Dropout Changes the Composition of the Observed Population Mean

Why a Pooled Mean Can Change When Subjects Drop Out

At a given visit, a pooled concentration mean summarizes the subjects whose concentrations are recorded at that visit. If subjects with particular concentration profiles stop contributing measurements, the set of subjects behind the mean changes. The later pooled mean can therefore differ from an earlier mean for two reasons: concentrations changed within subjects, and the observed subjects are no longer compositionally the same group.

These are distinct explanations. A pooled mean is not automatically the mean trajectory of a fixed group of subjects. To interpret it, identify the visit-specific denominator and consider how the subjects who remain differ from those who dropped out. The calculations below use hypothetical data; the concentrations and dropout patterns are illustrative, not study results.

For comparison, the fixed-cohort mean at visit \(j\) would average the concentrations for the entire original cohort, whether or not every value was recorded. When some values are missing, that mean cannot be calculated directly from observed concentrations alone. It is a different target from the observed-case mean and requires additional information or assumptions to estimate.

Separating Within-Subject Change from Composition

Consider a cohort that had baseline measurements for everyone. At a later visit, divide the original cohort into subjects still observed and subjects who have dropped out. Let \(S\) be the set still observed at the later visit and \(D\) the set no longer observed. Write \(\bar{C}_{S,0}\) for the baseline mean among eventual survivors, \(\bar{C}_{D,0}\) for the baseline mean among eventual dropouts, and \(\bar{C}_{S,1}\) for the later mean among survivors. The baseline mean for the full cohort is a weighted average of the two baseline subgroup means.

Add and subtract \(\bar{C}_{S,0}\) from the observed later-minus-baseline difference. This gives an exact decomposition, not an approximation:

The first term is the mean within-subject change among the subjects who remain observed, because the same subjects contribute at both visits. The second term is a baseline composition difference: it compares the eventual survivors’ baseline mean with the full cohort’s baseline mean. If survivors started with higher concentrations, this term is positive; if they started lower, it is negative. The decomposition explains why a pooled change need not equal the change experienced by a typical subject in the original cohort.

This calculation has a practical assumption: baseline concentrations are available for the subjects being compared, and the later observed subjects can be identified as members of the baseline cohort. It does not recover the unobserved later concentrations of dropouts. In particular, the decomposition describes the observed survivors and their baseline composition; it does not prove what the full cohort’s later mean would have been.

How the Dropouts’ Profiles Determine the Direction

A second useful identity compares the observed later mean with the full-cohort mean that would be available if every subject had a value at that visit. Let \(q=n_D/n\) be the dropout fraction, and let \(\bar{C}_{D,1}\) denote the mean later concentration among dropouts. The full-cohort mean is the weighted average of the survivor and dropout means. Subtracting it from the survivor mean gives:

Thus, if dropouts would have had lower later concentrations than survivors, the observed-case mean is higher than the full-cohort mean; if they would have had higher concentrations, it is lower. If the two groups have equal means, this particular compositional difference is zero. The identity is exact, but in real dropout data \(\bar{C}_{D,1}\) is usually unobserved. The direction and size of the difference cannot be confirmed from the observed later concentrations alone.

When Dropout Does and Does Not Shift the Mean

Dropout by itself does not guarantee a compositional shift. The issue is whether the probability of remaining observed is related to concentration profiles relevant to the mean. If, at a visit, subjects are lost in a way unrelated to their concentrations at that visit, then the observed-case mean can be unbiased for the full-cohort mean under suitable repeated-sampling assumptions. That does not mean the observed mean must equal the full-cohort mean in every realized sample: random selection can produce a difference by chance, especially when few subjects remain.

In contrast, when dropout is associated with concentration level or trajectory, the observed group can systematically differ from the original cohort. For example, if subjects with low concentrations are more likely to stop contributing measurements, the observed mean tends to be higher than the full-cohort mean. If higher-concentration subjects are more likely to drop out, the direction may reverse. A reason for dropout need not be concentration itself: a clinical event or burden of participation may be related to a profile and thereby alter which subjects remain.

Keep the target clear when comparing summaries. An observed-case mean answers, “What was the mean among subjects measured at this visit?” A fixed-cohort mean answers, “What was the mean for the original cohort at this visit?” These questions are not interchangeable. A mean based only on completers can be a useful description of completers, but it should not silently be interpreted as a description of everyone enrolled.

Diagnostics, Alternatives, and Limits

Start by reporting the number of subjects contributing at every visit. Then examine whether the subjects who later drop out differ from those who remain in characteristics already observed, such as baseline concentration or early concentration changes. A plot of individual profiles can help reveal whether measurements end disproportionately among subjects with particular levels or trajectories. These checks can identify warning signs, but they cannot establish the unobserved concentrations that dropouts would have had at later visits.

A few alternatives answer different questions rather than automatically solving the same one. The observed-case mean is transparent and requires no substituted values, but describes only those observed. A baseline-cohort comparison can show whether eventual survivors started differently from the full cohort, as in the decomposition above, but it does not supply their missing later values. A standardized summary can combine subgroup means using fixed, stated weights, but its interpretation depends on the subgroups and weights chosen and on having the information needed to calculate the subgroup means. No arithmetic choice alone reveals missing outcomes without assumptions.

A useful failure check is to ask whether the displayed mean and denominator refer to the same target at each visit. If the denominator shrinks, a change in the mean may reflect changing composition as well as concentration change. If the dropout pattern is related to concentration profiles, treating the later observed mean as if it were the original cohort’s mean is especially difficult to justify. Conversely, changing denominators alone do not prove bias: the relevant issue is how observation relates to the concentrations of interest.