Applying Complete-Case Restrictions to Means

From a Complete-Case Restriction to a Cohort Mean

In a longitudinal study, some participants may stop contributing measurements before the final visit. The mean at that visit is directly observed for completers, but not for the participants who dropped out earlier. A pattern-mixture analysis describes outcomes separately by dropout pattern. To obtain an overall mean, it must also specify how the unobserved outcomes in each pattern relate to outcomes that were observed.

A mean-level complete-case restriction supplies one such link. It equates the conditional mean of a missing outcome in an incomplete pattern to the corresponding conditional mean among complete cases, given a specified observed history. The restriction does not say that every participant who dropped out would have had the same outcome as a completer. It says that conditional means are equal at the chosen history level. The overall mean is then a weighted average of the pattern-specific means, with weights determined by how many participants belong to each pattern.

For a two-visit study with dropout before visit 2, the history \(H\) might be the visit-1 outcome, possibly together with baseline covariates. If the histories are discrete strata, estimate the conditional complete-case mean in each stratum and average those stratum means using the dropout pattern’s stratum proportions. If a regression model is used to represent the conditional mean, the same principle applies: predict at each dropout participant’s observed history and average the predictions. The restriction concerns the conditional mean, not necessarily the full conditional distribution.

Why the History Distribution Matters

A common shortcut is to assign every dropout participant the crude mean among complete cases. That shortcut is justified only under additional conditions—for example, if the conditional complete-case mean is constant across histories, or if the relevant history distributions happen to produce the same average. In general, dropout participants and completers can have different observed histories. The conditional restriction therefore requires averaging the complete-case conditional means over the dropout group’s histories, rather than over the completers’ histories.

The calculation follows from the law of total expectation. For a discrete history \(H\), partition the dropout pattern into history strata. Within each stratum, use the complete-case conditional mean under the restriction; then weight by the proportion of the dropout pattern in that stratum:

This is a pattern-specific mean. To reconstruct the cohort mean at visit \(t\), combine each pattern’s mean at that visit using its cohort proportion \(n_r/N\), where \(n_r\) is the number of participants in pattern \(r\) and \(N\) is the total cohort size:

Here \(\hat{\mu}_{t,r}\) is observed for a pattern that has measurements at visit \(t\), and it is restriction-based when the pattern lacks that visit. Pattern weights are not the number of participants with an observed visit divided by the number observed at that visit. The target is the mean for the original cohort, so the weights sum over the full cohort.

Applying the Restriction Sequentially

With more than two visits, a participant may have observed several outcomes before dropping out. The history used for a missing visit can then include the earlier observed outcomes. For example, when estimating visit 3 for participants observed through visit 2, one can condition on their visit-1 and visit-2 history and use the complete-case conditional visit-3 mean. This requires overlap in those histories: the complete cases must provide information at histories relevant to the dropout pattern.

For participants missing more than one visit, a mean at a later visit may require a sequence of conditional calculations. A restriction for the next missing outcome can be applied given the history available at that point; subsequent predictions must account for the distribution of histories that includes the earlier missing outcome. This tutorial’s calculations use settings where the history needed for each mean is observed in the pattern being estimated. They should not be extended to an unobserved conditioning variable as if it were known.

Check the Pattern Counts and the Arithmetic

Pattern-specific means must be paired with their own pattern counts. Confusing a stratum count with a pattern count changes the mixture, even if each stratum mean is correct. A useful check is to write each pattern’s contribution as \(n_r\hat{\mu}_{t,r}\), verify the multiplication separately, add the contributions, and divide by the full cohort size.

Assumptions, Diagnostics, and Alternatives

A mean-level complete-case restriction is an identifying assumption, not a consequence of observing the data. Its usefulness depends on defining the patterns and conditioning history in advance and defending the conditional equality as plausible for the scientific setting. It does not require dropout to be unrelated to outcomes without conditioning; rather, it asserts equality of the relevant conditional means across the incomplete pattern and complete cases. The data alone generally cannot verify the equality for missing outcomes.

• Check overlap. Every history represented in a dropout pattern should have comparable complete cases. If a history occurs only among dropouts, its conditional complete-case mean cannot be estimated from that stratum without extrapolation.
• Check the conditioning set. A history must be observed for the pattern to which the restriction is applied. Adding a variable that is itself missing does not solve the problem unless its distribution is separately handled.
• Check the pattern definition. Misclassifying dropout times changes both pattern weights and the histories used for the restriction.
• Check the arithmetic. Pattern counts must add to \(N\); each pattern contribution must use that pattern’s count; and the final mean must be in the outcome’s units.
• Check sensitivity. Because the restriction cannot be tested directly for missing outcomes, compare conclusions under defensible alternative assumptions rather than treating the restricted mean as assumption-free.

The complete-case-only mean uses outcomes from completers and avoids imputing outcomes in dropout patterns, but it describes the complete cases unless additional conditions justify interpreting it as a cohort mean. The mean-level restriction instead uses complete cases to supply conditional means and then restores the dropout patterns with cohort weights. A fully specified model for outcomes by pattern can impose a different set of assumptions; the choice should reflect the scientific question, available overlap, and plausibility of the required extrapolation. In particular, a restriction-based estimate can be unstable when complete cases are sparse within important histories, even if the overall number of complete cases appears adequate.

The essential distinction is between two kinds of weights. Within an incomplete pattern, history proportions determine how its restricted conditional means are averaged. Across patterns, pattern proportions in the original cohort determine how pattern-specific means are combined. Mixing these weighting steps—or using the completer history proportions for the dropout group—can produce a numerically precise but substantively different quantity.