Representing Visits With Counting Processes

Why Represent Visits as a Process?

In longitudinal follow-up, patients can have different numbers of visits, and their visits can occur at irregular times. A list of visit times records when measurements were taken, but counting-process notation gives a compact way to describe how the observed visit history accumulates during follow-up. It also distinguishes times when a patient is under observation from times outside the observation interval.

This representation is useful because a visit rate is not just a count divided by total study duration. The relevant rate can vary with time and with the patient’s observed history. A counting process makes the visit events and the time during which they could be observed explicit, without assuming that visits occur at a constant rate or that their timing is unrelated to the patient’s history.

If a patient has visits at times 1.2, 2.0, and 2.7 months, then \(N_i^V(t)=0\) before 1.2 months, equals 1 from the first visit until the second, equals 2 until the third, and equals 3 afterward. The process is nondecreasing and integer-valued. For a simple visit process, no two visits occur at exactly the same instant, so each jump is one. The visit history through time \(t\) includes the observed jump times up to \(t\), or, equivalently, the path of \(N_i^V(s)\) for \(s\le t\).

Follow-Up and the At-Risk Interval

Let \(L_i\) denote the start of patient \(i\)’s observable follow-up and \(C_i\) the end of that follow-up. A patient contributes time to the visit process only while visits could be recorded under the study’s observation system. A convenient convention is to count visits in \((L_i,C_i]\). The associated at-risk indicator is $$ Y_i(t)=I(L_i<t\le C_i). $$ It indicates whether the patient is under observation at time \(t\) under this endpoint convention. The treatment of a visit exactly at entry or exit should follow the study protocol; the displayed convention is one explicit choice.

Intensity calculations use time integrals. The values assigned to an at-risk indicator at isolated endpoints do not change an integral with respect to time, because single time points have zero duration. However, endpoint conventions do matter when deciding whether a recorded event at that exact time belongs in the data. State that rule separately rather than relying on an intensity formula to decide it.

Visit Intensity and Integrated Intensity

A visit intensity describes the instantaneous rate of visit occurrence conditional on the observed history just before time \(t\). Write \(\mathcal{F}_{i,t-}\) for the information available for patient \(i\) immediately before \(t\), including the prior visit history and other observed information specified by the model. When the patient is at risk, denote the visit intensity by \(\lambda_i^V(t\mid\mathcal{F}_{i,t-})\). Its units are visits per unit of time. It is a conditional rate, not a probability and not necessarily a constant over follow-up.

The conditional expectation statement follows directly from the martingale property: \(E[M_i^V(b)-M_i^V(a)\mid\mathcal{F}_{i,a}]=0\). Substituting the definition of \(M_i^V\) and rearranging gives $$ E[N_i^V(b)-N_i^V(a)\mid\mathcal{F}_{i,a}] = E\left[\int_a^b Y_i(s)\lambda_i^V(s\mid\mathcal{F}_{i,s-})\,ds\ \middle|\ \mathcal{F}_{i,a}\right]. $$ The right-hand side is generally an expectation because future intensity can depend on history that has not yet been observed at time \(a\). It is not generally correct to replace it with an integral of a future, random intensity as if that entire path were already known at \(a\).

At times strictly inside a risk interval, a familiar local approximation can also be used under suitable regularity conditions. If the patient is at risk throughout a sufficiently short interval after \(t\), the intensity is locally well behaved, and the process is simple with negligible probability of multiple jumps in that short interval, then $$ P\{N_i^V(t+h)-N_i^V(t)=1\mid\mathcal{F}_{i,t-}\} = \lambda_i^V(t\mid\mathcal{F}_{i,t-})h+o(h). $$ This is a local statement away from changes in risk status; it is not an endpoint formula. In general, the integrated at-risk intensity is the appropriate expression across an interval that may contain entry or exit.

Boundary Care, Assumptions, and a Short Justification

The intensity representation requires a well-defined observed history, an agreed rule for which visits are counted, and a rate that is nonnegative and integrable over the at-risk time. The process should be adapted to the history being modeled: whether a visit has occurred by time \(t\) must not depend on information from after \(t\). For the usual martingale interpretation, the intensity must describe the conditional event rate relative to that history and the resulting increments must satisfy the relevant integrability conditions.

A common mistake is to apply the interior local approximation at an entry or exit boundary. For example, at entry \(t=L_i\), the expression \(Y_i(t)\lambda_i^V(t)h\) can be zero under the convention \(Y_i(t)=I(L_i<t\le C_i)\), even though visits can occur during \((L_i,L_i+h]\). At exit \(t=C_i\), a pointwise convention might instead leave a positive leading term even though the patient has no observed follow-up after \(C_i\). Neither result is a sound general endpoint approximation.

To see why integration resolves the boundary issue, consider an interval beginning before entry and ending after entry. The risk indicator is zero before entry and one during the portion of the interval when the patient is observed. Therefore the accumulated rate is contributed only by that observed portion. If the patient enters at \(L_i\), for example, the contribution from \((a,b]\) is the integral over the overlap of \((a,b]\) with the at-risk interval, not the value of the indicator at the single instant \(a\), \(b\), or \(L_i\). This is the appropriate way to relate risk time to expected counts across a boundary.

What the Representation Does—and Does Not—Tell Us

Counting-process notation is more flexible than a single visit count or an average visits-per-month summary. It preserves the ordering and timing of visits and allows the rate to vary over follow-up. A basic total rate, such as total visits divided by total observed person-time, can be a useful descriptive summary, but it averages over time and patients. It does not encode a changing conditional intensity.

A fixed-visit schedule is often simpler to describe with visit indicators at prespecified times. A counting process is more natural when visit times are irregular or the number of visits varies, because each visit is represented as a jump at its actual time. Conversely, reducing data from a planned schedule to a continuous-time process may obscure the design’s discrete structure. The representation should match how observations could occur and be recorded.

The counting process itself is descriptive bookkeeping; it does not explain why visits occurred. A specified intensity may condition on prior observed history, but the notation alone neither determines which history variables belong in a model nor establishes that the modeled rate is appropriate. It also does not resolve measurement error, unrecorded visits, or ambiguity in the definition of follow-up. In practice, check that visit dates are ordered, event counts increase by the number of recorded visits, the risk interval matches observation records, and no modeled intensity is applied outside time at risk.

Common Pitfalls

• Treating an intensity as a probability. A rate of 0.2 visits per month is not a 20% probability of a visit in a month. A probability over a specified interval depends on the process assumptions and the interval’s accumulated intensity.
• Ignoring the at-risk interval. Time before entry or after follow-up ends must not contribute to observed visit exposure.
• Using an endpoint value as an interval rate. At entry and exit, the at-risk indicator can change. Use the integrated at-risk intensity across such intervals rather than an unqualified pointwise approximation.
• Assuming a Poisson process from an intensity alone. The intensity specifies a conditional rate, not independent increments or a Poisson distribution for counts.