Combining Slope and Cumulative Exposure

Why Combine Slope and Cumulative Exposure?

A marker’s current direction of change and its accumulated burden describe different features of a subject’s history. A rising marker may indicate a recent worsening trend even when its cumulative exposure is modest; a persistently high marker may produce a large cumulative exposure even if it is currently flat or declining. A joint model can put both features in the event-hazard link, allowing their associations with the event rate to be estimated separately.

The distinction is useful only if each component is defined from the latent marker trajectory, rather than treated as a noise-free feature of the observed measurements. The slope describes the trajectory’s instantaneous rate of change at event time. Cumulative exposure summarizes the trajectory from a specified time origin up to that time.

The Slope-plus-Cumulative Hazard Link

One conditional proportional-hazards specification is

Here \(h_i(t\mid b_i,\mathbf{w}_i)\) is subject \(i\)’s event hazard at time \(t\), conditional on the subject’s latent random effects \(b_i\) and survival covariates \(\mathbf{w}_i\). The common baseline hazard is \(h_0(t)\), \(\boldsymbol{\gamma}\) is the covariate coefficient vector, and \(\alpha_s\) and \(\alpha_A\) are the slope and cumulative-exposure association parameters. The hazard is conditional on being event-free immediately before \(t\). The link is additive on the log-hazard scale, so each association contribution is dimensionless.

If the marker is measured in marker-units and time in years, then \(\alpha_s\) has units years per marker-unit, the reciprocal of the slope’s units. The parameter \(\alpha_A\) has units per marker-unit-year. These units ensure that \(\alpha_s m_i'(t)\) and \(\alpha_A A_i(t)\) are dimensionless. Changing the marker or time scale changes the numerical coefficients unless the coefficients are rescaled accordingly.

For two subjects \(i\) and \(j\) compared at the same time, their conditional hazard ratio is

The common baseline hazard cancels. When the survival covariates are equal, the first term in the exponent is zero. The resulting contrast is conditional on the latent trajectories and random effects; it is not automatically a marginal hazard ratio obtained by averaging over random effects.

How the Two Contributions Enter

For a linear latent trajectory \(m_i(u)=a_i+q_i u\), where \(a_i\) is the marker value at time zero and \(q_i\) is a constant slope, differentiation and integration give

Thus the slope contribution is \(\alpha_s q_i\), while the cumulative contribution is \(\alpha_A(a_i t+q_i t^2/2)\). Even for a linear trajectory, the cumulative contribution generally changes with time. More generally, the slope captures the local trend at \(t\), whereas the integral depends on the whole path from time zero to \(t\). Neither component can generally be recovered from the other.

The coefficients describe conditional associations on the log-hazard scale. Holding the cumulative exposure and covariates fixed, a one-unit increase in slope changes the conditional hazard by a factor of \(\exp(\alpha_s)\), with the slope unit specified. Holding the slope and covariates fixed, a one-unit increase in cumulative exposure changes it by \(\exp(\alpha_A)\). These interpretations do not assert that changing a marker trajectory would cause the hazard to change.

Contrasts When Trend and Burden Differ

A hazard contrast can be driven in opposite directions by the two components. It is the sum of their contributions on the log-hazard scale that determines the fitted contrast. It is not correct to infer the total contrast from the sign of the slope alone or the cumulative exposure alone.

Assumptions, Diagnostics, and Alternatives

The formula is meaningful when the latent trajectory is defined on the chosen time scale and is differentiable at the time where the slope is used. The cumulative exposure must use a clearly stated origin and integration interval. In estimation, the longitudinal model must represent the trajectory adequately enough to estimate both its derivative and its integral. Because the derivative is especially sensitive to local curvature and measurement noise, using raw observed changes in place of a model-based latent slope can produce a very noisy association measure.

The hazard specification also assumes that the two association components enter linearly and additively on the log-hazard scale, with constant coefficients \(\alpha_s\) and \(\alpha_A\) over the modeled period. Standard joint-model estimation additionally requires an appropriate event-time model and a defensible account of censoring; for example, censoring should be conditionally independent of the event process given the modeled information, or its dependence must be modeled. These assumptions are not verified just by calculating a hazard ratio.

Useful checks include inspecting whether fitted longitudinal trajectories follow the observed repeated measurements without treating those measurements as the latent path; checking that the slope is stable to reasonable changes in the trajectory model; verifying cumulative integrals numerically or analytically; and confirming that coefficient units match the marker and time scales. Uncertainty in the fitted longitudinal trajectory and in both association coefficients should be reflected in inference for hazard contrasts. A plausible point estimate alone is not evidence that the slope has been estimated reliably.

This link differs from a slope-only association, which omits accumulated history, and a cumulative-only association, which omits the local trend. Compared with a current-value-plus-cumulative-exposure link, it replaces the current level as the instantaneous component with the current rate of change. The choice should be guided by the scientific question and supported by model checks; these alternatives are not interchangeable simply because all use the same marker trajectory.

A common mistake is to say, “U’s slope is larger, so U must have the higher hazard.” That reasoning ignores the cumulative term and any survival-covariate contrast. The correction is to calculate each term in the log hazard ratio, add them with their fitted coefficients, and then exponentiate the total. Also, a hazard ratio compares instantaneous event rates among subjects still event-free at that time; it is not a ratio of event probabilities over an interval.