Conditions For Ignoring The Visit Process

Why the Visit Process May Matter

In longitudinal studies, biomarker values are often observed only when a visit occurs. If visits are more likely when a participant’s health is worsening, the observed records may not represent the outcome trajectory of interest. The key question is not whether visits are irregular, but whether, after accounting for what is already recorded, the visit process carries additional information about outcomes that have not yet been observed.

A visit schedule can be ignored for a specified outcome analysis only under conditions that make its contribution to inference separable from the outcome model. This is a conditional claim, not a property of a dataset in the abstract: the conditions may hold for one outcome estimand or model and fail for another. The earlier material on observation intensities and shared random effects provides useful context for expressing the dependence involved.

Definition: For participant $i$, let $N_i(t)$ count visits through time $t$, and let $Y_i^*(t)$ denote the outcome that would be available at time $t$ if it were measured. The full outcome trajectory is $Y_i^*=\{Y_i^*(t):0\leq t\leq \tau\}$. Let $X_i$ denote measured covariates, and let $\mathcal H_i^o(t-)$ be the observed history immediately before $t$: measured covariates, prior visit times, and outcome values recorded at those visits. The unobserved portion of $Y_i^*$ consists of trajectory values not recorded in the observed data.

The Conditional-Independence Condition

A standard sufficient condition for ignoring the visit process in an outcome likelihood is that, conditional on measured covariates and the observed history, the visit mechanism has no additional dependence on the full outcome trajectory. In words, once the recorded past is known, the schedule must not reveal further information about unobserved outcomes. This condition allows visits to respond to recorded information, such as a previously measured biomarker or a recorded symptom.

Condition: For the period and outcome analysis under consideration, require $$ p_\psi\{N_i \mid Y_i^*,X_i\} = p_\psi\{N_i \mid \mathcal H_i^o,X_i\}, $$ where the right-hand side is a visit mechanism depending only on observed history and measured covariates. In an intensity formulation, a corresponding full-trajectory condition is $$ \lambda_i^N(t\mid \mathcal H_i^o(t-),X_i,Y_i^*) = \lambda_i^N(t\mid \mathcal H_i^o(t-),X_i) $$ for times at risk. The condition concerns the full trajectory, not only the current outcome value.

The intensity equation is a local expression of the restriction: given the recorded past and covariates, the instantaneous visit rate cannot additionally depend on any part of the unobserved trajectory. Replacing $Y_i^*$ with only $Y_i^*(t)$ would be weaker. It could miss dependence on a latent factor that predicts a future outcome, even when the current outcome is unaffected by that factor.

This conditional-independence condition is not enough by itself to justify omitting the visit model from a likelihood. The outcome-model parameters and visit-model parameters must also be distinct: the same unknown parameter cannot govern both outcome and visit distributions in a way that links their estimation. The outcome model must be correctly specified for the target analysis, and the observation pattern must provide enough information to estimate that target. Ignoring a visit mechanism does not repair a misspecified outcome model or create measurements where there are none.

Why the Likelihood Can Factor

The justification is clearest from the observed-data likelihood. Suppose the full-data outcome density is $f_\theta(Y_i^*\mid X_i)$, with outcome parameter $\theta$. Under the condition above, the visit-process likelihood for a realized record depends on the recorded history, but not on missing trajectory values once that history is fixed. Denote this factor by $g_\psi(N_i\mid\mathcal H_i^o,X_i)$. The joint full-data contribution is then the outcome density multiplied by this visit factor.

L_i^{\mathrm{obs}}(\theta,\psi) = \int f_\theta(Y_i^*\mid X_i)\, g_\psi(N_i\mid\mathcal H_i^o,X_i)\,dY_{i,\mathrm{mis}}^* = g_\psi(N_i\mid\mathcal H_i^o,X_i) \int f_\theta(Y_i^*\mid X_i)\,dY_{i,\mathrm{mis}}^*.

Here $Y_{i,\mathrm{mis}}^*$ denotes the unobserved trajectory values, and the integral leaves a density for the observed outcomes. The second equality holds because $g_\psi$ does not depend on those missing values. If the parameters are distinct, the visit factor can be omitted when making likelihood-based inference about $\theta$, provided the outcome likelihood is formed for the stated observed-data analysis. This is the sense in which the visit process is ignorable; it does not mean that visits are random or that the visit factor is literally constant.

