Neighboring-Case Restrictions in a Three-Visit Trial

Why Neighboring Patterns Matter

In a three-visit trial with dropout, each participant's observed visit pattern helps determine which outcomes are available to estimate directly. A restriction is needed to say how a missing outcome in one pattern relates to outcomes in other patterns. A neighboring-case restriction makes that link using patterns adjacent in the dropout sequence. The assignment must be done visit by visit: the nearest pattern in the sequence may itself lack the outcome, so a restriction can need to pass through more than one pattern.

The calculations below use a monotone pattern: once a participant misses a visit, later visits are also missing. The restrictions are assumptions about pattern-specific means, not facts established by the observed data. Their purpose is to supply values for means that the data alone do not reveal.

For these three patterns, the adjacent steps are \(111\leftrightarrow110\leftrightarrow100\). Thus, the missing visit 3 mean in pattern \(110\) can be linked to the observed visit 3 mean in pattern \(111\). In pattern \(100\), visit 2 can be linked to the observed visit 2 mean in pattern \(110\). Visit 3 in pattern \(100\) can be linked to the restricted visit 3 mean in pattern \(110\), which is itself linked to the observed mean in pattern \(111\). This is a chain of identifying assumptions; it does not make the visit 3 outcomes for patterns \(110\) or \(100\) observed.

Set Up the Restrictions Before Calculating

For a simple marginal mean analysis, the neighboring-case equalities are

The first equality supplies the visit 3 mean for pattern \(110\); the second supplies the visit 2 mean for pattern \(100\); and the third supplies its visit 3 mean. Combining the first and third gives \(\mu_{3,100}=\mu_{3,111}\). The observed means that anchor the chain are the visit 3 mean among pattern \(111\) participants and the visit 2 mean among pattern \(110\) participants.

The mean equalities are marginal over the participants in each pattern. They do not assert that participants in different patterns have identical observed histories, nor do they state equality conditional on those histories. The restriction is most credible when the adjacent patterns are sufficiently comparable for the outcome and target population at issue. In a real analysis, that comparability requires subject-matter justification; the equalities cannot be verified for outcomes that are missing in the relevant patterns.

From Pattern Means to Visit Means

Once each pattern has a mean for a visit, the overall mean at that visit is a pattern-weighted average. Let \(n_r\) be the number of participants in pattern \(r\), and let \(n=\sum_r n_r\). The pattern proportion is \(\hat{p}_r=n_r/n\). For visit \(j\), the restricted mean estimate is

This formula follows by partitioning the trial population into its mutually exclusive patterns. If \(R\) denotes a participant's pattern, then \(E(Y_j)=\sum_r P(R=r)E(Y_j\mid R=r)\). Replacing the population pattern proportions and conditional means by their sample estimates gives the displayed estimator. The identity itself is a partitioning result; the values for missing outcomes still depend on the restrictions.

Assumptions, Diagnostics, and Alternatives

The arithmetic is simple, but its interpretation relies on several conditions. First, the dropout patterns must be defined consistently, and the monotone sequence must be appropriate for the data being analyzed. Second, every observed anchor mean must be based on participants with the stated pattern and visit. Third, the neighboring equalities must be defended as identifying restrictions for the target population and outcome scale. Finally, pattern proportions should reflect the population whose overall mean is being estimated; if the trial has sampling or analysis weights, the corresponding weighted proportions and means may be needed.

Useful checks include confirming that the pattern counts sum to the analysis sample size, verifying that every assigned value has a stated source, and recomputing each weighted mean from the pattern contributions. Compare the observed histories and baseline characteristics across adjacent patterns: large differences do not disprove a marginal equality, but they can make it less plausible. Also examine sensitivity to the anchor means. If a small change in an assumed equality substantially changes the overall estimate, the result is assumption-sensitive and should be reported as such.

A complete-case analysis uses only pattern \(111\), so it does not use the observed outcomes from dropout patterns to estimate the visit means. Available-case summaries use the outcomes observed at each visit but do not, by themselves, specify the missing pattern-specific means needed to describe a common target population. Neighboring-case restrictions instead supply those pattern-specific means through explicit cross-pattern equalities. These approaches answer different questions and rely on different assumptions; none makes the missing outcomes empirically observed.

A common mistake is to assign pattern \(100\)'s visit 3 mean directly from the visit 3 observations in pattern \(111\) and then describe that as a direct neighboring comparison. In the dropout sequence, pattern \(110\) lies between them and itself lacks observed visit 3 values. The correct account is the chain \(\mu_{3,100}=\mu_{3,110}=\mu_{3,111}\), with each equality stated as an assumption. In this particular three-pattern setup, the chain gives the same numerical value as direct equality to \(111\), but the distinction matters: it makes the required restrictions visible and prevents the borrowed value from being mistaken for observed evidence.

The method is limited by the strength and scope of its restrictions. Marginal mean equality does not imply equality of full outcome distributions, and a mean restriction does not automatically justify conclusions about variances, quantiles, or individual trajectories. Nor does a good fit to observed data validate an equality involving an unobserved mean. The resulting estimates should therefore be described as conditional on the selected neighboring-case restrictions, rather than as estimates made assumption-free by the pattern structure.