Why the Observation Schedule Matters
A joint model uses longitudinal measurements and event-time information together. The measurement occasions are not merely labels attached to marker values: the decision about when to measure may itself contain information about a subject’s health. If every subject is measured at prespecified times, it is often reasonable to condition on those times and model the recorded marker values and event history. If visits are scheduled more often for subjects whose health is worsening, the observed schedule may also reveal something about the unobserved marker trajectory or event risk.
The issue is whether the observation schedule can be treated as given when forming the likelihood for the longitudinal and event outcomes. This is an assumption about how measurements are collected, not a consequence of fitting a shared-random-effects model. It should be justified by the study protocol and the scheduling process.
The definition is conditional: a schedule can depend on recorded information without being informative about what remains unobserved, provided that dependence is handled appropriately. Conversely, a schedule that looks regular may still be informative if a clinician quietly changes visit timing in response to symptoms that are not included in the model. The distinction is whether the schedule depends on information about the outcomes that is not already represented in the conditioning information.
Likelihood Factorization and the Omission Condition
Write \(T_i\) for event time, \(\delta_i\) for the event indicator, and \(X_i\) for modeled covariates. Let \(\theta\) denote parameters of the longitudinal and event submodels and the random-effects distribution. The outcome likelihood conditional on the schedule and random effects can be written \(p(y_i,T_i,\delta_i\mid S_i,b_i,X_i;\theta)\). If the schedule itself is modeled, write its distribution as \(p(S_i\mid b_i,X_i;\psi)\), where \(\psi\) denotes schedule-process parameters.
A factorization of the observed-data likelihood is then
A simple sufficient condition for conditioning on the schedule and omitting its model from likelihood-based inference about \(\theta\) is that the schedule distribution does not depend on \(b_i\) or on \(\theta\), given the modeled covariates. In that case,
so this factor can be taken outside the integral. If \(\psi\) is distinct from \(\theta\), it is constant with respect to estimation of \(\theta\), leaving the conditional likelihood
This is the precise basis for conditioning on observed visit times. The schedule must not change the distribution of the latent effects or outcomes in a way omitted from the model. In particular, conditional independence of measurements and event outcomes given \(b_i\) is not enough if the schedule itself depends on \(b_i\): the schedule factor then remains inside the integral and changes how the latent effects are weighted.
Worked Example: Prespecified Protocol Visits
Suppose a hypothetical protocol schedules marker measurements at weeks 0, 4, 8, and 12 for every enrolled subject, regardless of health status. Assume the protocol is followed and that event-related termination of follow-up is handled in the event-time likelihood. For a subject whose follow-up permits all four measurements, \(S_i=(0,4,8,12)\) is fixed by design.
Conditional on this schedule and the subject’s random effects, suppose the marker model is \(y_{ij}=m_i(t_{ij};b_i)+\epsilon_{ij}\), with \(\epsilon_{ij}\sim N(0,\sigma^2)\), and the event model has its specified conditional likelihood. The subject’s conditional outcome contribution is the product of the marker and event contributions at these fixed times. The protocol contributes no additional outcome information because it did not use the subject’s marker history or latent health to select the times.
For example, if the schedule probability under the protocol is \(p(S_i\mid X_i)=1\) for this planned schedule, the likelihood factorization gives \(1\times\int p(y_i,T_i,\delta_i\mid S_i,b_i,X_i;\theta)p(b_i\mid X_i;\theta)\,db_i\). The factor 1 does not alter the likelihood for \(\theta\). Conditioning on the schedule is justified by the design, not by the fact that the joint model includes random effects.
Adaptive Scheduling: When Observed History Is Enough
Some studies deliberately schedule a follow-up measurement based on a value already observed. This is not automatically a violation of the noninformative-schedule assumption. If the decision uses only recorded history and a known scheduling rule, then, conditional on that history, it need not reveal additional information about the unobserved future marker process or event outcome. In likelihood terms, a schedule factor based only on the observed past can be conditioned on when its mechanism is distinct from the outcome parameters and does not depend on unobserved information.
