Visit Process Models With Treatment-Dependent Monitoring

Why Treatment Belongs in a Visit-Process Model

In a longitudinal study, treatment can change not only a participant’s health but also how often the participant is monitored. A protocol may require more frequent checks for one treatment arm, clinicians may intensify follow-up after a treatment change, or treatment assignment may be associated with a pattern of contact. If visit rates differ by treatment, a model that gives every participant the same monitoring rate can misrepresent how the biomarker records were collected.

The key distinction is between treatment-dependent monitoring and monitoring associated with a participant’s evolving health. A treatment variable can describe different visit rates across assigned arms or treatment states. But if symptoms or prior biomarker values also affect scheduling, treatment alone does not describe the whole observation process. The treatment coefficient is a conditional rate contrast under the specified model, not by itself evidence that treatment caused the visits or that monitoring is noninformative for the outcome.

Definition: For participant \(i\), let \(N_i^V(t)\) count the number of observed visits by time \(t\). The visit-process intensity \(\lambda_i^V(t\mid\mathcal{H}_i(t-))\) is the instantaneous rate of a visit at time \(t\), conditional on the observed history \(\mathcal{H}_i(t-)\) immediately before \(t\). It has units of visits per unit time. Formally, for a small interval of length \(h\), the conditional probability of a visit in that interval is \(\lambda_i^V(t\mid\mathcal{H}_i(t-))h+o(h)\), when the intensity is finite and the usual regularity conditions hold.

Here \(Z_i\) denotes treatment assignment at baseline, and \(A_i(t-)\) denotes treatment being received just before time \(t\). They need not be the same: a participant assigned to one arm can discontinue, switch, or otherwise depart from the assigned treatment. The history \(\mathcal{H}_i(t-)\) may also include past visits, measured health variables, and other observed predictors available before \(t\). Using information from before \(t\) makes the predictors predictable: they do not use future information to explain the current visit rate.

Models for Treatment-Dependent Monitoring

A simple model for monitoring that varies by randomized assignment is

\lambda_i^V(t\mid\mathcal{H}_i(t-)) = \lambda_0^V(t)\exp(\gamma_Z Z_i).

The baseline intensity \(\lambda_0^V(t)\) is the rate for the reference assignment when \(Z_i=0\). If \(Z_i\) is binary, \(\exp(\gamma_Z)\) is the ratio of the conditional visit intensities for assigned treatment versus reference, at the same time \(t\). This ratio compares rates in visits per unit time; it is not a difference in the number of visits, nor a rate ratio for a terminal event.

When treatment received can change, one can instead include its predictable current value:

\lambda_i^V(t\mid\mathcal{H}_i(t-)) = \lambda_0^V(t)\exp\{\gamma_A A_i(t-)\}.

This describes a rate contrast between treatment states, conditional on the other model terms. It does not automatically identify the effect of treatment on monitoring: treatment changes may be responses to worsening health, and those same health changes may influence visits. If both assignment and current treatment are relevant, both can enter the model. A time-varying assignment contrast can be represented by

\lambda_i^V(t\mid\mathcal{H}_i(t-)) = \lambda_0^V(t)\exp\{\gamma_Z Z_i+\gamma_{Zt}Z_i g(t)\},

where \(g(t)\) is a specified time function, such as follow-up time in years. The conditional intensity ratio for \(Z_i=1\) versus \(Z_i=0\) at time \(t\) is \(\exp\{\gamma_Z+\gamma_{Zt}g(t)\}\). The time function and baseline intensity must be chosen so that the model can represent plausible changes in monitoring over follow-up.

These specifications rely on a correctly defined visit process and appropriate observation of predictors: the recorded events must correspond to the visits being modeled, follow-up time must be measured consistently, and covariates used at time \(t\) must be available before that time. The model also requires a suitable intensity form and adequate overlap in treatment histories to estimate the desired contrasts. If visit times are not observed completely or the relevant predictors are missing, the fitted intensity may not represent the intended process.

Worked Examples

Worked Example: Comparing Visit Rates by Assigned Arm

Suppose a hypothetical two-arm study records visits per participant-year. A fitted model has reference-arm intensity \(0.50\) visits per year and assignment coefficient \(\hat{\gamma}_Z=0.40\), with \(Z_i=1\) for the treatment arm and \(Z_i=0\) for reference. Assume the specified model has a common baseline intensity at the time being compared and includes no additional predictors.

