Conditioning on Event-Free Biomarker Visits

Why Event-Free Visits Change the Data

In a longitudinal study, a biomarker may be measured at scheduled visits only while a participant is alive and has not yet experienced the event of interest. A hospitalization, relapse, or other terminal event can therefore stop the sequence of measurements. At a later visit, the recorded biomarker values do not represent everyone who entered the study: they represent those who remained event-free long enough to be measured.

This distinction matters even when every scheduled measurement is obtained from every eligible participant. The eligible group changes over time as events occur. Its average biomarker value can change because individuals' biomarkers change, because the people remaining in the group differ from those who experienced an event, or both. A comparison that ignores this changing group can describe the observed survivors while being mistaken for a trajectory in the original cohort.

The value \(T_i\ge t\) means the participant has had no event before time \(t\); a study must specify how it handles an event occurring exactly at a scheduled measurement time. For simplicity, the examples below use visits at which event status is determined before the biomarker measurement. The central point is unchanged: the distribution measured at time \(t\) is conditional on eligibility at that time.

The Conditional Distribution Among Those Still Event-Free

Write \(f(y)\) for the biomarker distribution in the original cohort at a given time, if the biomarker is meaningfully defined for the target population, and let \(p(y)=P(R(t)=1\mid Y(t)=y)\). The distribution among event-free participants is weighted by the chance of being event-free at each biomarker value:

The denominator is the overall probability of remaining event-free. Unless event-free probability is constant across biomarker values, this conditional distribution need not match the original distribution. The same point applies when event-free status is related to other participant characteristics associated with the biomarker: the observed group may have a different mix of people, even if the biomarker is not itself the reason an event occurs.

A direct consequence is that the average biomarker among event-free participants is generally a weighted average, not the average in the original cohort:

This expression follows by taking the expectation of \(Y(t)\) under the conditional distribution above. It also shows why the direction of the change is not fixed: if higher biomarker values are less likely to remain in the observed group, the event-free mean tends to be lower; if they are more likely to remain, it can be higher. No direction can be concluded from the fact of selection alone.

Conditional Event Associations Have a Specific Target

A biomarker recorded at a visit can be used to describe or predict events occurring after that visit among participants who are event-free then. Such an analysis has a defined starting population: the participants who reached the visit without the event and had a biomarker measurement. It does not, by itself, describe the risk from study entry for everyone enrolled. Participants whose events occurred before the visit are absent from that analysis, and their visit biomarker values are unavailable under the measurement rule.

The distinction is between different questions, not simply between a correct and an incorrect calculation. A post-visit risk comparison among event-free participants may be useful for decisions made at that visit. A whole-cohort cumulative event risk from entry is a different quantity. Calling the former an estimate of the latter, without accounting for the different time origin and eligible group, changes the interpretation.

How Survivor Composition Can Mask Within-Person Change

A sequence of observed means at successive visits can combine two processes: change in biomarker values within people and change in which people contribute measurements. The following example makes both contributions visible. It does not require the biomarker to cause the event or the event to cause a biomarker change; it is an illustration of who is included in each mean.

Assumptions, Diagnostics, and Alternatives

The calculations above assume that the stated counts and biomarker values are measured correctly, and that the only reason a scheduled measurement is unavailable is an event before that visit. In practice, some event-free participants may also miss visits, withdraw, or have an unusable specimen. Then the measured sample is selected both by event-free status and by attendance or measurement availability. Interpreting its mean as the mean for all event-free participants requires additional assumptions about those other forms of missingness.

Useful checks begin with the observation process itself. At every scheduled time, report how many participants were enrolled, how many experienced the event before the visit, how many remained event-free, and how many actually supplied a biomarker. Plot visit means together with the number measured and, when relevant, summaries by baseline characteristics. A sharp decline in sample size or a changing mix of measured participants is a warning that the observed mean may not represent a stable cohort. These checks reveal the extent of selection; they cannot establish what the unobserved biomarker values would have been.

The main alternatives answer different questions. A survivor average deliberately summarizes participants who remain event-free at each time. A landmark analysis starts at a chosen visit and studies subsequent outcomes among those event-free and measured at that landmark. A whole-cohort trajectory targets change in the original cohort, but cannot be obtained simply by averaging the available values when event-related termination changes who is observed. Joint modeling provides a framework for analyzing longitudinal biomarker and event processes together; it does not make the distinction between these targets disappear.

A common mistake is to say, “The mean among those measured at 12 months is the cohort’s 12-month mean because the study began with a cohort.” That reasoning treats the people contributing to the 12-month mean as if they were still the original group. The correction is to state the conditioning explicitly: this is the mean among participants event-free and measured at 12 months. Whether that is the desired summary depends on the scientific question.