Deriving Neighboring-Case Constraints Across Visits

From Adjacent Patterns to Constraints Across Visits

With monotone dropout, a participant is observed through some visit and then has no later outcomes recorded. A restriction that links only adjacent dropout patterns addresses the first missing visit after dropout. To specify restrictions for later missing visits too, we need a systematic way to find an observed counterpart for each one. The rule here is direct: for a missing outcome at visit \(j\), use its marginal distribution in the nearest longer-followed pattern that observed visit \(j\).

The rule is an identifying assumption, not a consequence of monotone dropout. It can supply missing pattern-specific distributions for population summaries, but it does not say that dropout is random, nor does it determine how the missing outcomes relate jointly to one another or to observed outcomes for the same participant.

Patterns are ordered by how long participants were followed. For a missing visit \(j\) in pattern \(c<j\), patterns \(c+1,\ldots,j-1\) are longer-followed than pattern \(c\) but still do not observe visit \(j\). Pattern \(j\) is the first pattern in that ordering that does observe it. This is why the donor pattern is \(j\), rather than necessarily the immediately adjacent pattern \(c+1\).

The equality is marginal: it concerns \(Y_j\) alone within each pattern. In precise terms, for every measurable set \(A\),

If the outcome has a finite mean, equality of distributions also equates the pattern-specific means. The rule does not, by itself, equate conditional distributions such as the distribution of \(Y_j\) given \(Y_1\), or specify the joint distribution of \(Y_{c+1},\ldots,Y_J\).

Deriving the Visit-Specific Population Margins

Let \(\pi_c=P(C=c)\) be the proportion in pattern \(c\), with \(\sum_{c=1}^J\pi_c=1\). For visit \(j\), patterns \(c\geq j\) observed that visit, while patterns \(c<j\) did not. Applying the law of total probability first gives the population distribution as a weighted sum of all the pattern-specific distributions. The constraints replace each unobserved distribution in that sum with its donor distribution:

The first term combines all patterns that dropped out before visit \(j\); under the restriction, each contributes the same visit-\(j\) marginal as pattern \(j\). The remaining terms use distributions observed in their own patterns. If \(\mu_{j\mid c}=E(Y_j\mid C=c)\) exists, the corresponding population mean is

These expressions clarify what the restriction does: it fills in missing pattern-specific margins, after which the population margin follows by weighting patterns by their population proportions. The restriction is not a statement that all observed pattern means are equal. Patterns that observed visit \(j\) retain their own observed distributions in the sum.

Worked Implications

Assumptions, Diagnostics, and Alternatives

The construction requires a valid monotone pattern classification: pattern \(c\) must mean that visits through \(c\) are observed and later visits are not. It also requires a nonempty donor pattern \(j\) for every visit \(j\) whose missing margins are to be supplied. In a sample, an observed donor pattern is needed to estimate the donor distribution; a very small donor group can make that estimate unstable. For mean calculations, the relevant means must exist. These conditions make the procedure usable, but they do not make its identifying equality true.

A practical diagnostic is to check whether the observed-data layout really follows the assumed monotone pattern structure and whether every needed donor pattern has adequate data. Check the visit labels carefully: a one-visit indexing error changes which distribution is borrowed. Also distinguish pattern proportions from donor sample size. The population mean weights each dropout pattern by its own \(\pi_c\); it does not give every pattern equal weight merely because one pattern supplied the donor distribution.

The key untestable part is the equality involving a missing outcome: the data do not reveal \(F_{j\mid c}\) when \(c<j\). Observed data can describe \(F_{j\mid j}\) and the pattern weights, but cannot verify that the missing pattern has that same visit-\(j\) distribution. Results should therefore be understood as conditional on the restriction, rather than as empirical confirmation of it.

The earlier material on complete-case and available-case restrictions concerns different ways of constraining or combining pattern information. Here, the distinctive feature is the visit-by-visit donor match across patterns. Compared with a restriction limited to adjacent dropout patterns, this rule specifies a missing visit directly from the nearest pattern that actually observed that visit. It still supplies only marginal distributions. Additional assumptions would be needed to specify dependence among multiple missing visits or dependence between missing and observed outcomes.