Poisson Models For Recurrent Visit Times

Why Model Repeated Visits as a Process?

A patient’s visit record contains more than a total number of appointments. The times between visits and the distribution of visits across follow-up can matter when monitoring is related to health status. A Poisson-process model gives a simple starting point: it specifies how visits accumulate over time and makes the assumptions behind a constant visit rate explicit. It is a model for the observation schedule, not by itself a model for a biomarker or a terminal event.

Earlier material on representing visits with counting processes defines a counting process as a cumulative record of event occurrences. Here, the event is a visit. Let $N_i(t)$ be the number of visits for subject $i$ from the start of follow-up through time $t$, measured in a specified unit such as months. For a subject followed through time $T_i$, the total observed count is $N_i(T_i)$. The model applies over the defined observation window; it does not assume that visits continue after follow-up ends.

Definition: A homogeneous Poisson process with rate $\lambda_i>0$ for subject $i$ has $N_i(0)=0$, independent counts in disjoint time intervals, and a visit rate that is constant over time. For an interval of length $u$, its count has distribution $N_i(t+u)-N_i(t)\sim\operatorname{Poisson}(\lambda_i u)$, so $$ P\{N_i(t+u)-N_i(t)=k\} = \exp(-\lambda_i u)\frac{(\lambda_i u)^k}{k!}, \qquad k=0,1,2,\ldots. $$ The rate $\lambda_i$ has units of visits per unit time. The model also implies that two visits do not occur at exactly the same time with positive probability.

The expected number of visits over an interval of length $u$ is $\lambda_i u$, and its variance is also $\lambda_i u$. Thus, $\lambda_i$ is not a probability of a visit in a time period: it is a rate. For a very short interval of length $h$, the probability of one visit is approximately $\lambda_i h$, while the probability of two or more visits is of smaller order than $h$.

What the Constant-Rate Assumption Implies

A homogeneous Poisson process makes two related claims. First, the expected number of visits grows linearly with elapsed time: doubling the length of an interval doubles the expected count. Second, the counts in nonoverlapping intervals are independent under the model. In particular, a high count in one interval does not change the distribution of the count in another interval, provided the rate and the conditioning information are fixed.

The assumptions are stronger than merely saying that the average number of visits per month is stable across a cohort. The model says that, for a given subject and fixed rate, there is no systematic within-window change in visit intensity, and no dependence between counts in disjoint intervals. It does not require every observed interval to contain the same number of visits. Random variation includes occasional clusters and long gaps; the question is whether their pattern is compatible with the model rather than whether the sequence looks perfectly regular.

Worked Example: Counts Under a Constant Rate

Suppose, hypothetically, that a patient’s visits follow a homogeneous Poisson process with rate $0.8$ visits per month for a three-month observation window. Assume the rate is constant during follow-up and that disjoint interval counts are independent. Find the probability of no visits and the probability of exactly two visits.

The expected count over three months is $\lambda T=(0.8)(3)=2.4$ visits. Therefore,

P\{N(3)=0\} = \exp(-2.4)\frac{2.4^0}{0!} = 0.09072.

For exactly two visits,

P\{N(3)=2\} = \exp(-2.4)\frac{2.4^2}{2!} = (0.09071795)(2.88) = 0.2613.

Under this model, the probability of no visits in the three months is about 9.1%, and the probability of exactly two visits is about 26.1%. These are model-based probabilities for the stated window, not estimates of the probability that a particular patient will ever attend a visit.

A Short Justification of the Count Distribution

The Poisson count formula follows from the small-interval behavior of a constant-rate process. Write $p_k(t)=P\{N(t)=k\}$. Over a short increment $h$, the chance of one visit is $\lambda h+o(h)$, and the chance of two or more is $o(h)$. For $k\geq 1$, the probability of being at count $k$ at time $t+h$ is therefore the probability of already being at $k$ and having no visit, plus the probability of being at $k-1$ and having one visit, up to terms $o(h)$. Subtracting $p_k(t)$, dividing by $h$, and taking $h$ to zero gives

p_k'(t)=-\lambda p_k(t)+\lambda p_{k-1}(t).

The no-visit probability satisfies $p_0'(t)=-\lambda p_0(t)$, with $p_0(0)=1$, so $p_0(t)=\exp(-\lambda t)$. Solving the recurrence successively, using $p_k(0)=0$ for $k>0$, gives $p_k(t)=\exp(-\lambda t)(\lambda t)^k/k!$. This derivation relies on the constant rate and the negligible probability of multiple events in an arbitrarily short interval; it does not establish that those assumptions suit a particular visit record.

Adding Subject-Level Covariates

A common model allows the constant rate to differ across subjects according to measured covariates. Let $x_i$ be a covariate vector that is fixed for subject $i$ over the modeled period, and let $\beta$ be a coefficient vector. The exponential link keeps the rate positive:

\lambda_i=\exp(x_i^\mathsf{T}\beta).

For a one-unit increase in a covariate $x_{ij}$, holding the other covariates fixed, the rate is multiplied by $\exp(\beta_j)$. If time is measured in months, this compares expected visits per month; it does not describe a probability increase of $\beta_j$, nor does it imply a causal effect. Such an interpretation requires a suitable study design and assumptions beyond this process model.

Worked Example: Interpreting a Covariate Effect

Consider a hypothetical model in which a baseline symptom score $x_i$ predicts the visit rate in visits per month: $\lambda_i=\exp(\beta_0+\beta_1x_i)$. Suppose $\beta_0=\log(0.5)$ and $\beta_1=0.35$. Compare the modeled rates at symptom scores 0 and 2, and calculate the expected six-month counts.

