Exogenous And Endogenous Observation Times

Why the Visit Schedule Is Part of the Data-Generating Process

Longitudinal measurements are made at particular times, and those times may be chosen by a protocol, by practical constraints, or in response to a patient's health. The distinction matters because a set of observed measurements can represent a different slice of the underlying health trajectories when sicker patients are measured more often. In that situation, the observation schedule is not just a neutral calendar attached to the outcome data.

A regular schedule is not automatically exogenous, and an irregular schedule is not automatically endogenous. The key question is whether the timing process is conditionally independent of the patient's outcome process, given the factors specified as conditioning information. This is a statement about how the schedule is generated, not simply about how often visits occur.

Definition: Let $Y_i(t)$ denote the underlying longitudinal outcome for individual $i$ at time $t$, and let $N_i(t)$ count that individual's visits up to time $t$. Let $C_i$ denote relevant baseline or design variables. The schedule is exogenous conditional on $C_i$ if the entire visit process and the entire outcome trajectory are conditionally independent: $$ N_i(\cdot)\perp Y_i(\cdot)\mid C_i. $$ It is endogenous relative to $C_i$ when the conditional distribution of the visit process depends on the outcome trajectory or on health information that is associated with that trajectory. The classification is always relative to the stated conditioning variables and follow-up period.

Here, $Y_i(\cdot)$ means the complete trajectory over the period under consideration, including values that were not measured. The definition is therefore stronger than saying that visit times are unrelated to the measurements already recorded. A schedule can depend on a patient's current or past health even when the future outcome values remain unseen.

Conditional independence is commonly expressed by a factorization. For any schedule event $A$ and outcome-trajectory event $B$, exogeneity requires $$ P\{N_i(\cdot)\in A,\;Y_i(\cdot)\in B\mid C_i\} = P\{N_i(\cdot)\in A\mid C_i\} P\{Y_i(\cdot)\in B\mid C_i\}. $$ This condition must hold for all such events, not only for one visit or one summary of health. In practice, the complete trajectory is not observed, so the condition is primarily a model and design question rather than something verified by a single data check.

Protocol-Driven and Health-Driven Timing

Suppose a study schedules assessments at months 1, 3, and 6 for every participant, with the same calendar rules regardless of health. If the schedule is indeed generated by those rules, it is exogenous conditional on any relevant design factors, such as study site or randomized schedule assignment. The outcomes may still differ greatly between participants; exogeneity says that those differences do not determine the visit process after conditioning on the specified factors.

By contrast, a clinician may arrange an extra assessment when a patient's symptoms worsen or a marker appears concerning. The chance and timing of a visit then depend on health. Even if the trigger is a recorded past measurement, the resulting schedule depends on the outcome history. If visits are prompted by a current condition that is not fully observed, timing may depend on latent health as well.

The conditioning set is essential. If site-specific protocols cause different visit schedules, the schedule may be independent of outcomes conditional on site but associated with outcomes marginally if site is related to health. Likewise, a schedule that depends on an unmeasured severity factor may appear to depend on the outcome process even if it would be independent conditional on that factor. The relevant question is not whether timing and health are associated in an unadjusted table, but whether the proposed conditioning information adequately accounts for how timing is generated.

Worked Example: A Randomized Assessment Calendar

In a hypothetical study, 100 participants are assigned before follow-up to a calendar that determines whether they receive a month-3 assessment. The assignment is generated independently of their health trajectories, conditional on study site. For illustration, among 50 participants with a high underlying month-3 outcome and 50 with a low underlying outcome, 30 in each group receive the assessment.

Let $V=1$ indicate a month-3 assessment and $H$ denote the underlying outcome group. The observed conditional probabilities are $$ P(V=1\mid H=\text{high})=\frac{30}{50}=0.60,\qquad P(V=1\mid H=\text{low})=\frac{30}{50}=0.60. $$

For this binary summary, the joint probabilities factor: $P(V=1,H=\text{high})=30/100=0.30$, while $P(V=1)P(H=\text{high})=0.60(0.50)=0.30$. The same equality holds for the other cells. The design mechanism, not these sample counts alone, justifies treating the schedule as exogenous conditional on site. Equal observed rates are consistent with exogeneity, but do not prove independence of the full visit process from the full outcome trajectory.

Worked Example: Extra Visits Triggered by Current Symptoms

In a hypothetical clinic, patients with severe symptoms at a scheduled contact are offered an additional assessment during the following week. Suppose the additional assessment occurs for 45 of 60 patients with severe symptoms and for 8 of 40 patients without severe symptoms.

