How Informative Visits Distort Estimated Trajectories

Why the Observed Trajectory May Not Be the Population Trajectory

A longitudinal data set records outcomes when measurements are made, not necessarily at times chosen independently of patients’ health. If worsening symptoms prompt a biomarker test, for example, the measurements collected at a given time may disproportionately represent patients with higher biomarker values. A regression through the recorded values can then describe the trajectory among visits rather than the average trajectory among the population of patients.

The distinction is about the observation schedule, not simply missing measurements. A useful first calculation is to ask: at a fixed time, what outcome distribution would we see if we sampled a visit rather than a patient? Once visits are more likely at some outcome values than others, those two distributions need not agree.

For the calculations below, assume patients are under observation and eligible for a visit at the time being considered, the outcome is defined for each eligible patient, and the stated intensity correctly describes how visits arise. We compare the population mean at a fixed time with the mean outcome attached to visits at that time. This comparison isolates selection by the visit process; it does not by itself address dropout, measurement error, or terminal events.

How Outcome-Triggered Visits Tilt the Mean

At a fixed time \(t\), suppose the visit intensity is \(\lambda(t\mid Y)\), with \(Y\) denoting the outcome across eligible patients. A patient with outcome \(Y=y\) contributes visits in proportion to \(\lambda(t\mid y)\). Thus the outcome distribution among visits is the population outcome distribution reweighted by the intensity. Its mean is

Subtracting the population mean \(\mu(t)=E\{Y(t)\}\) gives a useful expression for the difference:

To verify this, write \(E(Y\lambda)=E(Y)E(\lambda)+\operatorname{Cov}(Y,\lambda)\), divide by \(E(\lambda)\), and subtract \(E(Y)\). When higher outcomes correspond to higher visit intensity, the covariance is positive and the mean among visits exceeds the population mean. The formula also shows that a visit-sampled mean equals the population mean if the intensity is constant across outcome values, or more generally if the covariance is zero.

A Worked Linear-Trajectory Distortion

Consider a hypothetical cohort with two equally sized patient groups. Their outcomes follow exact linear trajectories, with time \(t\) measured in years and the outcome measured in biomarker units. Group A follows \(Y_A(t)=2t\); Group B follows \(Y_B(t)=10\). The population mean trajectory is therefore \(\{2t+10\}/2=5+t\) units per patient. Its intercept is 5 units, and its slope is 1 unit per year.

Suppose a patient has probability \(0.1(1+Y/10)\) of receiving a measurement in a small, standardized observation window. This probability is positive for the outcomes considered and increases with the current outcome. Assume 100 patients in each group. For a transparent illustration, take the realized visit counts to equal the expected counts at \(t=0\) and \(t=1\).

Why the Slope Distortion Is Not Automatic

Outcome-dependent sampling need not change a trajectory’s slope in every setting. In particular, suppose patients have parallel linear trajectories \(Y_i(t)=a_i+\beta t\), where \(a_i\) varies across patients but the slope \(\beta\) is common. If the intensity is \(\lambda_i(t)=\exp\{\gamma Y_i(t)\}\), then at a fixed time it can be written as

The first factor is the same for all patients at that time, so it cancels when calculating the visit-sampled mean. The distribution of the \(a_i\)'s among visits is tilted by \(\exp(\gamma a_i)\), but that tilt does not vary with time. The visit-sampled mean is consequently a shifted intercept plus \(\beta t\): the level is distorted, but the common slope is preserved. This argument relies on parallel trajectories and this separable intensity; it is not a general guarantee for outcome-triggered observation.

Interpreting the Fit and Checking Its Weak Points

An ordinary regression or mixed-effects model for outcomes can be useful, but its interpretation depends on how observations enter the analysis and on the assumptions connecting observed records to the target population. If visit times are independent of outcomes, conditional on the variables represented in the model, the observed measurements may represent the intended population trajectory under suitable modeling conditions. If the visit intensity still depends on the current or latent outcome, the records may instead overrepresent some parts of the trajectory. A random-effects model for outcomes does not, by itself, remove selection caused by an unmodeled visit process.

In an actual analysis, consider whether visits cluster after unusual or worsening outcomes, and whether patients with different outcome levels contribute different numbers of records. Compare the measurement schedule and the target estimand: an average across patients at a specified time is not automatically the same quantity as an average across visits. These observations can flag a possible problem, but they cannot prove that the visit process is noninformative or identify its dependence on unobserved outcomes.

The main alternatives address different aspects of the problem. A design based on scheduled measurements limits outcome-driven differences in measurement opportunities. Inverse-intensity weighting uses a model for the visit process to reweight observed records toward the target population, subject to correct specification and adequate positivity. A joint model represents the outcome and visit processes together, potentially linking them through shared latent quantities. These approaches are not interchangeable guarantees: each requires assumptions, and results can be sensitive to how the observation mechanism is represented.

The central limitation is that the visit-sampled distribution alone may not reveal how many measurements would have been made for patients who were observed less often, particularly when visit dependence involves an unobserved current outcome. Consequently, a fitted trajectory can be precisely estimated for the recorded visits yet still differ from the population trajectory of interest. The distinction is about the estimand and the observation mechanism, not simply sample size.