At score 0, the rate is $\exp(\log(0.5))=0.5$ visits per month. At score 2,

\lambda_i = \exp\{\log(0.5)+(0.35)(2)\} = 0.5\exp(0.7) = 1.007\text{ visits per month}.

A one-point increase multiplies the rate by $\exp(0.35)=1.419$, or about 42% higher, holding other modeled covariates fixed. The expected counts over six months are $(0.5)(6)=3.00$ visits at score 0 and $(1.0069)(6)=6.041$ visits at score 2. These are expected counts under the model, not guaranteed individual counts. The comparison is an association unless additional causal assumptions are justified.

If subject-specific latent factors are included, a process may be Poisson conditional on those factors without being a simple Poisson process after they are averaged out. For example, earlier material on shared random effects explains how a shared subject effect can connect visit intensity to a biomarker trajectory. Conditional independence given the effect does not mean that visits are unconditionally independent across intervals: subjects with a persistently high latent rate tend to have more visits in multiple intervals.

Checking Whether a Constant Rate Is Plausible

A constant-rate model is most credible when the observation window has a clear start and end, the visit definition is consistent, and there is no strong reason to expect the rate to change systematically within the window. Useful first checks include plotting visit times, comparing counts across equal-length intervals, and examining rates across follow-up periods and across subjects. These are diagnostics, not automatic proof of a model or a formal test.

For example, a steady increase in visits near a clinical deterioration, a sharp drop after a protocol change, or regular visits imposed by a study schedule would make one common constant rate questionable. A large variance of interval counts relative to their mean can also signal heterogeneity or dependence, although pooling subjects with different rates can create this pattern even when each subject’s process is conditionally Poisson. Clinical knowledge and the way visits were scheduled should therefore be considered alongside the observed counts.

Worked Example: A Clustered Visit Pattern

Suppose, hypothetically, one patient has four visits in one month and none in the other five months of a six-month window. Fit the constant rate by the overall count divided by exposure: $4/6=0.6667$ visits per month. Under that fitted rate, the probability of four visits in one particular month and zero in the other five months is

\left[\exp(-2/3)\frac{(2/3)^4}{4!}\right] \left[\exp(-10/3)\right] = \exp(-4)\frac{16/81}{24} = 0.0001507.

This calculation assumes independent Poisson counts in the six one-month intervals. It shows that this exact allocation to a prespecified month is unusual under the fitted constant-rate model. It is not, by itself, a valid significance test: the month was identified after observing the pattern, the rate was estimated from the same data, and many other possible patterns could be considered. The cluster is a reason to investigate symptom changes, scheduling rules, or other sources of time variation, not proof of a particular explanation.

What the Model Does and Does Not Replace

A Poisson model for the total count can be convenient when only the number of visits and the exposure time are retained. But a count model alone does not preserve the ordering or exact timing of visits. Conversely, modeling event times with a Poisson process specifies both interval counts and their distribution across time. A scheduled-visit design may be better described by its protocol than by a random constant-rate process, while a process with a rate that changes over follow-up requires a model that permits that change. More detailed formulations of how intensity depends on calendar time or time since a prior visit are separate modeling choices.

The homogeneous model also has limitations. A fixed rate can be unrealistic when symptoms, treatment, access to care, or study procedures change over time. A single rate across patients can conceal genuine heterogeneity. And even if the visit process is well described, that does not establish whether visit timing is informative for the longitudinal outcome. A model for visits should be interpreted together with the scientific question and the outcome model, rather than treated as a complete analysis of the joint data.

Common pitfall: “The average is one visit per month, so visits follow a Poisson process.” This is not sufficient. A rate alone does not establish independent increments or a constant rate within the observation window. The correction is to state the process assumptions, inspect visit patterns and scheduling information, and describe a Poisson model as an approximation when those assumptions are not fully credible.

Define the event and exposure window.
Specify what qualifies as a visit, the time unit, and each subject’s period at risk of having a recorded visit.

State the rate model.
Give the constant-rate or covariate-dependent rate and its units; identify what is held fixed or conditioned on.

Assess the assumptions.
Compare counts and timing across follow-up, and consider whether clinical or administrative processes imply changing rates or dependence.

Interpret within scope.
Translate coefficients into rate comparisons and expected counts, while avoiding causal claims or conclusions about the longitudinal outcome that the visit model alone cannot support.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Under a homogeneous Poisson process with rate $0.4$ visits per week, what is the expected number of visits in five weeks?
For a rate of $1.2$ visits per month, calculate the probability of no visits in two months.
A covariate coefficient is $0.20$ in a log-rate model. What is the rate ratio for a one-unit increase, approximately?
Why does observing the same average number of visits per month across a cohort not establish a homogeneous Poisson process?
What does a cluster of visits in one short period suggest, and why is it not conclusive evidence against a constant-rate model?

Solutions. (1) The expected count is $\lambda T=(0.4)(5)=2.00$ visits. (2) The mean count is $(1.2)(2)=2.4$, so $P\{N(2)=0\}=\exp(-2.4)=0.09072$. (3) The rate ratio is $\exp(0.20)=1.221$, or about 22% higher per one-unit increase, holding other covariates fixed. (4) The cohort average does not establish constant within-subject rates or independent counts in disjoint intervals; different subjects can also have different rates. (5) A cluster may point to time-varying intensity, dependence, or a scheduling or clinical change, but Poisson variation can produce clusters by chance. Its interpretation depends on the full pattern, the observation process, and how the cluster was identified.