2)=12-5=7\), so the difference in association terms is \(0.1(7)=0.7\), and the ratio is \(e^{0.7}=2.014\).
Thus, from year 2 to year 4, the cumulative association multiplier rises by a factor of about 2.014, the current-value multiplier by about 1.492, and the slope-link multiplier does not change. These factors still do not include the baseline-hazard ratio between years 2 and 4.
How to compare the links responsibly
A useful comparison keeps the trajectory and reference times fixed, states the link and its units, and reports the association-multiplier ratio separately from any full hazard ratio. A current-value link reacts to changes in marker level. A slope link can distinguish a rising from a falling trajectory, but a constant slope has no changing association multiplier under a constant coefficient. A cumulative link retains the history encoded by the integral, so its association term can continue changing even when the current marker level changes slowly.
These distinctions matter because the same observed pattern of event risk may be described differently under different links. They should not be treated as interchangeable summaries of one coefficient: the predictors have different meanings and units. In particular, the numerical coefficients above cannot be compared as though a larger coefficient necessarily represented a stronger association. The model’s link, scale, and predictor units all matter.
The illustrative calculations also have limitations. The linear trajectory has constant slope and does not show what a slope link does when the marker curves. The cumulative exposure here begins at time zero and uses an unweighted integral; changing the origin or the definition of accumulation changes the quantity being modeled. In actual joint modeling, the trajectory is latent and estimated with uncertainty. Plugging in an estimated trajectory and treating it as known can understate uncertainty; the calculations here are conditional illustrations, not an uncertainty analysis.
Diagnostics and common pitfalls
Before interpreting a comparison, check that the trajectory is defined over the interval, that the slope exists where required, and that the cumulative integral uses the intended time origin and units. Check calculations by differentiating the stated trajectory and integrating it independently. A result with units inconsistent with its coefficient often signals an error: \(\alpha_Vm_i(t)\), \(\alpha_Sm_i'(t)\), and \(\alpha_CA_i(t)\) must each be dimensionless inside the exponential.
A flat \(R_S(t)\) on a linear trajectory is not, by itself, evidence that the slope link has no association. In this example, the slope-link multiplier is \(e^{0.3}\), not 1; only its ratio to its own time-zero value is 1. Similarly, a rising \(R_C(t)\) does not establish a rising full hazard: the full hazard also depends on \(h_0(t)\), and the association ratio compares a model component rather than the entire event rate.
- Wrong: “The current-value model’s hazard ratio at year 2 versus time zero is 1.492.” This treats the association-multiplier ratio as the full conditional hazard ratio.
- Correction: The association-multiplier ratio is 1.492. The full conditional hazard ratio is \(\{h_0(2)/h_0(0)\}(1.492)\), if \(h_0(0)>0\).
- Wrong: “The slope link has no association because \(R_S(t)=1\).”
- Correction: \(R_S(t)=1\) means its multiplier does not change from its time-zero value on this constant-slope trajectory. The multiplier itself is \(e^{0.3}\).
Finally, a pattern that looks implausible may reflect an incorrect time origin, units, trajectory shape, or link definition rather than a calculation error. Conversely, arithmetic consistency does not prove the link is scientifically appropriate. Model adequacy requires examining the fitted model and its assumptions, not only drawing conclusions from one illustrative trajectory.
Check Your Understanding
Answer each question using the definitions above. Solutions follow the questions.
- For the stated trajectory, calculate \(A_i(3)\) and \(R_C(3)\).
- Why is \(R_S(t)=1\) for every \(t\) in this example, and what does that result not imply?
- If \(h_0(2)/h_0(0)=0.8\), what is the full conditional hazard ratio at year 2 versus time zero under the current-value link?
- Which trajectory feature determines the instantaneous change in \(\log R_C(t)\)? State the relationship.
Solutions. (1) \(A_i(3)=2(3)+0.25(3^2)=6+2.25=8.25\) marker-unit-years. Thus \(R_C(3)=e^{0.1(8.25)}=e^{0.825}=2.282\). (2) The slope is constantly \(0.5\), so \(\eta_S(t)=0.6(0.5)=0.3\) at all times and its difference from the time-zero term is zero. This does not imply that the slope association multiplier is one or that there is no slope association; the multiplier is \(e^{0.3}=1.350\). (3) The full ratio is \(0.8(1.492)=1.194\), using the current-link association ratio from the worked example. (4) The current marker level determines it: \(d\log R_C(t)/dt=\alpha_Cm_i(t)\). Here that rate is \(0.1(2+0.5t)\) per year.