This qualification matters. A clinician may act on a symptom, examination, or informal judgment not recorded in the analysis data. Calling the process “adaptive” does not make it ignorable: if the information driving the decision is absent from the model and predicts the marker trajectory or event risk, visit timing can be informative. Similarly, a schedule rule based on an unobserved latent severity is not made harmless merely because the decision is operationally understandable.
Worked Example: A Visit Triggered by a Recorded Marker
Consider a hypothetical study in which a second visit is scheduled if the first recorded marker value is at least 10 units. The rule is fully documented: after \(y_{i1}\geq 10\), an extra visit occurs with probability 0.90; after \(y_{i1}<10\), it occurs with probability 0.10. A subject has \(y_{i1}=11.2\) units and receives the extra visit.
Given the recorded first value, the schedule factor is \(P(\text{extra visit}\mid y_{i1}=11.2)=0.90\). For inference about the marker and event parameters, this known factor does not vary with \(\theta\), provided the rule uses only the recorded first value and the documented randomization and does not depend on unrecorded clinical information. Thus, conditional on \(y_{i1}\) and the realized schedule, the outcome likelihood for \(\theta\) can omit the factor 0.90.
The assumption does not say the schedule is statistically unrelated to all observed marker values: it plainly depends on \(y_{i1}\). It says the schedule adds no information beyond the recorded history used by the rule. If the extra visit was instead prompted by an unrecorded symptom that also predicts event risk, the stated justification would fail.
Why Dependence on Latent Effects Is Different
Suppose visit frequency depends on \(b_i\), such as a latent severity effect shared by the marker and event submodels. Even if the marker and event outcomes are conditionally modeled given \(b_i\), the schedule now changes the distribution of \(b_i\) among subjects with a particular observed schedule. The schedule factor cannot generally be removed from the integral. Ignoring it may therefore distort estimation of both the marker trajectory and the association between the marker process and event risk.
Worked Example: Scheduling Depends on Latent Severity
For a transparent hypothetical calculation, let \(b_i\) take values \(+1\) and \(-1\), each with prior probability 0.50. Let \(Y_i\) be a binary marker measurement with \(P(Y_i=1\mid b_i=+1)=0.80\) and \(P(Y_i=1\mid b_i=-1)=0.20\). Suppose the observed schedule category \(S_i=2\) (two visits) occurs with probability 0.80 when \(b_i=+1\), and with probability 0.20 when \(b_i=-1\).
The probability of observing two visits is \(P(S_i=2)=0.50(0.80)+0.50(0.20)=0.50\). Bayes’ rule therefore gives \(P(b_i=+1\mid S_i=2)=0.50(0.80)/0.50=0.80\); the schedule has shifted the latent-effect distribution from 0.50 to 0.80 for \(b_i=+1\).
For a subject with two visits, the correct conditional probability of \(Y_i=1\) is \(P(Y_i=1\mid S_i=2)=0.80(0.80)+0.20(0.20)=0.68\). If the schedule is ignored and the original 0.50/0.50 mixture is used, the calculation is \(0.50(0.80)+0.50(0.20)=0.50\). The discrepancy, 0.68 versus 0.50, is not a minor bookkeeping issue: it results from using the wrong mixture over latent severity. A model for the schedule, or a defensible alternative that accounts for its dependence on observed information, is needed.
Justification and Practical Checks
The omission result follows directly from factorization. Under a schedule mechanism independent of \(b_i\) and \(\theta\) given \(X_i\),
For fixed observed \(S_i\), the first factor is constant in \(\theta\), so maximizing the full likelihood over \(\theta\) gives the same result as maximizing the conditional likelihood. But if \(p(S_i\mid b_i,X_i)\) depends on \(b_i\), it cannot be taken outside the integral. That algebra identifies exactly which assumption is doing the work.