For the reference arm, the fitted conditional rate is \(\hat{\lambda}^V( t\mid Z=0)=0.50\) visits per year. For the treatment arm,

\hat{\lambda}^V(t\mid Z=1) = 0.50\exp(0.40) = 0.50(1.4918) \approx 0.7459 \text{ visits per year}.

The fitted rate ratio is \(\exp(0.40)\approx1.492\). Under this model, the treatment-assigned group has an estimated conditional visit rate about 49.2% higher than the reference group at the same modeled time. This describes the fitted monitoring pattern; it does not show that assignment caused the difference unless the design and analysis support that causal interpretation. The calculation also assumes the model’s reference rate and assignment effect are appropriate for the comparison.

Worked Example: Expected Visits After Treatment Starts

Consider a hypothetical participant whose treatment begins at year 1 and remains in effect through year 3. Suppose the visit intensity is \(0.400\) visits per year before treatment and is multiplied by \(\exp(0.600)\) after treatment. Assume the intensity is constant within each period, there are no other covariates or rate changes, and follow-up is complete through year 3.

For a counting process with intensity \(\lambda_i^V(t)\), the integrated intensity over follow-up is its compensator. Its expected count equals the expected integrated intensity; here the intensity is specified as deterministic within each period, so the expected count is the integral of that rate. During year 0 to year 1, the expected count is \(0.400\times1=0.400\) visits. During years 1 to 3, the rate is

0.400\exp(0.600) = 0.400(1.8221188\ldots) = 0.7288475\ldots \text{ visits per year}.

That rate applies for two years, giving \(2(0.7288475\ldots)=1.4576950\ldots\) expected visits after treatment starts. The total expected count is therefore

0.400(1)+2\{0.400\exp(0.600)\} = 0.400+1.4576950\ldots = 1.8576950\ldots \approx1.858 \text{ visits}.

This is an expected count, not a prediction that this particular participant will have 1.858 visits. The calculation depends on the assumed constant rates within the two periods and the specified treatment effect.

Worked Example: Checking Whether Treatment Alone Describes Scheduling

Suppose a hypothetical fitted model includes both current treatment \(A_i(t-)\) and a binary symptom indicator \(S_i(t-)\), which equals 1 if a symptom was recorded before time \(t\):

\lambda_i^V(t\mid\mathcal{H}_i(t-)) = 0.30\exp\{0.25A_i(t-)+0.70S_i(t-)\}.

Assume these covariates are observed before each interval being modeled and the model form is adequate. For a participant not currently treated and without the symptom, the rate is \(0.30\) visits per year. For a participant currently treated but without the symptom, it is

0.30\exp(0.25) = 0.30(1.2840) \approx0.3852 \text{ visits per year}.

For a treated participant with the symptom, the rate is

0.30\exp(0.25+0.70) = 0.30\exp(0.95) = 0.30(2.5857) \approx0.7757 \text{ visits per year}.

At the same symptom status, the treatment-associated rate ratio is \(\exp(0.25)\approx1.284\). At the same treatment status, the symptom-associated rate ratio is \(\exp(0.70)\approx2.014\). The larger symptom coefficient suggests that measured symptom status is strongly associated with visit intensity in this fitted model. It does not prove that symptoms caused visits, and it does not rule out unmeasured health-related triggers.

Justification: From Intensity to Expected Visit Counts

The link between an intensity and an expected number of visits follows from the definition of conditional intensity. Let \(dN_i^V(t)\) indicate the number of visits in a very short interval around \(t\), and let the history immediately before that interval be given. The conditional expected increment is approximately \(\lambda_i^V(t\mid\mathcal{H}_i(t-))dt\). Summing these increments across a partition of follow-up and taking the limit gives the compensator

\Lambda_i^V(\tau) = \int_0^\tau \lambda_i^V(t\mid\mathcal{H}_i(t-))\,dt.

The counting-process difference \(N_i^V(\tau)-\Lambda_i^V(\tau)\) has mean zero under the usual conditions for a valid intensity, so

E\{N_i^V(\tau)\}=E\{\Lambda_i^V(\tau)\}.

If the intensity is a known, fixed function of time, the expectation reduces to the integral of that function, as in the piecewise-rate example. If it depends on random, evolving history, the expected count is instead the expectation of the integrated conditional intensity; one cannot generally substitute an average covariate into the exponential and expect the same result.

