Deriving Selection Bias Under Outcome-Triggered Visits

Why Visit-Triggered Measurements Can Be Unrepresentative

When a high biomarker value makes a clinical visit more likely, the measurements recorded at visits need not represent the biomarker states patients occupy over time. A patient who spends a small fraction of follow-up in a high state may nevertheless contribute a large fraction of observed measurements from that state if visits occur much more frequently while the patient is high. This is a sampling distortion caused by the observation schedule; it does not by itself show that the biomarker causes any clinical outcome.

The central calculation compares two distributions: the distribution of states across patient-time and the distribution of states across visit events. The distinction matters because patient-time is accumulated continuously, whereas a visit-level data set gives more weight to states that generate more visits. The derivation below uses long-run occupation times and state-specific event rates, rather than assuming that long-run occupation proportions describe the state at every particular instant.

Assumptions and Long-Run Derivation

For the basic result, assume the state is well defined just before each visit, visits form a simple counting process, and the conditional visit intensity is \(\lambda_H\) in state \(H\) and \(\lambda_L\) in state \(L\). Thus, over a short interval of length \(dt\), conditional on the history, the chance of a visit is approximately \(\lambda_sdt\) when the current state is \(s\); the chance of two or more visits is negligible relative to \(dt\). Assume the long-run occupation proportions exist and that state-specific visit counts track their accumulated conditional rates over long follow-up. A standard sufficient setting is a recurrent process with suitable regularity and finite intensities, so the visit-count martingale fluctuations divided by \(T\) vanish as follow-up grows.

The intensity assumption gives a cumulative conditional rate of \(\lambda_s A_s(T)\) visits while in state \(s\). Under the long-run conditions just stated, \(N_s(T)/T\) approaches \(\lambda_s\pi_s\). Therefore, if the overall long-run visit rate is positive, the fraction of visits made in state \(H\) approaches the high-state rate per unit follow-up divided by the total rate:

The positivity condition is essential: if both state-specific contributions to the overall rate are zero, there are no long-run visits and a visit-level state proportion is undefined. The corresponding low-state fraction is \(q_L=1-q_H\). The formula weights each state by both its share of patient-time and its visit rate while occupied.

Subtracting the time-based high-state proportion from the visit-based proportion gives a direct measure of the distortion:

When both states occupy positive fractions of follow-up and the total visit rate is positive, this difference is positive exactly when \(\lambda_H>\lambda_L\). If the state-specific rates are equal and positive, the two state distributions agree. If one state has zero occupation, there is no two-state mixture to distort. These statements concern the limiting distribution across visits; with a finite number of visits, the realized fraction varies around that long-run quantity.

Worked Examples

Interpreting the Bias and Checking the Assumptions

For two possible biomarker values \(y_H\) and \(y_L\), the difference between the visit-weighted mean and the time-weighted mean is

If \(y_H>y_L\) and \(\lambda_H>\lambda_L\), this difference is positive: the mean calculated from visits exceeds the time-weighted mean under this model. The interpretation is a difference between sampling distributions. It is not a causal effect of the biomarker on itself, nor does it establish that a clinical outcome is biased in the same direction.

In practice, checking the calculation requires attention to how states and exposure are defined. If the state is known over follow-up, one can compare the amount of time spent in each state with the number of visits occurring in each state. Dividing state-specific visit counts by state-specific exposure gives empirical state-specific rates; their values can be compared with the proposed \(\lambda_H\) and \(\lambda_L\). If measurements are available only at visits, the intervening state may be unknown, so the calculation may not be directly checkable from those measurements alone. A scheduled assessment or other information about the trajectory may be needed to assess the trigger mechanism.

The simple model can fail if visit intensity varies within a state, depends on additional history, or differs across patients in ways associated with their state histories. In those settings, the relevant long-run rate is based on accumulated conditional intensities, not necessarily one common \(\lambda_s\) multiplied by one pooled occupation proportion. A lag between a triggering value and the visit can also mean that the state at the visit differs from the state that generated the visit. The model must then distinguish the trigger state from the measured state.

Comparison, Limitations, and a Common Mistake

A measurement collected at a fixed time chosen independently of the biomarker state targets a time-based distribution, subject to other selection mechanisms. A measurement collected whenever a state-dependent visit occurs targets the visit-based distribution derived above. These are different sampling schemes. The formula isolates one reason for their difference; it does not account for nonattendance, dropout, terminal events, measurement error, or other selection mechanisms unless those are explicitly incorporated into the assumptions.

The two-state calculation is useful for understanding the mechanism but is deliberately restrictive. Real biomarkers are often continuous, and visit rates may vary smoothly with the current value, with past values, or with patient characteristics. The result also gives a long-run mixture, not a guarantee about the proportion in a short study or a particular calendar-time interval. Finite samples, changing state distributions, and changes in clinical practice can all cause observed proportions to differ from the long-run expression.

• Common mistake: “High values occupy 25% of patient-time, so 25% of observed visits should be high.” This ignores the different visit rates across states.
• Correction: Calculate the state-specific visit contributions \(\pi_H\lambda_H\) and \(\pi_L\lambda_L\), add them, and divide the high-state contribution by the positive total visit rate.
• Another mistake: Use long-run occupation proportions as the state probabilities at every particular time. The alternating-state example shows why this is not valid without additional assumptions such as stationarity at the time considered.

This derivation quantifies the source of the distortion, but it does not itself correct visit-weighted analyses. Any adjustment requires additional modeling and assumptions about the observation process; the long-run formula is a way to understand what needs correcting, not a universal correction procedure.