Worked Example: Visits Scheduled Independently of the Trajectory

Consider a hypothetical study in which each participant’s follow-up visit times are generated in advance by a scheduling system, independently of the participant’s outcome trajectory and measured covariates. The outcome is a biomarker measured at each completed visit. Suppose the scheduling parameters are separate from the biomarker-model parameters.

Conditional on $X_i$, the visit schedule has the same distribution for all possible values of $Y_i^*$. Thus $p_\psi(N_i\mid Y_i^*,X_i)=p_\psi(N_i\mid X_i)$, which is a special case of the condition that permits dependence only on observed history and covariates. Since the parameters are distinct, the observed likelihood factors into a schedule term and the outcome likelihood. The schedule term can be omitted when estimating the biomarker-model parameters.

This conclusion relies on the stated design: a schedule fixed in advance and not changed in response to health information. If staff instead add visits after seeing concerning measurements, independence from the trajectory no longer follows from the original scheduling plan. The actual visit mechanism must be assessed.

Worked Example: Dependence on Recorded History

In a hypothetical monitoring program, a participant has a recorded symptom score at month 2. The program schedules a month-3 visit if that score is at least 6 and otherwise schedules a month-6 visit. For this example, assume the rule is followed exactly, the score is fully recorded, and future visits do not depend on unrecorded symptoms or latent health status.

The rule makes visit timing dependent on prior outcome-related information, so visits are not independent of health history. But the trigger is part of $\mathcal H_i^o(t-)$: it is known before the next visit time is determined. Under the example’s assumptions, the visit mechanism depends on recorded history rather than additionally on the unobserved future trajectory. The conditional-independence condition can therefore hold.

For an outcome analysis that models the future biomarker conditional on the recorded month-2 score, the visit factor can be omitted if the outcome and scheduling parameters are distinct and the other likelihood assumptions hold. This does not imply that a simple average of measurements at all observed visits estimates the population mean at a fixed calendar time. The latter calculation may overrepresent people with certain recorded histories. The target, model, and conditioning information must be stated.

How Latent or Future Dependence Breaks the Condition

A visit can depend on information that is not in the recorded history, even if it does not appear to respond to the current measured outcome. That dependence makes the schedule informative about unobserved outcomes and invalidates the factorization above. In particular, a current-value check alone cannot rule out dependence on a latent predictor of future outcomes.

Worked Example: A Latent Trigger Reveals a Future Outcome

Consider a hypothetical population with a latent binary health state $U_i$, where $P(U_i=1)=0.5$. The current outcome is fixed at $Y_i^*(0)=5$ for everyone, but the future outcome is $Y_i^*(1)=10$ if $U_i=1$ and $Y_i^*(1)=0$ if $U_i=0$. A visit before time 1 occurs with probability 0.8 when $U_i=1$ and 0.2 when $U_i=0$. Assume no recorded history reveals $U_i$.

The overall probability of a visit is $$ P(\text{visit})=0.5(0.8)+0.5(0.2)=0.5. $$ By Bayes’ rule, among those with a visit, $$ P(U_i=1\mid\text{visit})=\frac{0.5(0.8)}{0.5}=0.8. $$ Their mean future outcome is therefore $0.8(10)+0.2(0)=8$, whereas the population mean future outcome is $0.5(10)+0.5(0)=5$.

The current outcome is constant, so conditioning on $Y_i^*(0)$ cannot remove the dependence between visits and $Y_i^*(1)$. The visit process depends on the latent $U_i$, which predicts the unobserved future outcome; with $U_i$ absent from $\mathcal H_i^o$, the full-trajectory condition fails. An intensity comparison based only on the current outcome could miss this failure.

Event-Time Selection Is Not the Same as Finite-Interval Inclusion

A related diagnostic pitfall is to treat the probability of at least one visit during a finite interval as proportional to the visit rate. That is not generally exact. If a constant Poisson rate is $\lambda$ over an interval of length $\Delta$, the probability of at least one visit is $1-\exp(-\lambda\Delta)$. Rate-proportional selection applies to sampling at an event time, or as a short-interval approximation, not automatically to finite-interval inclusion.

