Deriving the Cumulative Effect of a Marker Trajectory

From a Marker Trajectory to Accumulated Exposure

A cumulative-association model uses more than the marker value at one instant. It links the event hazard to the marker’s accumulated trajectory up to that time. To use the model for a particular subject, we must first turn the specified latent trajectory into an area: the integral of the marker level across time. We can then compare two subjects by taking the difference in their areas and applying the association parameter.

This tutorial works through that calculation. The emphasis is on deriving the area from a given trajectory and carrying it correctly into the hazard model. The earlier material on Cumulative Association: Definition and Hazard Ratio introduced the general interpretation; here we make the trajectory-to-hazard calculation explicit.

Model, Notation, and Conditions

Let \(m_i(s;b_i)\) be subject \(i\)’s latent marker value at time \(s\), conditional on that subject’s random effects \(b_i\). For a fixed lower limit of zero and a comparison time \(t\), define the cumulative marker exposure \(A_i(t)\) as the area under this latent trajectory from zero to \(t\).

Consider the cumulative-association hazard model

where \(\lambda_0(t)\) is the baseline hazard, \(w_i\) is a vector of covariates, \(\gamma\) is its coefficient vector, and \(\alpha\) is the cumulative-association parameter. The hazard is conditional on the subject’s latent trajectory, represented by \(b_i\), and the covariates. In this model, a one-unit increase in cumulative exposure changes the log hazard by \(\alpha\), holding the covariates and other terms fixed.

The calculation requires the trajectory to be defined over the full integration interval and the cumulative link to be the model being used. The units must also be coherent: if \(m_i(s)\) is measured in marker units and time in years, \(A_i(t)\) is in marker-years, so \(\alpha\) is in inverse marker-years. Comparisons that use the same baseline hazard and equal covariates allow those terms to cancel. If covariates differ, their contribution must be retained. These are conditional hazard comparisons, not causal effects.

Why the Difference in Areas Gives the Hazard Ratio

For two subjects with the same covariates, divide their conditional hazards at time \(t\). The common baseline hazard and covariate term cancel. Taking the logarithm then gives the log-hazard contrast directly.

More generally, if the subjects have different covariates, the ratio is

This derivation concerns the instantaneous hazard at the specified time. The ratio is not a ratio of survival probabilities or cumulative hazards. Its value can change with \(t\), because the accumulated areas can change with time.

Integrating a Polynomial Trajectory

Suppose a trajectory is given as a polynomial, \(m_i(s)=\sum_{k=0}^{K}\beta_{ik}s^k\). Integrating each term gives

The key is to integrate with respect to the trajectory’s time argument \(s\), and then evaluate the resulting expression at the comparison time \(t\). For a contrast, integrate both trajectories over the same interval before subtracting their areas.

An Exponential Trajectory and a Nonlinear Area

The same procedure applies when the trajectory is not polynomial. For an exponential term, use its antiderivative and evaluate at both interval boundaries. The resulting area need not be linear in time, so the hazard contrast may evolve at a nonlinear rate.

A Contrast With an Exact Area Difference

Small differences in the integrated areas can matter when multiplied by the association parameter. Keep adequate precision through the multiplication and exponentiation, then round the final result. The following calculation illustrates why the percentage interpretation should be computed from the hazard ratio rather than rounded prematurely.

Checking the Calculation and Its Limits

A sound calculation has two distinct parts: the area implied by the trajectory, and the hazard contrast implied by that area and \(\alpha\). Check them separately. For an analytic integral, differentiate the antiderivative to recover the original trajectory. For a numerical integral, compare the result with a finer numerical approximation when feasible. In either case, verify that the area has marker-times-time units, while \(\alpha A_i(t)\) is dimensionless, as required inside an exponential.

The contrast also depends on the model assumptions. The trajectory used must represent the latent marker path over the interval, rather than an arbitrary substitute that changes the quantity being integrated. The comparison must account for covariate differences, and the interpretation is conditional on the specified trajectories and model. If trajectories or model parameters are estimated, a contrast calculated from point estimates alone does not express its uncertainty. Confidence intervals or other uncertainty summaries must incorporate relevant estimation uncertainty.

This cumulative link differs from the current-value and slope links covered earlier: those use the marker level or its rate of change at a particular time, whereas this model uses an integral over the trajectory. These alternatives answer different modeling questions. A good fit cannot be established merely by obtaining a plausible area or hazard ratio; model adequacy and trajectory plausibility need to be assessed separately.

• Wrong: “A’s marker is higher at year 3, so its cumulative exposure must be higher.” A value at one instant does not determine the area over the entire interval. Correction: integrate each trajectory over the specified interval and compare the resulting areas.
• Wrong: subtract the trajectories at time \(t\) and call that the cumulative contrast. Correction: subtract the integrals, \(\int_0^t m_A(s)\,ds-\int_0^t m_B(s)\,ds\).
• Wrong: interpret a hazard ratio of 0.7805 as a 0.7805% reduction. Correction: the percentage lower is \((1-0.7805)\times100\%=21.95\%\).
• Wrong: treat the conditional hazard ratio as a survival-probability ratio or a causal effect. Correction: it is a model-based ratio of instantaneous hazards conditional on the stated covariates and latent trajectories.