Common Errors In Modeling Informative Visit Schedules

Why Small Modeling Errors Matter

In longitudinal studies, a visit is not always just a convenient occasion to collect a biomarker. Symptoms may prompt extra visits, missed visits may reflect a patient's health, and follow-up may end for reasons related to the outcome. A model that treats the observed visit schedule as an unimportant backdrop can therefore describe a selected set of measurements rather than the intended longitudinal process.

This tutorial focuses on three errors that can be easy to make even when a visit-process model has been specified: treating realized visit times as fixed without justification, confusing visit intensity with dropout, and using an unobserved current outcome as though it were directly measured. The earlier material on conditions for ignoring the visit process and inverse-intensity weighting supplies the broader context; here the emphasis is on recognizing these particular modeling mistakes.

A typical observation at visit time \(t_{ij}\) might be written \(Z_{ij}=m_i(t_{ij})+\epsilon_{ij}\), where \(Z_{ij}\) is the recorded value and \(\epsilon_{ij}\) is measurement error. Between visits, \(m_i(t)\) need not be observed. The distinction between \(Z_{ij}\) and \(m_i(t)\) is central: information recorded at an earlier visit is not automatically the current biomarker value.

Error 1: Treating the Observed Visit Times as Fixed

A fixed-schedule analysis conditions on the visit times as if they had been set independently of the outcomes. This can be reasonable when visits are determined by a protocol and deviations from that protocol do not carry outcome-related information, subject to the model's other assumptions. It is not justified merely because the analyst has a list of recorded dates.

If worsening health increases the chance of a visit, people with worsening health may contribute more observations during the periods when their biomarker is high. The observed visit schedule then contains information about the outcome process. Treating it as fixed may yield a biased description of the longitudinal trajectory, and ordinary standard errors may fail to reflect the modeled dependence. This is not a claim that every irregular schedule is informative: the issue is whether visit timing depends on relevant observed or unobserved outcome information after accounting for the specified model.

A useful justification is to describe the visit mechanism and explain why it carries no additional information about the outcome after conditioning on variables included in the analysis. If that claim is not credible, possible alternatives include explicitly modeling the visit process or using an inverse-intensity-weighting approach under its own assumptions. Neither alternative is a guarantee: misspecification, inadequate measurement of predictors, or poor overlap in visit intensity can still undermine inference.

Error 2: Confusing Visit Intensity with Dropout

A visit intensity describes the rate of visits while a subject remains under observation and eligible for a visit. Dropout, by contrast, is an event that ends or interrupts follow-up. It is often represented by a dropout time \(D_i\) and a dropout hazard, the instantaneous rate of dropout among subjects who have not yet dropped out. A person can have a high visit rate and a low dropout hazard, or the reverse.

The risk sets and units must match the process being modeled. Visits are recurrent events and are counted; dropout is typically a terminal event and is counted at most once. A high number of visits does not mean that a patient is less likely to drop out, nor is the visit intensity a substitute for a dropout model. A terminal event such as death may also end biomarker observation, but its process is still distinct from the rate of nonterminal visits.

When follow-up can end for several reasons, define the observation period and event of interest explicitly. For example, administrative study end is not necessarily patient dropout, and a missed appointment is not necessarily permanent dropout. Misclassifying a temporary gap as terminal dropout changes both the risk set and the process being modeled.

Error 3: Treating an Unobserved Current Outcome as Measured

Suppose a model says that visits become more likely when the current biomarker is high. That statement may be scientifically reasonable, but it does not mean the current biomarker is observed at every time. Between visits, an analyst cannot simply insert \(m_i(t)\) as a measured covariate. A previously recorded \(Z_{ij}\) is available history; it may be a useful predictor, but it is not generally equal to the current \(m_i(t)\).

There are two distinct modeling choices. One can define the visit intensity using observed history, such as the last recorded biomarker, provided the timing and interpretation of that history are specified. Alternatively, one can let the visit intensity depend on a latent current value and account for uncertainty about that value when fitting the model. The second choice requires a model for the unobserved outcome process and an appropriate likelihood or integration; it cannot be implemented by pretending latent values were recorded.

There is also a timing issue. Predictors in an intensity should be based on information available immediately before the visit. A biomarker value measured at a visit can plausibly predict later visits, but using that same measurement to claim it caused or predicted the visit that produced it requires a clearly specified ordering or a model for their joint timing.

A Short Justification: Why the Mean Plug-In Can Fail

For the exponential intensity above, if \(M\sim N(\mu,\sigma^2)\), the normal moment-generating function is \(E\{\exp(aM)\}=\exp(a\mu+\tfrac{1}{2}a^2\sigma^2)\). Therefore $$ E\{\exp(\alpha+\gamma M)\} =\exp(\alpha+\gamma\mu+\tfrac{1}{2}\gamma^2\sigma^2), $$ whereas evaluating the intensity at the mean gives \(\exp(\alpha+\gamma\mu)\). Their ratio is $$ \frac{E\{\exp(\alpha+\gamma M)\}}{\exp(\alpha+\gamma\mu)} =\exp(\tfrac{1}{2}\gamma^2\sigma^2)\geq 1. $$ The quantities are equal only when \(\gamma=0\) or \(\sigma^2=0\). Thus uncertainty about a latent current value can matter even when its mean is known. This calculation relies on the stated normal distribution and exponential intensity; it is not a universal formula for all visit-process models.

Practical Checks and Alternatives

Before fitting or interpreting a model, check whether its process definitions, time ordering, and assumptions match the data collection mechanism. A check cannot prove ignorability or identify an unobserved trigger, but it can reveal inconsistencies that need explanation.

• Check visit timing: compare protocol dates with actual dates, and distinguish scheduled visits, patient-initiated visits, and missed appointments where possible. Look for clustering after symptoms or treatment changes.
• Check the risk set: confirm that each subject contributes time only while eligible for the process being modeled. Do not count visits after observation ends or count terminal dropout as a recurrent visit.
• Check covariate availability: label each predictor as recorded before the visit, measured at the visit, or latent between visits. Do not describe a latent value as an observed covariate.
• Check model behavior: inspect fitted rates over time and across observed histories, and compare simulated visit patterns with the study's actual pattern. Extreme rates or implausible clusters may indicate a poor specification.
• Check sensitivity: if visit triggers are unrecorded or current values are uncertain, assess how conclusions change under plausible alternatives. Diagnostics based only on observed data cannot rule out dependence on unobserved information.

The main alternatives answer different questions. Conditioning on observed visit times can be efficient when the schedule is ignorable under the specified model. Inverse-intensity weighting reweights observed measurements to address certain forms of unequal observation, but requires a sufficiently well-specified intensity and adequate positivity. A joint model represents outcome and visit processes together, which can accommodate specified dependence on latent outcomes, but depends on the correctness and identifiability of that joint specification. None automatically solves informative dropout: that process must be represented or justified separately.

Common Mistakes

• “We know the visit dates, so they are fixed.” Knowing realized dates does not establish that they were independent of the outcomes. Describe how those dates arose and justify conditioning on them.
• “More visits mean less dropout.” Visit intensity and dropout hazard describe different events. Define and model each process using its own event count and risk set.
• “The latest measurement is the current value.” The latest recorded value is historical information. It may be used as such, but it is not automatically the unobserved value between visits.
• “A diagnostic proves the visit process is noninformative.” Checks can identify visible problems, but observed data alone may not reveal dependence on unmeasured triggers. State the remaining assumptions and assess sensitivity where appropriate.