Visit Rates Driven By Latent Trajectories

Why Model Visits Through a Latent Trajectory?

A patient’s disease severity can change between clinical assessments. If worsening health prompts extra contact, then visits may become more frequent precisely when the underlying condition is changing, even if that condition is not measured at every visit. The observed schedule is therefore potentially informative about a disease trajectory that is only partially seen.

The key distinction is between a state that is observed at a visit and an underlying state that evolves continuously or frequently over follow-up. The earlier material on current-state-triggered visit intensities treats a visit rate as depending on a current state available to the model. Here the trigger is a latent trajectory: the disease state at time \(t\) may affect visits even when no measurement is taken at that time. A latent state is not automatically the same as the most recent recorded measurement.

A common model for patient \(i\)'s instantaneous visit rate is

Here \(X_i\) is a vector of covariates, \(\lambda_0^V(t)\) is a baseline visit intensity, and \(\alpha\) describes how the latent state is associated with the visit rate. The intensity is a rate per unit time, conditional on the specified trajectory and covariates. If \(\alpha>0\), higher latent values correspond to higher conditional visit rates; if \(\alpha<0\), they correspond to lower rates. This is an association in the model, not by itself evidence that changing the disease state would cause a change in visits.

Assumptions and Interpretation

An intensity model needs a well-defined follow-up interval and a visit-counting process. The latent trajectory must be specified on that interval, including times between recorded visits, and the model must yield a nonnegative, integrable intensity. In the log-linear form above, the exponential ensures nonnegativity; integrability over a finite interval is still needed for a finite expected number of visits. The trajectory and intensity must also be defined using information available up to time \(t\), so the rate at \(t\) does not depend on future values in a way that would make it non-predictable.

For a one-unit increase in \(M_i(t)\), holding time and covariates fixed, the conditional visit-rate ratio is \(\exp(\alpha)\). If \(M_i(t)\) is measured in a meaningful unit, this ratio is per one-unit increase in that latent scale. The interpretation depends on how the latent state is scaled and oriented: reversing which direction represents worse disease changes the sign of \(\alpha\).

The conditional intensity describes the rate given the latent trajectory. It does not imply that the trajectory is observed, nor does it by itself specify the full distribution of visit times. In particular, an intensity alone should not be treated as proof that visit counts have independent increments. When a calculation uses a conditional inhomogeneous Poisson process, that is an additional assumption and will be stated explicitly.

From a Trajectory to an Integrated Visit Rate

For a fixed latent path \(M_i(t)\), the expected number of visits over an interval \([a,b]\), conditional on that path and the model covariates, is the integrated intensity

This quantity accumulates the rate over time. A high rate for a short period and a moderate rate over a longer period can produce the same integrated intensity. If, additionally, visits form a conditional inhomogeneous Poisson process, then the conditional visit count on \([a,b]\) is Poisson with mean \(\Lambda_i^V(a,b\mid M_i,X_i)\). In that case, the probability of no visits is \(e^{-\Lambda_i^V}\), and the probability of at least one is \(1-e^{-\Lambda_i^V}\).

The no-visit formula follows directly from the Poisson probability at count zero. For \(N_i^V(b)-N_i^V(a)\) with conditional mean \(\Lambda_i^V\),

Thus \(P\{N_i^V(b)-N_i^V(a)\geq1\mid M_i,X_i\}=1-e^{-\Lambda_i^V}\). This probability calculation is not valid from an intensity alone if the conditional Poisson assumption has not been made.

Why Replacing the Latent State by Its Average Can Fail

When the trajectory is unobserved, it can be tempting to substitute its mean into the rate function. In a nonlinear model, this generally does not produce the average rate. For the log-linear model, conditional on covariates, \(E[\lambda_i^V(t)\mid X_i]=\lambda_0^V(t)e^{\gamma^\top X_i}E[e^{\alpha M_i(t)}\mid X_i]\), which is generally different from \(\lambda_0^V(t)e^{\gamma^\top X_i}e^{\alpha E[M_i(t)\mid X_i]}\).

The difference matters because visit patterns reflect both the relationship between visits and the latent state and the distribution of latent states in the population. A latent value at an observed visit does not reveal the whole trajectory for that patient, and people with different latent paths can contribute different amounts of follow-up at different rates.

Model Checks, Alternatives, and Limitations

A latent-trajectory visit model is useful when a plausible disease process may influence contact even between measurements. It also raises a challenge: the key predictor is not fully observed, so a fitted relationship can depend on how the trajectory is represented and inferred. A model that fits recorded measurements well may still describe the unobserved path poorly.

Useful checks include comparing observed visit counts and timing with patterns predicted across follow-up, examining whether discrepancies are concentrated in periods or patient groups, and assessing whether conclusions change under reasonable alternative trajectory specifications. Check that the modeled trajectory scale has a clinically interpretable direction and that fitted rates are plausible. Systematic underprediction of visits during apparent deterioration, for example, can indicate that the latent path or its association with visits is inadequate. Because the trajectory itself is partly unobserved, such checks cannot conclusively verify the trigger mechanism.

The main alternatives answer different questions. An externally scheduled visit model is appropriate when the schedule is fixed independently of evolving health, while a model based on an observed current state uses information actually available at that time. A latent-trajectory model allows the underlying health state to matter between recorded assessments, but requires assumptions about that unobserved state. If its assumptions are implausible or weakly supported by the data, a simpler visit model and sensitivity analysis may be more defensible than a highly detailed latent specification.

• Do not treat the latest recorded value as the true state throughout the gap. That substitution is an additional approximation, not a consequence of a latent-trajectory model.
• Do not infer a causal effect from \(\alpha\). A nonzero trajectory coefficient describes conditional dependence in the specified model; unmeasured factors may influence both disease progression and visits.
• Do not translate intensity into a visit probability without assumptions. The conditional Poisson formula requires a conditional Poisson process, not just a stated conditional intensity.
• Do not assume the latent trajectory is identified by visit times alone. Limited measurements or few visits may leave many plausible combinations of trajectory and visit dependence.