Worked Example: Comparing Scheduled And Triggered Follow-Up

Why Compare Follow-Up Schedules?

A biomarker is observed only when a measurement is taken. A fixed protocol, such as a visit every 30 days, determines observation times in advance. Under symptom-triggered monitoring, visits occur when symptoms or concerns prompt contact, so observation times can cluster in periods when a patient's health is changing. Consequently, the data collected under the two schedules may differ in both when measurements are made and which biomarker values are represented often in the record.

A difference between summaries from the two schedules is not, by itself, evidence that the underlying patients or biomarker trajectories differ. It may reflect different visit timing, different numbers of visits per person, or both. The comparison therefore needs to specify the summary's target. For example, an average across all recorded visits gives greater weight to people with more visits. An average that first summarizes each person and then averages across people gives each person equal weight. Neither is universally preferable; each answers a different question.

A Hypothetical Side-by-Side Comparison

Suppose four participants are followed for 60 days, with the biomarker measured in arbitrary biomarker units. Under the protocol-based schedule, all four are measured on days 0, 30, and 60. Under symptom-triggered monitoring, they are measured at baseline and then at the times shown below. These data are hypothetical; they illustrate how the schedules can produce different records for the same small cohort.

How Visit Timing Changes the Record

Under the protocol schedule, each participant has gaps of 30 days between successive visits. There are eight such gaps across the four participants. Under triggered monitoring, the observed gaps are 20, 20, and 20 days for A; 30 and 30 days for B; and 60 days each for C and D. The triggered gaps therefore have counts of three at 20 days, two at 30 days, and two at 60 days. Their mean is

The mean gap alone conceals an important difference: the triggered data include several short intervals and two long ones, whereas every protocol gap is 30 days. Visit counts by calendar day also differ. The protocol records four observations on each of days 0, 30, and 60. The triggered records have four observations on day 0, one on day 20, one on day 30, one on day 40, and four on day 60. Thus the triggered observations are not distributed across calendar time in the same way as the protocol observations.

This comparison is descriptive. The average of protocol measurements at days 0, 30, and 60 targets those planned measurement occasions. A pooled mean across triggered visits targets the values represented in that triggered visit record. Neither is automatically the average biomarker over all participants and all times in the 60-day window. Making that broader target explicit matters, especially if the biomarker tends to rise or fall over follow-up.

Why Triggered Visits Can Overrepresent Some Values

The mechanism behind this imbalance can be expressed using the visit intensity. Let \(Y(t)\) denote a participant's biomarker at time \(t\), and let \(\lambda(t\mid H(t))\) denote the instantaneous visit rate given the observed history \(H(t)\). Informally, over a short interval of length \(dt\), the probability of a visit is approximately \(\lambda(t\mid H(t))dt\). If visits become more frequent when the biomarker or related symptoms are high, then high-biomarker periods contribute more recorded observations.

For a simple pooled person-time comparison, suppose follow-up is defined over a common time window, the visit intensity is well-defined conditional on the history, and the average intensity is positive. For any function \(g\) of the biomarker, the expected value of \(g(Y)\) among visits is proportional to its person-time expectation weighted by the visit intensity:

To see why, consider a short interval \(dt\). The expected contribution of \(g(Y(t))\) to a visit total is approximately \(\lambda(t\mid H(t))g(Y(t))dt\). Its expected contribution to the visit count is approximately \(\lambda(t\mid H(t))dt\). Summing these contributions over the target person-time and taking the ratio gives the displayed intensity-weighted expectation. If \(\lambda\) is larger at times when \(Y\) is high, then the numerator gives those times extra weight.

This argument describes a possible source of distortion, not a claim that all symptom-triggered schedules are informative in the same way. A trigger may depend on symptoms without depending on the biomarker after conditioning on recorded history. Conversely, a protocol with extra patient-initiated visits may include informative observations even if it also has regular planned visits.

Checks, Alternatives, and Limitations

Before interpreting a schedule comparison, inspect the observation process as well as the biomarker values. Useful checks include visit counts per participant; distributions of calendar visit times and inter-visit gaps; visit counts across biomarker or symptom levels; and the fraction of participants with long gaps or no follow-up measurements. Plotting individual visit histories against biomarker values can reveal patterns that a single mean gap or pooled mean misses. These checks describe the data; they do not establish that a particular visit mechanism has been fully measured.

The main alternatives answer distinct questions. A scheduled-visit summary describes measurements at planned times and can be compared at matching occasions. A pooled triggered-visit mean describes the average among recorded visits and therefore reflects the trigger pattern. A participant-averaged summary gives each participant equal weight but still averages over the times each person happened to be observed. Inverse-intensity weighting aims to represent a specified person-time target when the intensity can be modeled adequately and the required observation conditions hold. The earlier material on inverse-intensity weighting and conditions for ignoring the visit process gives the broader foundations for these approaches; here the key issue is to match the summary to its target.

A simple comparison has important limitations. The hypothetical calculations do not adjust for changes in who remains under observation, differences in follow-up duration, measurement error, or differences in the underlying cohorts. If the schedules are applied to different groups, schedule and group differences may be entangled. Even in the same group, symptoms, treatment changes, or other time-varying factors may influence both visits and biomarker values. Weighting is not a cure for unrecorded triggers, poorly estimated intensities, or a mismatch between the target and the observed follow-up.

A common mistake is to say, “The triggered mean is higher, so the biomarker must be higher in the triggered group.” In the example, the observed triggered visit mean is higher, but the same participants and different observation times produced the records. The correction is to state what was averaged, describe how visits were distributed, and avoid interpreting a visit-weighted difference as a group or causal difference without an analysis that addresses those issues.