Sparse Visits and Event Timing in Link Comparisons

Why Visit Density and Event Timing Matter

A joint model relates a subject’s event hazard to a feature of the subject’s underlying biomarker trajectory. That feature might be the current value, the rate of change, or accumulated exposure. These are different summaries of the trajectory, so they need not be equally well supported by the measurements. With few visits, a fitted trajectory may rely strongly on the longitudinal model’s shape assumptions. If an event occurs soon after a visit, the last measurement may be temporally close to the event, but it is still a noisy measurement rather than the exact latent value at event time.

The key distinction is between the visit schedule and the event’s position relative to that schedule. Sparse visits limit how much the data reveal about changes between observations. An event close to a visit can help anchor the trajectory near its event time, provided the measurement precedes the event and the measurement process is appropriately represented. It does not, by itself, establish a reliable slope or cumulative history.

For example, a proportional-hazards specification may write the hazard for subject \(i\) at time \(t\), conditional on the latent trajectory and covariates \(X_i\), as

where \(h_0(t)\) is the baseline hazard, \(\gamma\) is a vector of covariate coefficients, \(\alpha\) is the association coefficient, and \(g_i(t)\) is the chosen trajectory feature. For a current-value link, \(g_i(t)=m_i(t)\); for a slope link, \(g_i(t)=m_i'(t)\); and for a cumulative link, \(g_i(t)=\int_0^t m_i(s)\,ds\). These models require a correctly specified or sufficiently adequate longitudinal trajectory model, an appropriate event-time model, and a defensible account of how measurement error and within-subject dependence are handled. The event time and censoring process must also be represented appropriately for the scientific design.

What Sparse Visits Leave Uncertain

A small number of visits can have different implications for the three features. A current-value link near an observed visit may be reasonably anchored by that measurement, although measurement error and the gap to event time remain relevant. A slope link is more demanding: it depends on how the trajectory changes locally, and two observations cannot reveal whether the change between them was linear, curved, or interrupted. A cumulative link uses information over an interval, but sparse measurements still leave the path between visits to be supplied by the fitted trajectory model.

Sparse sampling does not always mean that a two-point slope has a larger measurement-error standard error. If the two observations have independent errors with the same variance, a wider time interval can reduce the error in their average rate of change. But a wide interval cannot reveal local changes, and it can hide curvature. Thus, a precise average slope over a long interval is not necessarily a precise estimate of the instantaneous slope at event time.

This example illustrates why a link comparison should be read alongside the visit pattern. A slope-based result can reflect the assumed shape between visits, not just a locally observed rate. A cumulative feature can likewise be numerically well-defined under a fitted trajectory while remaining dependent on assumptions about the unobserved path.

Events Soon After Visits

If a measurement is taken shortly before an event, it may be a useful anchor for the current-value feature. However, it is not the latent value at event time: measurement error remains, and the biomarker may change during even a short interval. A measurement taken after an event cannot be treated as a pre-event predictor without a model that justifies doing so; in many settings, the event itself changes whether or how later measurements can be collected.

Timing is also informative about what can be estimated. When many events occur soon after visits, current values near event times may be comparatively well supported. Yet the event-time slope may still be weakly determined if there are few measurements bracketing that time. The cumulative feature draws on the whole history up to the event, but its accuracy depends on how well that history is known, including the unobserved portions between visits.

How Different Links Can Respond to the Same Trajectory

The following calculation isolates the difference among link features. It is not a model-fitting result: the coefficients are hypothetical and are chosen only to illustrate their units and implications. Suppose the latent trajectory is \(m_i(s)=8+0.5s\) biomarker units, where \(s\) is in months, and consider an event at month 6.2. Compare this trajectory with a reference trajectory that remains at 8 units, has zero slope, and therefore has no cumulative exposure above that reference.

Assumptions, Diagnostics, and Alternatives

A useful comparison keeps the longitudinal model, covariate adjustment, and event-model structure as comparable as possible while changing the association feature. Each feature must be defined on the same time scale and in scientifically interpretable units. For the slope link, the trajectory must be differentiable where the slope is evaluated. For cumulative exposure, the integration start time and exposure window must be specified. If visit times depend on a subject’s health status, treating the observed schedule as uninformative may be inappropriate; the measurement process and study design must be considered.

Before interpreting a difference among link results, inspect how much information the schedule provides near event times and over the history used by the cumulative feature. Useful summaries include the number and spacing of visits before events, the time from the latest pre-event visit to the event, and whether events cluster in periods with especially sparse measurement. Plot observed biomarker values and fitted trajectories against time, paying attention to long unobserved intervals and to whether the model is extrapolating beyond the measurements.

When measurements are sparse, alternatives include collecting more frequent measurements where feasible, using a simpler trajectory justified by the design, or reporting that conclusions depend on the assumed trajectory. Carrying the last observation forward is not automatically a sound substitute: it assumes the biomarker remains constant between visits. A spline or other flexible trajectory may represent curvature, but flexibility cannot manufacture information absent from the visit schedule. These strategies address different problems and should be selected based on the data and scientific question rather than because one link produces a preferred association.

A further limitation is that event timing can leave little information about biomarker evolution after the last pre-event measurement. This is especially important for slope links, which ask about a local rate, and for cumulative links, which integrate the modeled history. Close timing can help with event-time anchoring, but it does not validate the trajectory shape, remove measurement error, or make results from distinct link models directly comparable on a single coefficient scale.

Common Pitfall

The practical conclusion is not that sparse visits always invalidate link comparisons. Rather, sparse schedules change what each comparison can support. A close pre-event visit may strengthen the current-value information, while a slope or cumulative result may remain sensitive to the assumed trajectory between visits. Interpret any difference in association in light of those distinct information requirements.