Likelihood Contributions From Observed Visit Times

Why the Whole Follow-Up Interval Matters

A record of visit times contains more information than the number of visits. If two patients each have three visits, but one has visits spread across a year and the other has all three in the first month, those schedules are different observations of the visit process. The time between visits—and the time from the last visit to the end of follow-up—also informs how plausible a proposed visit intensity is.

This tutorial develops the likelihood contribution from the observed visit times and the intervals in which no visit occurred. It is the part of a joint analysis that describes the observation schedule. When visit intensity depends on a biomarker or a latent trajectory, this contribution can connect the visit schedule to the outcome model. The earlier material on Representing Visits With Counting Processes and Observation Intensities Conditional On Patient History provides the relevant process and intensity language.

Definition: For subject \(i\), let \(N_i(t)\) count visits by time \(t\), and let \(\mathcal V_i=(t_{i1},\ldots,t_{in_i})\) be the observed visit times in the at-risk interval \((L_i,C_i]\). The visit intensity \(\lambda_i(t\mid\mathcal H_i(t-),b_i)\) is the instantaneous event rate at time \(t\), conditional on the history \(\mathcal H_i(t-)\) available just before \(t\) and, when included, subject-specific latent quantities \(b_i\). It has units of visits per unit time.

The at-risk interval begins at \(L_i\) and ends at \(C_i\), which might be an administrative study end or another observation endpoint. The history immediately before \(t\), written \(\mathcal H_i(t-)\), must contain only information available before \(t\). This is the predictability condition: an intensity cannot use a measurement that is only obtained at the visit occurring at time \(t\) to explain that same visit's instantaneous rate. It may, however, depend on measurements from earlier visits.

The standard event-time likelihood below assumes a simple point process with a correctly specified, nonnegative, predictable intensity; the observed visits are recorded without unmodeled omissions within the stated at-risk interval; and the interval endpoints and censoring are handled appropriately. In examples with a fixed administrative endpoint, the endpoint is treated as given. If stopping follow-up is itself informative, its mechanism may require additional modeling.

The Event-Time Likelihood

For one subject, conditional on the specified history and any latent quantities, the visit-process likelihood contribution is

L_{V,i} = \left\{\prod_{j=1}^{n_i} \lambda_i(t_{ij}\mid\mathcal H_i(t_{ij}-),b_i)\right\} \exp\left\{-\int_{L_i}^{C_i} \lambda_i(u\mid\mathcal H_i(u-),b_i)\,du\right\}.

Each observed visit contributes its intensity at its event time. The exponential term accounts for the entire at-risk interval, including every gap with no visit. If there are no visits, the product is empty and equals 1, while the exponential term remains: observing no visits can still be informative about the intensity.

Interpretation: The event-time likelihood balances intensity at the observed visit times against cumulative intensity over the full observation interval. The integral is dimensionless; with \(n_i\) observed visits, the likelihood density has units of time to the power \(-n_i\), relative to the event-time measure.

A short derivation shows where both factors arise. Over a small interval of width \(dt\), conditional on the history at its start, the probability of a visit is approximately \(\lambda(t)dt\), while the probability of no visit is approximately \(1-\lambda(t)dt\). Multiplying these contributions across a fine partition gives a factor \(\lambda(t_{ij})dt\) for each interval containing an event and factors \(1-\lambda(u)du\) for intervals without events. Taking the limit, the product of event rates remains, and the no-event factors become an exponential:

\prod_{\text{no-event intervals}}\{1-\lambda(u)du\} \ \longrightarrow\ \exp\left\{-\int_{L_i}^{C_i}\lambda(u)\,du\right\}.

The event intervals' \(dt\) terms specify the reference measure for the event times; the displayed likelihood omits that common measure notation. The argument also explains why discarding the no-visit intervals is not a harmless simplification: it removes the part of the likelihood that penalizes a model for predicting many visits where none occurred.

