Why Visit Timing Is Part of the Data
Longitudinal data are often described by the measurements collected at each visit. But the times at which those measurements are collected can also carry information. A patient whose health is worsening may be seen sooner or more often than a patient whose health is stable. In that case, the observed measurements do not simply represent the same fixed calendar schedule for every patient: the evolving health process helps determine when it is measured.
This matters because many analyses summarize measurements as if each observation had the same sampling meaning. If some health states lead to more visits, a summary over all recorded visits can give those states greater weight. Even when the analysis uses time explicitly, a visit that occurs in response to a patient's health may not be comparable to a visit assigned without regard to that health.
For notation, let \(Y_i(t)\) denote a patient's health-related longitudinal outcome at time \(t\), and let \(V_{ij}\) denote the time of that patient's \(j\)th visit. Let \(Z_i\) represent baseline or design variables that the analysis conditions on, such as a randomized schedule assignment. The central distinction is whether the process determining \(V_{ij}\) depends on the patient's evolving health, not merely whether the visit times vary between patients.
This equality expresses independence from the current health value, conditional on \(Z_i\), in a simplified single-time statement. In practice, scheduling may depend on past symptoms or previously measured values rather than the current unobserved value. That can still make the schedule informative about the health trajectory: past information may predict current health. Conversely, if the variables driving the schedule are fully represented in the conditioning set and the analysis appropriately accounts for them, timing may no longer be informative relative to that conditional target.
A schedule does not have to be irregular to be informative. A protocol may specify regular visits but permit extra assessments when symptoms worsen. Nor does irregularity alone establish informativity: visits can vary because of administrative logistics, travel, or randomized scheduling, without depending on evolving health. The reason for the timing, and its relation to the target outcome, is what matters.
How Health-Dependent Visits Change What Is Observed
A simple selection calculation shows why visit timing can affect summaries. Consider a single assessment opportunity and let \(R=1\) indicate that the outcome is observed at that opportunity. Suppose the probability of being observed depends on the health value \(Y\), and write \(q(Y)=P(R=1\mid Y)\). Assume \(P(R=1)>0\). The mean among observed values is then
The first equality follows by averaging each possible health value in proportion to its observation probability: \(E(YR)=E\{Yq(Y)\}\), while \(E(R)=E\{q(Y)\}\). The second follows from \(E\{Yq(Y)\}=E(Y)E\{q(Y)\}+\operatorname{Cov}\{Y,q(Y)\}\). Thus, if higher values of \(Y\) make observation more likely, the covariance is positive and the observed mean exceeds the population mean. If higher values make observation less likely, the direction reverses. This is a justification for concern, not a correction procedure: it applies to this simplified selection setup and does not by itself specify how to analyze a complex longitudinal schedule.
Worked Example: More Visits When Health Is Worse
Suppose, hypothetically, that at a particular follow-up time a cohort's health score \(Y\), in points, is 2 for 25% of patients and 8 for 75%. Higher scores indicate worse health. A clinic is more likely to obtain the assessment from patients with a high score: \(q(2)=0.20\) and \(q(8)=0.80\). Find the cohort mean and the expected mean among observed assessments.
Assumptions. Treat the two health values and their proportions as the distribution at this one assessment opportunity. Assume the stated observation probabilities apply to all patients with the corresponding score, and that an observed assessment records that score. There is a positive probability of observation in both groups.
The cohort mean is
Among all patients, the fraction both having score 2 and being observed is \((0.25)(0.20)=0.05\). The corresponding fraction for score 8 is \((0.75)(0.80)=0.60\). Therefore the total observed fraction is \(0.05+0.60=0.65\), and
The observed-assessment mean is about 1.038 points higher than the cohort mean. This is not evidence that the cohort's health worsened because visits occurred; it is a difference in which health values are represented among the observed assessments. The scheduling rule gives high-score patients more representation.
Worked Examples of Different Scheduling Mechanisms
Worked Example: Visits Assigned Independently of Health
In a hypothetical study, each patient is randomly assigned at enrollment to visits every 2 weeks or every 6 weeks. Assignment is completed before follow-up, and the protocol does not alter visit timing in response to symptoms or biomarker results. Suppose the outcome at week 6 is observed for 100 patients in each schedule group. The mean score is 5.1 points in the 2-week group and 5.0 points in the 6-week group. Does this difference show that the schedule depends on evolving health?
Assumptions. For this conclusion, take random schedule assignment as implemented, with no health-dependent changes to the assigned times. The 100 patients per group are the relevant observed groups at week 6. The comparison is descriptive; no claim is made that the two sample means must be identical.
No. The scheduling rule was assigned independently of evolving health, so a difference in the observed means does not establish health-dependent timing. The observed difference is \(5.1-5.0=0.1\) points, but finite samples can differ even when schedule assignment is independent of health. The design supports independence of the assigned schedule; it does not guarantee identical health outcomes or rule out later departures from the protocol. If clinicians add visits for patients whose health worsens, those added visits require separate consideration.
Worked Example: A Schedule Responds to Symptoms
In a hypothetical outpatient program, patients with new or worsening symptoms are asked to return within 3 days; patients without those symptoms keep a visit scheduled 4 weeks later. At a follow-up decision point, 30 patients have worsening symptoms and 70 do not. Suppose all 30 in the first group are seen within 3 days, while 14 of the 70 in the second group are also seen within 3 days for unrelated scheduling reasons. What fraction of the early visits comes from the worsening-symptom group?
Assumptions. Treat the counts as the complete hypothetical set of patients at that decision point. Define an early visit as one within 3 days. The symptom classification is made before the scheduling decision, and the counts refer to visits in that same window.