Distinguishing Treatment-Driven and Health-Driven Visits

A treatment-only model can represent differences in monitoring associated with assignment or treatment state, but it does not establish why those differences occurred. A randomized assignment may be useful for describing or comparing arm-specific monitoring patterns. In contrast, current treatment is often related to health history: for example, a clinician may start, stop, or change treatment because a participant’s condition has changed. If health history also predicts visits, omitting it can make the treatment coefficient absorb part of that association.

A model that includes observed health history can make the contrast more specific: it compares rates at the same values of those included predictors. It still cannot account automatically for unmeasured symptoms, clinician judgment, or other unobserved visit triggers. Nor does adjustment make an association causal without suitable assumptions about treatment assignment, time-varying confounding, measurement, and model specification. The earlier material on observation intensities conditional on patient history and visit rates driven by latent trajectories provides related modeling context; here the focus is on how treatment enters the observed visit process.

Approach	What it represents	Main caution
Treatment-only intensity model	Conditional rate differences by assigned arm or current treatment	May leave health-related variation in visit rates unexplained
Separate arm-specific baseline intensities	Potentially different time patterns of monitoring by arm	Requires enough observations in each arm to estimate the patterns
Treatment plus observed health history	Rate contrasts by treatment among participants with the same modeled history	Unmeasured or poorly measured visit triggers can remain
Joint model with latent health-related dependence	Can represent shared latent features of health and visits	Requires additional assumptions and may be weakly identified with sparse visits

Diagnostics, Pitfalls, and Limitations

Check whether the fitted model captures the main observed features of the schedule. Compare visit counts and timing by assignment and treatment state, and inspect whether fitted rates systematically miss intervals or groups. Examine whether the estimated treatment contrast changes when measured pre-visit health history is included. Strong changes can indicate that treatment and health history share information about scheduling; they do not, by themselves, identify the true mechanism. Also check for sparse treatment histories, near-empty treatment-by-time groups, and excessive influence from a few participants. These features can make contrasts unstable.

Using future treatment or health information. A predictor must describe information available just before the visit. Using a value recorded at the visit to predict that same visit can reverse the time ordering.
Calling a rate ratio a visit-count difference. The exponential of a coefficient compares conditional rates. Expected counts require integrating the intensity over the time each condition applies.
Treating assignment and treatment received as interchangeable. Assignment is usually fixed at baseline; treatment received can change over time and may respond to health.
Claiming treatment caused more visits from a fitted coefficient. A conditional association in the visit model alone does not establish the causal mechanism.
Assuming treatment explains all informative monitoring. Health changes can affect both treatment and visit timing, and unobserved triggers may remain after adjustment.

Treatment-dependent visit models are useful descriptions of an observation schedule, but they have limitations. A simple proportional-rate model may not capture treatment effects that vary over time, abrupt protocol changes, or treatment-history effects that persist after a switch. Estimation also depends on adequate visit information within the treatment patterns being compared. Finally, a good fit to observed visit counts does not prove that all relevant health-related dependence has been captured.

Key takeaway: State whether a model uses assigned treatment or treatment received, define every predictor using information available before the visit, and interpret treatment coefficients as conditional visit-rate contrasts. Then ask whether evolving health also helps explain monitoring.

Check Your Understanding

Answer each question, then compare with the short solutions below.

A model estimates a treatment-assignment coefficient of \(\log(1.25)\). Interpret its exponentiated value in context.
A participant has a visit intensity of \(0.60\) visits per year for 1.5 years, then treatment begins and multiplies the rate by 2 for the next year. Under constant rates within each period, calculate the expected count.
Why might the coefficient for current treatment change when a pre-visit symptom indicator is added to the model?
What is wrong with using a biomarker value measured at a visit to predict the intensity of that same visit?
Does a treatment coefficient above zero prove that treatment caused more frequent monitoring? Explain.

Solutions. (1) The conditional visit-rate ratio is \(\exp\{\log(1.25)\}=1.25\): at the same modeled time and values of other covariates, the assigned-treatment group has an estimated rate 25% higher than the reference group. (2) The first-period expected count is \(0.60(1.5)=0.90\) visits. The later rate is \(2(0.60)=1.20\) visits per year, giving \(1.20(1)=1.20\) visits. The total is \(0.90+1.20=2.10\) expected visits. (3) Symptoms may be associated with both current treatment and the visit rate, so including symptom status changes the conditional comparison represented by the treatment coefficient. This does not establish a causal explanation. (4) That value is not available before the visit; using it violates the required time ordering and can make the model circular. (5) No. The coefficient describes a conditional association under the model. Causal interpretation requires additional design and identification assumptions, and relevant health-related triggers may remain unmeasured.