Worked Example: Constant Visit Rate

Suppose a hypothetical patient is followed from year 0 to year 3 and has visits at years 0.5, 1.4, and 2.7. Assume a homogeneous Poisson visit process with constant rate \(\lambda(t)=0.4\) visits per year over the whole interval. This model assumes a constant rate, independent increments, and no additional history dependence. The likelihood for the three ordered event times, relative to their time measure, is

L_V=(0.4)^3\exp\{-0.4(3-0)\} =0.064\exp(-1.2) =0.064(0.301194\ldots) =0.0192764\ldots \approx 0.01928\ \text{year}^{-3}.

The visit-time product is \(0.4^3\), and the no-visit contribution over all three years is \(\exp(-1.2)\). The units are year\(^{-3}\) because there are three event times. This is a likelihood density, not a probability that the exact three times will occur. For comparison, if only the count \(N(3)=3\) were observed, its Poisson probability would be \(e^{-1.2}1.2^3/3!\). That count probability answers a different question because it does not use the actual event times.

Changing Rates and History-Dependent Rates

When the rate changes over time, the likelihood uses the rate applying at each visit and integrates the rate across each part of follow-up. A gap with no visit contributes through the integral just as an interval containing a visit does. With a history-dependent intensity, the rate must be evaluated using the history available at each time, so the cumulative-intensity integral can change at observed visits.

Worked Example: A Piecewise-Constant Intensity

In a hypothetical follow-up from year 0 to year 4, suppose the intensity is 0.2 visits per year on \([0,2)\) and 0.5 visits per year on \([2,4]\). The visits occur at years 0.8, 2.4, and 3.1. Assume these rates are fixed in advance and the process is Poisson within the stated rate model. One visit occurs in the first segment and two in the second, so the event-rate product is \(0.2(0.5)(0.5)\). The integrated intensity is

\int_0^4 \lambda(u)\,du =0.2(2)+0.5(2)=1.4.

Therefore,

L_V=0.2(0.5)(0.5)\exp(-1.4) =0.05(0.24659696\ldots) =0.0123298\ldots \approx 0.01233\ \text{year}^{-3}.

The model assigns a higher instantaneous rate after year 2, and the two visits in that segment contribute the two factors of 0.5. The exponential includes the exposure in both segments, including the final interval from year 3.1 to year 4 with no visit. Omitting that final gap would overstate the likelihood under this model.

Worked Example: Combining a Lagged Measurement and Visit Times

Consider a hypothetical subject followed for five weeks, with visits at weeks 1 and 4. Let the intensity be \(\lambda(t\mid\mathcal H(t-))=0.1\exp\{0.5z(t-)\}\) visits per week, where \(z(t-)\) is the most recent measured biomarker value available just before \(t\). The baseline value is 0; the measurement at week 1 is 2 and the measurement at week 4 is 1. The rate is therefore 0.1 on \([0,1)\), \(0.1e^1=0.271828\) on \([1,4)\), and \(0.1e^{0.5}=0.164872\) on \([4,5]\). At week 1, the event intensity uses the history before that visit, so it is 0.1; at week 4, it uses the earlier value 2, so it is 0.271828.

The integrated intensity and visit-process contribution are

\int_0^5\lambda(u)\,du =0.1(1)+0.271828(3)+0.164872(1) =1.080356,

L_V=0.1(0.271828)\exp(-1.080356) \approx 0.009228\ \text{week}^{-2}.

To illustrate the outcome contribution, suppose the two measurements are conditionally independent given the model, each has a normal distribution with mean 1 biomarker unit and standard deviation 0.5 biomarker units, and these parameters are treated as known. The measurement density at week 1 is

f(2)=\frac{1}{0.5\sqrt{2\pi}}\exp\left\{-\frac{(2-1)^2}{2(0.5)^2}\right\} \approx 0.107982\ \text{unit}^{-1},

and at week 4 it is \(f(1)=0.797885\ \text{unit}^{-1}\). Under these stated conditional models, the illustrative joint contribution is the visit-process factor times the two outcome densities:

L_{\text{joint}} \approx 0.009228(0.107982)(0.797885) \approx 0.0007951\ \text{week}^{-2}\text{unit}^{-2}.