Writing $V=1$ for an additional assessment and $S$ for symptom status gives $$ P(V=1\mid S=\text{severe})=\frac{45}{60}=0.75,\qquad P(V=1\mid S=\text{not severe})=\frac{8}{40}=0.20. $$ The risk difference is $0.75-0.20=0.55$, or 55 percentage points, and the ratio of these probabilities is $0.75/0.20=3.75$. Thus, an additional assessment is substantially more likely in the severe-symptom group in this hypothetical data.

Because current symptoms influence the schedule, timing is endogenous relative to the outcome process that includes symptoms, unless the relevant dependence is accounted for in the conditioning set. A visit's occurrence is not simply a random selection of times at which to observe health. The calculation shows a difference in visit probabilities; by itself it does not establish that symptoms caused the visits, since other factors could also contribute.

Latent Health Can Link Visits and Outcomes

The health feature that prompts contact need not be measured in the analysis data. A clinician may respond to functional decline, an unrecorded symptom, or an overall impression of worsening health. Such a feature can also be related to the longitudinal outcome. The observed visit process and outcome trajectory can then be dependent even when the specific trigger is absent from the recorded measurements.

The following calculation illustrates the association created by a shared latent health factor. Let $U=1$ indicate severe latent health and $U=0$ indicate less severe latent health. Suppose $P(U=1)=0.40$, a marker value at a particular time is $Y=10+5U$ units, and the probability of a visit during a short window is 0.80 when $U=1$ and 0.20 when $U=0$.

Worked Example: Visits Associated With a Latent Health Factor

Let $V=1$ indicate a visit during the window. First compute the overall probability of a visit: $$ P(V=1)=0.40(0.80)+0.60(0.20)=0.32+0.12=0.44. $$ The probability that a visitor has severe latent health is $$ P(U=1\mid V=1)=\frac{P(V=1\mid U=1)P(U=1)}{P(V=1)} =\frac{0.80(0.40)}{0.44} =\frac{0.32}{0.44}\approx 0.7273. $$

For a person without a visit, $P(V=0)=1-0.44=0.56$, so $$ P(U=1\mid V=0)=\frac{P(V=0\mid U=1)P(U=1)}{P(V=0)} =\frac{0.20(0.40)}{0.56} =\frac{0.08}{0.56}\approx 0.1429. $$ Thus, severe latent health is much more prevalent among those visited during the window.

The marker is $Y=15$ units when $U=1$ and $Y=10$ units when $U=0$. Its mean is $E(Y)=0.40(15)+0.60(10)=12$ units. The joint mean is $E(VY)=0.40(0.80)(15)+0.60(0.20)(10)=4.8+1.2=6$ units. Therefore, $$ \operatorname{Cov}(V,Y)=E(VY)-E(V)E(Y)=6-(0.44)(12)=0.72 $$ in marker units. This nonzero covariance arises because the latent factor is related to both the marker and the visit. The schedule is endogenous relative to the recorded marker when $U$ is not included in the conditioning information.

A Short Justification: Why Unequal Conditional Probabilities Matter

For a binary visit indicator $V$ and a binary health status $H$, independence conditional on $C$ requires the visit probability to be the same across health statuses within each value of $C$, where both statuses have positive probability. To see this, suppress $C$ temporarily. Independence implies $$ P(V=1,H=h)=P(V=1)P(H=h). $$ Dividing both sides by $P(H=h)>0$ gives $$ P(V=1\mid H=h)=P(V=1). $$ The right side does not depend on $h$, so $P(V=1\mid H=1)=P(V=1\mid H=0)$. Conversely, if those conditional visit probabilities are equal, call their common value $q$, then $P(V=1,H=h)=qP(H=h)=P(V=1)P(H=h)$; the corresponding equality for $V=0$ follows by subtracting from $P(H=h)$. The joint probabilities therefore factor for both values of $V$ and $H$.

This argument explains why conditional visit probabilities are a useful way to describe a health-dependent schedule. It does not establish independence of full processes from equality at one time point: independence of $N_i(\cdot)$ and $Y_i(\cdot)$ requires the factorization for events involving the entire schedule and trajectory.

How to Classify a Schedule—and What the Classification Does Not Say

Specify the time window and outcome trajectory.
State which follow-up period and which health outcomes are relevant. A schedule may be independent of one outcome and dependent on another.

State the conditioning information.
Include design variables or other factors that genuinely determine the schedule. The meaning of “exogenous” changes when the conditioning set changes.

