Informative Omics Visit Processes

Why Visit Timing Can Change the Data

Longitudinal omics data are often collected at visits whose timing is partly determined by a participant’s health. A clinician may order an assay sooner after symptoms worsen, or monitor a patient more frequently when a previously measured marker is high. Consequently, the observed assay records may over-represent periods or people with poorer health. This is not the same as merely having fewer measurements: it is a feature of which times are observed.

If visit timing is unrelated to the evolving health and omics process, analyses can often treat the observation schedule as given, subject to the assumptions of the chosen longitudinal or survival model. When visit frequency depends on health status or prior omics values, that condition can fail. A trajectory estimated from observed records may then differ from the trajectory in the target population, and associations with outcomes can be distorted. The central question is not simply whether patients have unequal numbers of visits, but whether their timing carries information about their underlying state.

The intensity has units of visits per unit time. It describes the instantaneous rate of a visit, conditional on the modeled history and being under observation. For example, an intensity of 0.5 visits per month means the conditional chance of at least one visit in a very short interval \(dt\) is approximately \(0.5dt\), not that every participant has exactly one visit every two months.

Earlier material on repeated-measures estimands distinguishes a target such as the average trajectory over participants from a summary over recorded assays. Those targets need not agree when visit rates differ across health states. Mixed models describe longitudinal outcomes, and joint longitudinal-survival models connect trajectories to event risk; informative visit timing adds an observation-process component to this picture.

How Visit-Weighted Data Become Biased

A simple calculation shows the source of the problem. Suppose a participant’s state has value \(Y\), and the instantaneous visit rate in that state is \(\lambda(Y)\). At a randomly selected visit, states with larger rates are more likely to be represented. Under a stable population and a visit mechanism whose rate depends on state, the mean state observed at visits is

The second equality follows by writing \(E\{Y\lambda(Y)\}=E(Y)E\{\lambda(Y)\}+\operatorname{Cov}\{Y,\lambda(Y)\}\), then dividing by \(E\{\lambda(Y)\}\). If higher values of \(Y\) prompt more visits, the covariance is positive and the visit-weighted mean exceeds the population mean. If the visit rate is unrelated to \(Y\), the covariance is zero and this particular source of bias disappears. In longitudinal data, the same logic applies over time, with changing health, prior measurements, and time at risk all potentially affecting the intensity.

Modeling the Observation Process

A joint model represents the omics trajectory and the visit process together, often alongside the survival outcome. Let \(Y_{ij}\) be a measured omics value for participant \(i\) at visit time \(t_{ij}\). A longitudinal submodel might specify a trajectory \(m_i(t)\) using fixed effects and participant-specific random effects \(b_i\), with measurement error around that trajectory. A visit-process submodel then specifies the conditional intensity, for example:

Here \(\lambda_0^V(t)\) is a baseline visit intensity, \(Z_i(t)\) contains modeled health or history variables, and \(m_i(t)\) is the underlying trajectory. The parameter \(\eta\) describes how the latent trajectory is associated with the rate of visiting, conditional on the other terms. If \(m_i(t)\) increases by one unit, the intensity is multiplied by \(\exp(\eta)\), assuming the model’s other predictors stay fixed. A positive \(\eta\) means higher trajectory values are associated with more frequent visits; it does not establish that the marker causes visits.

If the survival outcome is also modeled, its event intensity may depend on the same latent trajectory or random effects. The likelihood then combines the observed longitudinal outcomes, visit times, and event-time information, and integrates over unobserved random effects. This allows the analysis to account for shared latent characteristics that can influence both measurements and observation times. The precise model depends on the estimand and data-generating mechanism; adding a visit model does not automatically correct every form of bias.

The model’s assumptions must match the data collection process. In particular, the conditional intensity must adequately represent visit timing, including relevant predictors and the time scale; the trajectory and outcome models must be sufficiently specified; and the random-effects distribution and dependence structure must be plausible. A shared-random-effects approach commonly assumes that, conditional on the random effects and included history, the remaining visit process, measurement errors, and event process have the specified conditional structure. This is a modeling assumption, not a fact guaranteed by having fit all three submodels.

An important distinction is whether visits depend on recorded history or on an unobserved current state. If the rate depends on a prior measured value, that history may be included directly in a visit-intensity model. If clinicians respond to an evolving but unmeasured health state that is related to the latent omics trajectory, modeling that trajectory jointly with visits may be more appropriate. If the relevant drivers are omitted, a fitted joint model can still be biased.

When Weighting Is an Alternative

A different strategy is inverse-intensity weighting. The basic idea is to give less weight to measurements made at times with a high modeled visit rate, because those states are over-represented among observed visits. In an estimating equation, this may involve weights proportional to the reciprocal of the estimated intensity. Such weights require a correctly specified visit model, adequate support (the intensity must not be effectively zero for relevant histories), and careful definition of the target estimand. Stabilized or truncated weights may reduce numerical instability, but truncation changes the adjustment and can leave residual bias.

Inverse-intensity weighting is attractive when the visit process can be modeled from observed history and the goal is to correct a specified estimating equation. A shared-random-effects joint model is useful when latent trajectory features plausibly affect both visits and outcomes, and the full parametric structure is scientifically defensible. Neither is universally superior. Weighting can become unstable when estimated rates are small; joint models can be sensitive to distributional and functional-form assumptions. A model that ignores visit timing is simplest, but is defensible only when the visit schedule is non-informative for the target analysis, conditional on variables already handled.

Diagnostics, Failure Modes, and Limits

Before selecting an adjustment, examine the scheduling mechanism. Plot visit counts or estimated visit rates against prior health measures and omics values, accounting for time under observation. Compare observed intervals between visits across clinically meaningful histories, and check whether visits cluster after changes in symptoms or prior assay values. These patterns can reveal dependence that a fixed-visit analysis would miss, although they do not prove that the fitted model captures the mechanism.

For a fitted visit model, assess whether predicted visit counts and timing resemble what was observed across time and relevant covariate strata. Inspect residual or calibration summaries for the observation model, and examine intensity and weight distributions for extreme values. For joint models, also check the longitudinal and survival submodels, and assess sensitivity to the trajectory form, random-effects distribution, and inclusion of plausible health-history predictors. If the observed timing remains systematically misrepresented, inferences that rely on the joint model are not trustworthy.

A common pitfall is to treat all observed assays as equally informative simply because the longitudinal model accounts for within-person correlation. Correlation modeling addresses dependence among repeated outcomes; it does not, by itself, correct selection of observation times. The correction is to ask whether visit intensity depends on health or latent trajectory features, and to model or otherwise adjust for that process when it affects the target estimand.

A second pitfall is to interpret an association between marker level and visit rate as proof that the marker caused extra monitoring. Scheduling may respond to symptoms, clinical decisions, or other unmeasured factors correlated with the marker. Visit-process models describe and adjust for a hypothesized observation mechanism; they do not identify causal effects without the additional assumptions and design needed for causal interpretation.

Finally, visit-process adjustment has limits. A misspecified intensity or latent trajectory model can leave bias; very sparse data may not identify the relationship between latent health and visit timing; and extreme inverse-intensity weights can produce high variance. The observation schedule may also be driven by unrecorded clinical judgment. Sensitivity analyses over plausible visit-process specifications are therefore important. Informative visit timing concerns when measurements occur; it should not be conflated with a participant permanently ceasing follow-up, which is a distinct observation problem.