Deriving Bias When High-Risk Participants Drop Out

Why Selective Loss Changes What We Observe

Suppose a cohort contains participants at both higher and lower risk of an adverse outcome. If higher-risk participants are less likely to remain under observation, the group still observed at a later visit is no longer composed in the same proportions as the original cohort. Even if the outcome within each risk group does not change, the overall observed mean or event rate can shift simply because the group mix changed. This is a basic calculation underlying selection bias among participants who remain under observation.

The calculation is deliberately simple: divide the cohort into two groups, specify each group’s outcome average and retention probability, and compare the original cohort value with the value among retained participants. It isolates the contribution of changing group composition. It does not, by itself, establish why participants dropped out or provide a complete analysis of longitudinal data.

For this calculation, assume that group membership and retention probabilities refer to the same cohort and target time. Also assume that within each group, retained participants have the group’s specified outcome mean: \(E(Y\mid G=H,R=1)=\mu_H\) and \(E(Y\mid G=L,R=1)=\mu_L\). In other words, retention may differ between the two groups, but this calculation does not include additional selection on outcome values within a group. These assumptions make the group-composition effect explicit; real settings can be more complicated.

Deriving the Retained-Sample Mean

Among the original cohort, the proportion both high risk and retained is \(p q_H\); the proportion both low risk and retained is \((1-p)q_L\). Thus the overall retention probability is \(P(R=1)=p q_H+(1-p)q_L\), and the proportion of retained participants who are high risk is

The remaining retained participants are low risk. Weighting the group means by their proportions among retained participants gives the observed mean:

Define the selection bias here as the retained-sample mean minus the original-cohort mean. Subtracting \(\mu=p\mu_H+(1-p)\mu_L\) and collecting terms gives a compact expression:

The signs matter. The formula predicts a downward shift only when the two contrasts have opposite signs. If the high-risk group has a lower value of \(Y\), selective loss of that group can instead raise the observed mean. The labels “high risk” and “low risk” alone do not tell us the direction of bias for every outcome; the meaning and scale of \(Y\) must also be specified.

Worked Calculations

Assumptions, Diagnostics, and Alternatives

The two-group formula is a decomposition, not a universal dropout correction. It is most useful when groups can be meaningfully defined before the target measurement, the original group proportions and retention rates can be described, and the outcome has a coherent meaning for the full cohort at the target time. For a fixed-horizon event rate, the event outcome must also be defined consistently, and the calculation needs an assumption about whether event status is observed for those who remain. If outcome values within a risk group also affect retention, the retained group mean may differ from the group mean used in the formula; the simple between-group expression then does not capture all selection.

Useful checks include comparing retention rates across baseline risk groups, comparing observed outcome summaries across those groups, and checking whether the retained group proportions differ from the original proportions. These checks can reveal the ingredients for a composition shift when the relevant groups and outcomes are observed. They cannot establish that unobserved outcomes among dropouts match outcomes among retained participants. A small difference in observed retention rates is not proof of no bias if group outcome contrasts are large, and similar group averages do not rule out selection within groups.

The independent-dropout benchmark provides a useful contrast: if \(q_H=q_L\), the group proportions are unchanged by retention, and the formula gives zero composition bias. This is a benchmark for the between-group mechanism, not a guarantee that every analysis is unbiased. Another possible comparison is to retain group-specific summaries rather than report only a pooled retained mean; that makes changes in group mix visible but does not recover unobserved outcomes. More elaborate models or adjustment strategies require additional assumptions and information; this two-group calculation alone does not select among them.

The key limitation is that the result depends on the chosen grouping. A broad “high-risk” category may conceal meaningful differences in risk or retention within that category. Also, the formula describes a difference in averages, not a causal effect of dropout, and it does not imply that individual outcomes changed. It quantifies how observed composition can move a pooled summary under stated assumptions.

Common Pitfall

A tempting but incorrect argument is: “Only 10% of participants dropped out, so the observed mean cannot be far from the cohort mean.” The amount of dropout alone does not determine the bias. If nearly all losses come from a small group whose outcome mean is very different, the retained mean can shift substantially. Conversely, even substantial dropout need not create this particular composition bias if retention is equal across groups or the group means are equal. The correction is to examine both contrasts—retention and outcome—and the group proportions, then use the formula rather than infer bias from the overall dropout percentage.