Worked Example: Event-Time Weighting Versus One-Month Inclusion

Suppose two equally common groups have outcome values 0 and 10, and constant visit rates of 1 and 3 visits per month, respectively. If a participant is selected at a visit event, the instantaneous event-time weighted mean is $$ \frac{1(0)+3(10)}{1+3}=\frac{30}{4}=7.5. $$

If instead participants are selected because they have at least one visit during a full month, their inclusion probabilities are $1-\exp(-1)\approx0.6321$ and $1-\exp(-3)\approx0.9502$. The mean outcome among included participants is $$ \frac{0.5(0.6321)(0)+0.5(0.9502)(10)} {0.5(0.6321)+0.5(0.9502)} = \frac{4.751}{0.7912} \approx 6.008. $$

Thus the event-time calculation gives 7.5, whereas finite one-month inclusion gives approximately 6.008. Neither calculation establishes that the visit process is ignorable; they describe different selection rules. Treating the finite-interval inclusion probability as proportional to the rate would give the wrong answer for this example.

Assessment, Alternatives, and Limitations

To assess whether a schedule can be ignored for a particular outcome analysis, first specify the target outcome and the relevant time period. Then ask what information determines visits, and whether every such trigger is recorded in $\mathcal H_i^o(t-)$ or $X_i$. A visit process driven by a previous measured value may satisfy the condition; one driven by an unrecorded symptom, an unobserved current biomarker, or a latent factor associated with future outcomes may not. The conditional-independence claim cannot generally be verified from observed data alone, because the crucial unobserved outcomes are unavailable at nonvisit times.

If the condition is plausible, an outcome likelihood that accounts for the actual observed measurements can be used without an explicit visit-process factor, subject to distinct parameters and adequate outcome-model assumptions. Inverse-intensity weighting is an alternative strategy that models the observation intensity and reweights observed contributions; it requires its own modeling and positivity conditions. A joint model can represent dependence between visit and outcome processes through shared latent variables or other specified structures, but introduces additional assumptions and parameters. These approaches are not interchangeable without regard to the target estimand and assumptions.

Even when visits are ignorable in the likelihood, the data may not identify an outcome at times or for covariate histories with virtually no observation. This is a positivity or support problem, not evidence that the visit process is informative. Likewise, a correctly specified conditional model for outcomes at observed times does not automatically justify an unadjusted mean across observed visits as an estimate of a fixed-time population mean.

Common mistake: “The visit rate depends only on the current measured outcome, so the schedule is ignorable.” This does not establish independence from the full future trajectory: a latent variable can affect future outcomes and visit rates while leaving the current outcome unchanged. The correction is to assess dependence on the full outcome trajectory conditional on recorded history, or to state and justify extra assumptions that make current-outcome dependence sufficient.

A useful conclusion is therefore conditional and specific: “For this outcome likelihood, the visit mechanism is ignorable given the recorded history and measured covariates, assuming distinct parameters, a correctly specified outcome model, and adequate observation support.” If those assumptions are doubtful, the appropriate response is to model or otherwise address the visit process, and to explain which assumptions that alternative requires.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A visit is scheduled in response to a biomarker value recorded at the previous visit. Does that fact alone violate the conditional-independence condition? State what else must be assumed.
Why is equality of the visit intensity conditional on the current outcome alone insufficient to establish independence from the full trajectory?
For a constant rate of 2 visits per month, calculate the probability of at least one visit in a 0.5-month interval.
State two assumptions, in addition to conditional independence, needed to justify omitting the visit-process factor in likelihood inference for outcome parameters.
Does an ignorable visit process guarantee that an unadjusted mean across observed visits estimates a population mean at a fixed time? Explain.

Solutions. (1) No. It may depend on recorded history, which is allowed; assume that visits have no additional dependence on unobserved or future outcomes given that history, and that the outcome analysis handles the recorded history appropriately. (2) A latent variable may predict future outcomes and affect visit intensity while being unrelated to the current outcome; conditioning on the current outcome then misses the dependence. (3) $1-\exp[-(2)(0.5)]=1-\exp(-1)\approx0.6321$, or about 63.21%. (4) The outcome and visit parameters must be distinct, and the outcome model must be correctly specified; adequate observation support for the target is also required. (5) No. Ignorability for a specified likelihood does not eliminate differences in who is observed at a given time, so the target and analysis must be aligned.