Describe how visits are generated.
Check the protocol and practice: are contacts fixed in advance, randomized independently of health, or prompted by symptoms, measurements, or clinical judgment?

Consider unrecorded health information.
Ask whether latent severity or another unmeasured feature could affect both visits and outcomes. A lack of association with recorded measurements does not rule out this possibility.

Separate classification from evidence.
Describe the schedule mechanism and the evidence supporting that description. Observed associations can flag possible dependence, but observed data alone rarely verify independence of complete processes.

Several distinctions prevent common misclassification. A schedule can be irregular because of weekends, site logistics, or randomized assessment dates and still be exogenous conditional on those design features. Conversely, a perfectly regular schedule can be endogenous if patients who deteriorate are systematically given more frequent appointments. A schedule that depends on a past measured outcome is still outcome-dependent timing, even though the triggering value was observed.

Endogenous timing is not synonymous with measurement error, dropout, or a terminal event. Those processes can coexist with visit timing, but the classification here concerns dependence between the observation schedule and the longitudinal health trajectory. Nor does identifying endogenous timing prove that an observed association is causal: a common health factor, clinical decision, or design feature may explain the dependence.

Diagnostics, Alternatives, and Limitations

A useful first check is to compare visit occurrence or frequency across clinically meaningful health groups, while accounting for known design factors. A strong pattern—such as the difference in the symptom example—raises concern that the schedule depends on health. Review of protocol documents, visit logs, reasons for contact, and clinician decision rules can clarify whether this pattern is expected by design. Such checks are informative but incomplete: unmeasured health can drive timing without leaving a clear pattern in recorded outcomes, and a recorded association may reflect an omitted design factor.

The main alternative is to treat observation times as fixed or exogenous. That is appropriate when the study design supports conditional independence, not merely because a simpler analysis is convenient. When contacts are triggered by health, the schedule should instead be recognized as a health-dependent part of the data-generating process. This tutorial's classification does not prescribe a particular model or adjustment procedure; those require additional assumptions about the visit process and the information available at each time.

The limitation is fundamental: the full $Y_i(\cdot)$ trajectory is not observed between visits, so independence from that trajectory cannot generally be confirmed from the observed measurements alone. A defensible classification combines design knowledge with careful description of observed patterns and explicit acknowledgment of what may be latent. The earlier material on informative visit timing motivates why this distinction matters; the essential refinement here is to state the dependence claim precisely and condition it on a clearly named set of variables.

Common pitfall: “Visits are irregular, so timing must be endogenous.” Irregularity describes the spacing of visits, not their relationship to health. A randomly assigned irregular calendar can be exogenous, while regular visits supplemented by symptom-triggered contacts can be endogenous. Classify the schedule by its conditional dependence on the outcome process, not by its appearance on a calendar.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A study assigns each participant a different assessment calendar by randomization before follow-up. The calendar is not affected by health. What condition supports classifying timing as exogenous, and what should be conditioned on?
Among 80 patients with worsening symptoms, 56 receive an extra visit; among 120 patients without worsening symptoms, 24 receive one. Calculate both visit probabilities and their difference. What does the contrast suggest?
In a hypothetical latent-health setting, $P(U=1)=0.25$, $P(V=1\mid U=1)=0.60$, and $P(V=1\mid U=0)=0.20$. Calculate $P(V=1)$ and $P(U=1\mid V=1)$. What does this say about the composition of the visited group?
Two groups have the same average number of visits, but one group tends to receive visits during periods of worsening health. Does equality in average visit counts establish exogeneity? Explain.
Give one example of an irregular but exogenous schedule and one example of a regular but endogenous schedule.

Solutions. (1) The schedule should be conditionally independent of the full outcome trajectory given the factors governing assignment, such as site and randomized calendar assignment. Randomization before follow-up supports this condition; the relevant design variables should still be stated. (2) The probabilities are $56/80=0.70$ and $24/120=0.20$. Their difference is $0.70-0.20=0.50$, or 50 percentage points. This suggests health-dependent timing, though the contrast alone does not rule out other explanations. (3) $P(V=1)=0.25(0.60)+0.75(0.20)=0.15+0.15=0.30$. Then $P(U=1\mid V=1)=0.25(0.60)/0.30=0.15/0.30=0.50$. Although only 25% of the full group has $U=1$, 50% of those visited do, so the visited group is enriched for $U=1$. (4) No. Equal average counts do not show that visit times are independent of health; timing within the follow-up period may still respond to worsening health. (5) Visits on randomly assigned, uneven calendar dates can be irregular but exogenous. Regular monthly visits plus extra contacts triggered by symptoms can be endogenous.