Choosing A Visit-Process Model For Irregular Follow-Up

Why the Visit-Process Model Is a Scientific Choice

Irregular follow-up can arise for very different reasons. Visits may occur at an approximately stable background rate, become more frequent after a recorded symptom or biomarker result, or be more common for people with an unobserved characteristic that also affects their biomarker trajectories. These mechanisms imply different statistical models, even when the observed visit-time records look similar.

Choosing a formulation is therefore not just a question of which model fits best. It requires asking what is known about the scheduling mechanism, what information was recorded between visits, and whether unobserved subject characteristics could plausibly connect monitoring to the outcome process. The earlier material on likelihood contributions from observed visit times and conditions for ignoring the visit process provides useful context: modeling visits explicitly is especially important when their timing carries information about the outcomes or when the scientific target concerns the observation mechanism itself.

Definition: For subject \(i\), let \(N_i(t)\) be the number of visits recorded by time \(t\), and let \(t_{ij}\) denote the time of the \(j\)th visit. The visit-process intensity \(\lambda_i(t\mid\mathcal{H}_i(t-))\) is the instantaneous rate of a visit at time \(t\), conditional on the information \(\mathcal{H}_i(t-)\) available just before \(t\). In a small interval of length \(dt\), the conditional probability of a visit is approximately \(\lambda_i(t\mid\mathcal{H}_i(t-))dt\); the probability of two or more visits is of smaller order than \(dt\).

The history immediately before \(t\) is important: a model for a visit at \(t\) may use past observations, but not a biomarker value first measured at that same visit as if it were already known. Each formulation below describes a different set of conditioning variables and makes assumptions about what remains associated with visits after conditioning.

Three Formulations and What They Assume

A fixed-rate formulation says that visits follow a specified rate that does not depend on a subject’s evolving observed outcome history or an unobserved subject effect. A common version is a Poisson process with constant intensity, possibly conditional on baseline covariates:

\lambda_i(t\mid Z_i)=\lambda_0(t)\exp(\gamma^\mathsf{T}Z_i).

Here \(Z_i\) contains recorded covariates, \(\lambda_0(t)\) is a baseline rate, and \(\gamma\) describes covariate associations with the visit rate. A constant-rate model is the special case in which \(\lambda_0(t)=\lambda_0\). Conditional on the specified covariates, the model assumes that visits do not depend on unmodeled recent outcomes or latent characteristics relevant to the outcome. An administrative schedule is not automatically a Poisson process: if visits are set at fixed appointment times, it may be more appropriate to represent those opportunities directly or condition on the schedule rather than impose a continuous-time random-rate model.

A history-dependent formulation allows the rate to depend on observed information available before the visit:

\lambda_i(t\mid\mathcal{H}_i(t-)) =\lambda_0(t)\exp\{\gamma^\mathsf{T}Z_i+\alpha^\mathsf{T}g(\mathcal{H}_i(t-))\}.

The function \(g\) might summarize the most recent recorded symptom, the previous biomarker measurement, or time since the last visit. The chosen summaries must be defined using past information and justified by the scientific mechanism. This model is appropriate when observed history is thought to trigger or otherwise predict monitoring, and that history is available to the analyst. It does not, by itself, account for dependence caused by an unobserved factor after conditioning on the recorded history.

A shared-latent-effect formulation connects the visit process and longitudinal outcome through a subject-specific latent quantity, often a random effect \(b_i\):

\lambda_i(t\mid b_i,Z_i) =\lambda_0(t)\exp(\gamma^\mathsf{T}Z_i+\xi^\mathsf{T}b_i).

In a joint model, the outcome distribution also depends on \(b_i\). The effect can represent an unobserved subject characteristic that influences both the outcome trajectory and monitoring frequency. This approach requires a specified distribution for \(b_i\), a model for how it enters each component, and an integration over its uncertainty. Conditional on \(b_i\) and the modeled covariates, the formulation commonly assumes that the visit process and outcome process are independent, unless additional dependence is explicitly included. That conditional-independence assumption is substantive, not guaranteed by using a random effect.

Worked Implications for Hypothetical Follow-Up

Worked Example: A Fixed Background Visit Rate

Suppose a hypothetical study follows each participant for 8 months. After accounting for baseline covariates, assume visits follow a Poisson process with constant rate \(\lambda=0.25\) visits per month. This assumption means that the conditional rate is stable over follow-up and does not depend on prior outcomes or a latent subject effect. It is plausible for this calculation only if the monitoring mechanism is consistent with those conditions.