This product is an example of how the schedule and measurements can contribute to one likelihood. It is not a claim that the schedule is independent of the biomarker: here the first measurement changes the rate for subsequent weeks. In a model with shared latent effects or an unobserved trajectory, the likelihood must account for those quantities, often by integrating their conditional joint contribution against their specified distribution.

How Visit and Outcome Contributions Fit Together

A joint likelihood specifies both how measurements arise and how their observation times arise. For example, conditional on a latent subject effect \(b_i\), one can model the visit times through \(L_{V,i}(b_i)\) and the observed outcomes through a conditional measurement likelihood \(L_{Y,i}(b_i)\). If those components are conditionally independent given \(b_i\) and the visit times, their conditional product is integrated over the distribution of \(b_i\):

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

If the visit intensity depends directly on past observed outcomes, the joint model must instead respect that sequential history: each rate uses only measurements observed before that time, and the outcome contribution describes the values observed at the visits. The precise factorization follows the assumed conditional model. In either case, the key idea is that observed visit times are data, not automatically fixed design points. Conditioning on them may discard information about a shared latent trajectory or about dependence between monitoring and outcomes.

Comparisons, Diagnostics, and Limitations

Approach	What it uses	What it leaves out or assumes
Full event-time likelihood	Visit times and all no-visit intervals in follow-up	Requires a specified intensity and at-risk interval
Count-only Poisson likelihood	Total number of visits	Does not use where visits fall within follow-up
Outcome likelihood conditional on observed times	Measurements at the recorded visits	Does not model the schedule unless the conditioning is justified for the question

Useful checks begin with the data and the risk interval: confirm that every recorded visit lies inside \((L_i,C_i]\), that the endpoint is correctly represented, and that gaps without visits have not been dropped. Then inspect whether the fitted intensity implies plausible numbers and timing of events across subjects and time. A model that predicts high rates throughout a long period with no visits will receive a small exponential survival contribution. Likewise, rates evaluated using future measurements, or intervals incorrectly treated as at risk, signal a modeling error.

The likelihood does not protect against a misspecified intensity, unrecorded visits, or an unsuitable assumption about follow-up termination. Nor does observing the event times alone necessarily distinguish a trajectory-driven rate from other sources of heterogeneity. Those limitations matter especially when the same latent features influence both biomarker values and monitoring frequency.

A Common Mistake

Pitfall: Multiplying the visit intensities at the observed times and stopping there seems reasonable because each observed visit is evidence for a high rate. But it rewards arbitrarily high rates and ignores the absence of visits elsewhere. The correction is to include \(\exp\{-\int_{L_i}^{C_i}\lambda_i(u)\,du\}\) over the full at-risk interval. The event factors reward intensity where visits occurred; the exponential accounts for where they did not.

Check Your Understanding

Try each question, then compare your reasoning with the solutions below.

A constant intensity of 0.3 visits per month applies over 4 months, with visits at months 1 and 3. Write the event-time likelihood and calculate it.
Why does a subject with no visits still have a visit-process likelihood contribution?
For a model based only on the total visit count, what information in the event-time likelihood is discarded?
A visit intensity depends on the biomarker measured at the current visit. What condition must be addressed before using that measurement to define the intensity at that visit?

Solutions. (1) The event-rate product is \(0.3^2=0.09\), and the integrated intensity is \(0.3(4)=1.2\). Thus \(L_V=0.09e^{-1.2}=0.02711\ \text{month}^{-2}\). (2) The no-event observation over the at-risk interval contributes \(\exp\{-\int_L^C\lambda(u)\,du\}\); it contains information about how many visits the model predicts, even when none occurred. (3) The count-only likelihood discards the locations of visits within follow-up and therefore cannot use the spacing or timing pattern. (4) The intensity must be predictable: it can use only information available just before the event. The current visit's measurement is not available before that visit, although it may inform rates after the measurement has been recorded.