There are \(30+14=44\) early visits. Of these, 30 are from patients with worsening symptoms, so
or about 68.18%. By comparison, the worsening-symptom group is only \(30/(30+70)=30\%\) of the patients at the decision point. Early visits therefore disproportionately represent patients with worsening symptoms in this hypothetical example. The rule directly connects timing to a health-related history, so treating these early assessments as if their timing were assigned independently of health would misdescribe how the data were collected. The 68.18% figure describes the early visits; it does not estimate a causal effect of symptoms on health.
Worked Example: More Frequent Visits Weight Some Patients More Heavily
Suppose, hypothetically, that a clinic uses a patient's last biomarker result to set the next month of follow-up. Four patients with stable results have values of 2, 3, 3, and 4 points and are seen once during that month. Four patients with worsening results have values of 7, 8, 8, and 9 points and are seen weekly, giving each four visits. For illustration, assume each patient's recorded value stays at the stated value throughout the month. Compare the mean across patients, giving each person equal weight, with the mean across all visits.
Assumptions. The values and visit counts are the complete hypothetical data for the month. Repeated values are held constant by assumption solely to make the weighting calculation clear. The scheduling rule uses the prior result, not an assertion that future values are known when visits are set.
The mean across the eight patients is
For the visit-level mean, the stable group contributes \(2+3+3+4=12\) points across 4 visits. The worsening group contributes \(4(7+8+8+9)=4(32)=128\) points across 16 visits. Thus
The visit-level mean is 1.50 points higher because the worsening-result group contributes four times as many visits per patient. This calculation does not say that either summary is universally correct: the patient-level mean weights patients equally, whereas the visit-level mean weights recorded occasions equally. The appropriate target must be specified before deciding which summary is meaningful.
Assessment, Comparisons, and Limitations
A practical assessment begins with the schedule's stated rules and the records of what actually happened. Useful questions include whether the protocol permitted additional assessments for symptoms, whether appointment gaps vary with recent biomarker values, and whether reasons for early or delayed visits were recorded. A table or plot of time to the next visit against the most recent observed health measure can reveal patterns worth investigating. Such checks are clues, not proof: a measured association may reflect a shared cause, and an absence of association with recorded history cannot rule out dependence on unmeasured current health.
Keep three situations distinct. Informative timing concerns when assessments occur in relation to evolving health. Dropout concerns follow-up ending or measurements no longer being collected; a patient can have informative visit timing without dropping out. A terminal event can also end the opportunity for later visits, but that fact alone does not establish that the preceding visit schedule depended on health. These mechanisms can occur together, and evidence about one does not automatically settle the others.
| Situation | What determines timing or observation? | Main interpretive issue |
|---|---|---|
| Fixed or independently assigned schedule | Protocol or randomized schedule, not evolving health | Irregular realized dates may occur without health-dependent scheduling |
| Health-related schedule | Symptoms, past measurements, or other health history | Observed visits can overrepresent the health histories that prompt them |
| Dropout | Follow-up ceases or observations are no longer collected | Later outcomes may be unavailable; this is distinct from visit spacing |
| Follow-up ended by a terminal event | The event removes the opportunity for later assessments | Do not confuse no later visit with a health-triggered visit schedule |
The comparison also indicates why simply counting visits or analyzing all recorded values as one pooled sample may be unsuitable for a target that gives each patient or each calendar time equal weight. A fixed schedule can simplify interpretation, but may be impractical or clinically inappropriate. When timing responds to health, an analysis may need to represent the observation process or make its target and assumptions explicit. This tutorial does not prescribe a particular model: the appropriate approach depends on what drives scheduling, what was measured, and the scientific quantity of interest.
Common Mistake and Practical Takeaway
Wrong reasoning: “Every patient was eventually measured, so visit timing cannot affect the analysis.” Correction: eventual measurement does not make the observation occasions equivalent. If worsening health leads to more frequent assessments, patients with that history may contribute more recorded values, as the visit-level calculation showed. Also, visits that occur at different times may capture different points on a changing trajectory.
The key distinction is not regular versus irregular dates, nor many versus few observations. Ask whether the rule or behavior that determines visit timing is related to the evolving health process relevant to the analysis, conditional on the design and variables being considered. Then state the target: a patient-level summary, a calendar-time trajectory, and a mean per recorded visit are not interchangeable.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A protocol randomizes patients to visits every 2 or every 8 weeks, with no health-based changes. Is the assigned schedule informative about evolving health? What could change that conclusion in practice?
- In a hypothetical cohort, 20% of patients have a score of 1 and 80% have a score of 6. Assessment probabilities are 0.10 and 0.50, respectively. Calculate the population mean and the mean among observed assessments.
- A clinician schedules an extra visit when the previous biomarker value is high. Why can this make the timing informative about current health even though the current value was not used to schedule the visit?
- All patients have the same number of visits, but some are seen early because of worsening symptoms. Does equal visit count establish a schedule independent of health?
Solutions. (1) The assigned schedule is independent of evolving health under the stated design. Health-based departures from assignment could make actual visit timing informative. (2) The population mean is \((0.20)(1)+(0.80)(6)=5.00\) points. The observed fractions are \((0.20)(0.10)=0.02\) and \((0.80)(0.50)=0.40\), total \(0.42\). Thus the observed mean is \(\{(0.02)(1)+(0.40)(6)\}/0.42=2.42/0.42=5.762\) points. (3) A previous value can predict the current value; if the underlying health process changes over time, scheduling based on that history can make visit timing related to current health. (4) No. Equal counts do not remove dependence of timing on health: the calendar dates can still be chosen in response to symptoms. The relevant question is how the timing was determined, not only how many visits occurred.