The expected number of visits in 8 months is the rate multiplied by exposure time:

E\{N_i(8)\}=0.25\times 8=2.00\text{ visits}.

For a constant-rate Poisson process, the probability of no visits over the interval is \(\exp(-\lambda t)\). Thus,

P\{N_i(8)=0\}=\exp(-0.25\times 8)=\exp(-2)=0.1353.

Under this model, the probability of at least one visit is \(1-0.1353=0.8647\). These are model-based probabilities, not evidence that a fixed-rate model is scientifically adequate. If actual appointments were scheduled in advance at specific months, the Poisson assumption would need justification or replacement by a schedule-based representation.

Worked Example: A Recorded Symptom Predicts Visits

Consider hypothetical monitoring in which a pre-visit symptom score \(S_i(t-)\) is recorded in the clinical history. Suppose the proposed intensity is \(\lambda_i(t)=0.20\exp\{0.35S_i(t-)\}\) visits per month. The model assumes the score is known before the next visit, the log-rate is linear in the score, and other relevant dependence is adequately controlled or absent.

At score 1, the estimated rate is

0.20\exp(0.35\times1)=0.20\exp(0.35)=0.2838\text{ visits per month}.

At score 3, it is

0.20\exp(0.35\times3)=0.20\exp(1.05)=0.5715\text{ visits per month}.

The rate ratio comparing score 3 with score 1 is \(\exp\{0.35(3-1)\}=\exp(0.70)=2.014\). Thus, under this model, a two-point higher recorded symptom score is associated with about 2.01 times the instantaneous visit rate, conditional on the model’s other terms. This is an association, not a causal effect of symptoms on visits. If scores are observed only at visits, the analyst also needs to define precisely which prior score is carried forward and for how long; the unobserved symptom level between visits cannot be treated as directly recorded history.

Worked Example: A Shared Subject Effect

Suppose a hypothetical joint model includes a subject effect \(b_i\) in both the biomarker trajectory and the visit rate. Let the visit intensity be \(0.15\exp(0.60b_i)\) visits per month, with \(b_i\) measured on the model’s latent scale. Assume, for illustration, that two subjects have latent effects \(b_i=1\) and \(b_i=-1\). The shared-effect model is justified only if a latent factor is scientifically plausible, its distribution is specified, and the outcome model also includes the shared effect.

For \(b_i=1\), the conditional rate is \(0.15\exp(0.60)=0.2733\) visits per month. For \(b_i=-1\), it is \(0.15\exp(-0.60)=0.08232\) visits per month. Their conditional rate ratio is

\frac{0.15\exp(0.60)}{0.15\exp(-0.60)} =\exp(1.20)=3.320.

The subject with the higher latent effect has a higher modeled rate, and—because the same effect enters the outcome model—also differs in the outcome distribution. These rates are conditional on hypothetical latent values, which are not observed data. Estimation must account for uncertainty about \(b_i\), rather than assigning each subject a known effect.

Why the Likelihood Distinguishes the Choices

For a visit process with conditional intensity \(\lambda_i(t)\) over an observed follow-up interval \([0,\tau_i]\), the visit-time likelihood contribution has the form

L_{N,i}=\left\{\prod_{j=1}^{n_i}\lambda_i(t_{ij})\right\} \exp\left\{-\int_0^{\tau_i}\lambda_i(u)\,du\right\}.

Here \(n_i=N_i(\tau_i)\) is the observed number of visits. The product rewards intensity at observed visit times; the integral term accounts for the time at which no visit occurred. The expression assumes the intensity is correctly specified over the observed risk interval, visit times are recorded as assumed, and the process has no unmodeled simultaneous jumps. Different formulations substitute different conditioning information into \(\lambda_i(t)\), so they make different predictions both for observed visit times and for intervals without visits.

With a shared latent effect, the visit likelihood is conditional on \(b_i\), which is unobserved. A joint contribution must average over its distribution. If \(L_{Y,i}(b_i)\) is the outcome contribution and \(f(b_i)\) is the latent-effect density, then a typical joint contribution is

L_i=\int L_{Y,i}(b_i)L_{N,i}(b_i)f(b_i)\,db_i.

This integral explains why a shared-effect model is not merely a history-dependent model with an extra measured covariate. The latent effect must be inferred from both the observed outcomes and visits and integrated out. A marginal visit rate obtained by averaging over the latent distribution is generally not enough to reproduce the full joint likelihood: after seeing a subject’s visit pattern and outcomes, the conditional distribution of that subject’s latent effect changes.

A Practical Basis for Choosing

Start with the design.
Determine whether visits were scheduled, generated by a background monitoring mechanism, or initiated in response to symptoms or previous measurements. Distinguish scheduled opportunities from realized visits, including missed appointments where that distinction is recorded.