In practice, inspect the protocol and the operational record of visits, not just the observed times. Ask whether appointment times were fixed in advance; whether changes were driven by recorded past marker values; whether unrecorded symptoms or clinician judgment affected scheduling; and whether missed visits or attendance depended on health. Also distinguish the planned schedule from the visits actually observed. A fixed plan does not guarantee a noninformative observed schedule if attendance is selective.
When an event ends observation, there are naturally no later marker measurements. This structural truncation is not, by itself, evidence that the pre-event visit process was informative. The event-time contribution and the observation process must be described consistently, and the event and censoring assumptions are separate from the assumption about how visits are scheduled while a subject remains under observation.
| Schedule situation | Can the schedule usually be conditioned on? | Reason or concern |
|---|---|---|
| Times fixed by protocol | Yes, if the protocol is followed and the outcome model is otherwise appropriate | Times do not select subjects based on their health or latent effects |
| Extra visits based only on recorded past outcomes | Often, under a known, distinct scheduling rule | Condition on the observed history that drives the decision |
| Visits prompted by unrecorded symptoms or latent severity | Not without further justification | The schedule may reveal unmodeled information about outcomes or random effects |
| Missed visits or attendance related to health | Not automatically | The realized measurement occasions may be selected, even if planned times were fixed |
Diagnostics, Alternatives, and Limitations
Useful diagnostics begin with process knowledge. Tabulate visit counts and gaps by baseline covariates, prior recorded marker values, and observed health indicators. Examine whether visits cluster just before events or whether changes in visit frequency follow recorded marker changes. Such patterns are warnings to investigate, not proof that the schedule is informative: protocol-driven timing or event truncation can create patterns for other reasons. Conversely, an absence of visible association does not establish independence from latent severity or unrecorded information.
If informative scheduling is plausible, the main alternatives are to model the visit process jointly with the marker and event processes, to use a design or weighting approach justified for the specific observation mechanism, or to conduct sensitivity analyses under plausible degrees of schedule dependence. A joint visit-process model can include observed predictors and latent effects, but it requires adequate information and additional assumptions about that process. Weighting methods likewise require a correctly specified observation mechanism and sufficient positivity: relevant histories must have a nonzero chance of receiving measurements. None of these options repairs an unknown, unrecorded scheduling mechanism automatically.
The limitation is fundamental: the observed data alone usually cannot verify that scheduling is independent of latent health. The strongest support may come from a protocol that fixed times before outcomes were known and documentation showing that deviations were unrelated to unobserved health. If scheduling depends on measured past values, a conditional analysis may be reasonable under the stated conditions; if it depends on latent severity or omitted clinical information, treating times as fixed can bias the outcome model.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A protocol fixes visits at months 0, 3, and 6. What makes it reasonable to condition on these times?
- A documented rule schedules an extra visit using only the last recorded marker value. Is this automatically informative? State the needed qualifications.
- In the latent-severity example, why is it incorrect to retain the original 0.50/0.50 random-effects mixture among subjects observed at two visits?
- Does observing fewer measurements before an early event, by itself, prove that the visit schedule is informative? Explain.
- Name two practical warning signs that would motivate investigation of the scheduling assumption.
Solutions. (1) The times were set by protocol rather than selected according to health, marker values, or latent effects; the outcome likelihood can therefore condition on them if the protocol is followed and other model assumptions hold. (2) No. Conditioning can be justified when the rule uses only recorded history, its mechanism is known or appropriately represented, its parameters are distinct from the outcome parameters, and no unrecorded information or latent state drives the decision. (3) Because \(P(b_i=+1\mid S_i=2)=0.80\), not 0.50; the observed schedule changes the mixture over latent effects, yielding \(P(Y_i=1\mid S_i=2)=0.68\), rather than 0.50. (4) No. An event ends the opportunity for later measurements, so fewer pre-event observations may be structural; it is not alone evidence of a health-dependent visit process. (5) Examples include visit frequency changing after high recorded marker values, unexplained visits clustered before events, documented scheduling changes prompted by unrecorded symptoms, or attendance varying with health.