Restriction Choices at the Final Unobserved Visit

Why the Final Visit Needs Careful Treatment

In a longitudinal study with monotone dropout, a participant who misses a visit does not return at a later visit. At the final visit, this structure creates a special boundary: the only participants with an observed final outcome are the completers. As a result, some restrictions that sound like different borrowing rules produce the same marginal assignment. Other versions differ because they condition on participants’ observed history.

This distinction matters when assigning a final outcome to people who dropped out earlier. An assigned mean is not recovered from their missing final outcomes; it follows from a restriction imposed on the pattern-specific distribution. Earlier material on neighboring-case restrictions establishes the general idea of borrowing across patterns. Here the focus is the final-visit boundary in a four-visit setting, where the available donor patterns are limited.

The restriction choices below are stated for means. A distributional equality would be stronger: it would constrain the entire distribution of \(Y_4\), not only its mean. Mean equality alone does not imply equal variances, quantiles, or shapes.

Three Ways to Assign a Final-Visit Mean

A marginal complete-case restriction assigns each dropout pattern the completers’ final-visit mean. An available-case assignment uses the mean among people whose final outcome is observed. Under monotone dropout, those are exactly the completers. Therefore, if both rules use the same analysis population and neither conditions on additional variables, they give the same marginal assignment.

A neighboring-case restriction instead relates adjacent dropout patterns. For example, the pattern \(D=3\) is adjacent to the complete-case pattern \(D=4\). A marginal equality between their final-outcome means assigns the completers’ mean to \(D=3\). To reach \(D=2\) or \(D=1\) by adjacent steps requires further restrictions linking those patterns in turn. Those intermediate final outcomes are missing too, so the chain is an assumption, not a sequence of observed donor means.

Conditioning on History Can Change the Assignment

The marginal equality does not say that people who dropped out and completers have the same distribution of observed history. If the relation between \(Y_4\) and that history matters, one can specify a conditional restriction instead. For \(D=3\), a natural shared history includes \(Y_1,Y_2,Y_3\); a neighboring-case restriction may equate the conditional mean of \(Y_4\) in \(D=3\) and \(D=4\) at comparable histories.

To calculate such an assignment from observed data, the relevant history groups must occur among completers. This is an overlap or positivity requirement: each history value being used for \(D=3\) must have completers from whom a conditional \(Y_4\) mean can be estimated. Even with overlap, equality of the conditional means is an identifying assumption about a missing outcome, not something verified by observing \(Y_4\) in both patterns.

Deriving the Marginal Coincidence

The equality between the complete-case and available-case means follows from the definition of monotone dropout, not from a statistical approximation. Let \(R_4=1\) indicate that \(Y_4\) is observed. Under monotone dropout, \(R_4=1\) if and only if \(D=4\). Therefore the available-case mean targets the same observed pattern mean as the complete-case mean:

If a marginal complete-case restriction sets \(\mu_{k4}=\mu_{44}\) for each \(k<4\), it assigns that same value to each missing pattern. If instead adjacent restrictions set \(\mu_{k4}=\mu_{k+1,4}\) for \(k=1,2,3\), repeated substitution gives \[ \mu_{14}=\mu_{24}=\mu_{34}=\mu_{44}. \] The neighboring chain reaches the same endpoint only because every link is assumed. A single restriction between \(D=3\) and \(D=4\) says nothing by itself about \(\mu_{24}\) or \(\mu_{14}\).

Diagnostics, Comparisons, and Limitations

Observable data can reveal whether a proposed calculation is feasible, but cannot generally verify the restriction on missing \(Y_4\). For a conditional assignment, examine the observed history distributions by pattern and check that the donor pattern has observations in the history groups used. A history group absent among completers has no corresponding observed \(Y_4\) mean to borrow. Sparse groups also make estimated conditional means unstable. Neither a good overlap table nor similar observed histories proves equality of the unobserved final outcomes.

The main alternatives answer different questions. A marginal complete-case restriction assigns the completer mean without adjusting for history. The marginal available-case calculation uses all observed \(Y_4\) values; under monotone dropout this is the completer sample, so it does not add another donor group. A neighboring-case restriction relates patterns locally and may be stated marginally or conditional on shared history. Conditional restrictions can account for observed-history composition, but require overlap and stronger modeling or stratification choices. A chain of neighboring restrictions can also accumulate assumptions as it moves from \(D=4\) to earlier dropout patterns.

These are identifying restrictions, not tests that establish equality for missing outcomes. Observed-data compatibility checks can flag contradictions with the observed-data structure, but agreement with observed data does not distinguish among restrictions that assign different values to unobserved \(Y_4\). It is prudent to state whether a restriction is marginal or conditional, which patterns it links, and which history variables it conditions on. When the final estimate depends on unsupported links or sparse histories, report that dependence and consider alternative assignments as a sensitivity analysis.