Inventory available history.
If a recorded, pre-visit history plausibly predicts monitoring, consider a history-dependent intensity. Check that its values and timing are actually observed and that the proposed summaries are defined without using future information.

Ask what remains unexplained.
If subject-specific latent characteristics could plausibly influence both outcomes and visit frequency, a shared-effect formulation may be scientifically appropriate. Its outcome model, latent-effect distribution, and conditional-independence assumptions must also be defensible.

Check whether the data can distinguish models.
Look for enough subjects, visits, variation in observed histories, and follow-up time to estimate the proposed dependencies. Compare model implications and diagnostics, but do not treat fit statistics as a substitute for a defensible mechanism.

Useful diagnostics include comparing observed and model-predicted visit counts, checking whether visit rates vary across time or recorded history in ways the model misses, and examining time-rescaling residuals when the intensity is specified. Simulation from the fitted model can reveal whether it reproduces visit-count distributions and patterns over follow-up. These checks can uncover misspecification, but none proves that the selected mechanism is correct: different models can generate similar observed records, especially when measurements are sparse.

The main alternatives answer different questions. A fixed-rate model is parsimonious and can be appropriate when monitoring is plausibly exogenous after measured covariates, but it cannot represent triggers omitted from those covariates. A history-dependent model describes dependence on recorded, evolving information, but it may not capture residual subject-level association. A shared-latent-effect model represents one structured form of residual dependence, but demands stronger distributional assumptions and more information for estimation. These models are not interchangeable, and a more elaborate model is not automatically preferable.

Identification, Limitations, and a Common Pitfall

With few visits per person, the data may not reveal whether frequent monitoring reflects recent observed symptoms, a stable latent tendency, or random variation. This weak identification is particularly concerning when the history is measured only at visits: the measured record may omit the very fluctuations that prompted a visit. A shared-effect model may then produce estimates that depend substantially on its distributional and functional-form assumptions. Sensitivity analysis across scientifically plausible formulations is useful when the mechanism is uncertain.

A frequent mistake is to choose a fixed-rate model because the average number of visits appears similar across subjects. Similar averages do not establish independence from outcome history or latent subject characteristics. The correction is to ask whether visit occurrence varies with pre-visit history, design features, or plausible shared causes, and then assess whether the study has enough information to represent those mechanisms. Conversely, a model should not be made history-dependent by using a measurement recorded at the visit to predict that same visit: the predictor must be available before the event being modeled.

No visit-process formulation can recover unrecorded triggers without assumptions, and model selection cannot turn observational association into causal evidence. The choice should be reported as a transparent working model, with its conditioning set, time scale, assumptions, and limitations stated. Where the data cannot distinguish plausible mechanisms, that uncertainty is part of the result rather than a reason to claim that one model has been established as true.

Key takeaway: Choose the simplest visit-process formulation that represents the scientifically plausible mechanism and is supported by the observed design and records: fixed-rate for conditionally exogenous monitoring, history-dependent for recorded pre-visit triggers, and shared-latent-effect for modeled residual subject-level dependence linking visits and outcomes.

Check Your Understanding

Answer each question, then compare with the solutions below.

A study has appointments at months 1, 3, and 6, with no symptom-triggered rescheduling. Why might a continuous-time fixed-rate Poisson model be a poor first representation?
A recorded symptom score measured at the last visit predicts the next visit. Which formulation directly represents this dependence, and what timing condition must be met?
In a shared-effect model, why is it generally not correct to estimate a subject’s latent effect as known and omit integration over its distribution?
Under a constant rate of \(0.10\) visits per month, what is the probability of no visits in 5 months, assuming a Poisson process?
Give one reason a history-dependent model may not adequately represent the association between visits and outcomes.

Solutions. (1) The appointments are predetermined opportunities, not necessarily events generated continuously at a random rate; represent the schedule or condition on it unless a Poisson approximation is justified. (2) Use a history-dependent formulation. The score must be observed before the visit whose intensity it predicts, and the rule for using the last score must be specified. (3) The latent effect is unobserved and uncertain; the likelihood must average the outcome and visit contributions over its specified distribution, as in \(\int L_{Y,i}(b_i)L_{N,i}(b_i)f(b_i)\,db_i\). (4) \(P\{N_i(5)=0\}=\exp(-0.10\times5)=\exp(-0.5)=0.6065\). (5) It may omit an unobserved subject characteristic that affects both monitoring and the outcome, leaving residual dependence after conditioning on recorded history.