Boundary Patterns and Missing Neighbors

Why Boundary Patterns Need Special Care

Neighboring-case restrictions transfer information between patterns with adjacent dropout times. For an interior pattern, there may be a neighbor on either side. At an endpoint, one of those neighbors is absent. This is not a minor bookkeeping issue: a restriction can assign a missing outcome only if the proposed donor pattern actually observes that outcome, or if an additional assumption supplies it.

Consider a study with three scheduled visits and monotone dropout: once a participant misses a visit, later visits are also missing. A boundary pattern is the earliest pattern that drops out, or the latest incomplete pattern. Completers are a special endpoint: they have no missing outcomes, but they can sometimes serve as donors for an incomplete pattern. Whether they are allowed as donors must be stated, not silently presumed.

The observed data identify \(\mu_{k,t}\) directly when \(t\leq k\). When \(t>k\), that pattern-specific mean is not observed. A neighboring-pattern restriction supplies an identifying assumption, for example \(\mu_{k,t}=\mu_{k+1,t}\) when pattern \(k+1\) observes visit \(t\). The equality is an assumption about the unobserved outcome distribution, not a fact established by the observed values.

What Counts as a Permissible Boundary Assignment?

A donor must observe the outcome being assigned. In the three-visit setting, pattern 1 can borrow \(Y_2\) from pattern 2 because pattern 2 observes \(Y_2\). But pattern 2 does not observe \(Y_3\). Consequently, borrowing \(Y_3\) for pattern 1 from pattern 2 is not a permissible direct assignment. Pattern 3 does observe \(Y_3\), so it could be used as a donor, but doing so asserts a direct equality between the relevant pattern-specific distributions. Alternatively, a chain of restrictions might connect patterns 1, 2, and 3; that chain requires each link to be stated and justified as an assumption.

At the other edge, pattern \(T-1\), the latest incomplete pattern, can borrow \(Y_T\) from pattern \(T\) if completers are permitted donors. If a proposed restriction is limited to dropout patterns and excludes completers, then the latest incomplete pattern has no later dropout-pattern donor. The analysis must then use another explicit assumption or acknowledge that the outcome is not identified by the proposed neighboring restriction.

Worked Assignments at the Two Edges

Why the Mixture Formula Works

For a fixed visit \(t\), the full population is partitioned into mutually exclusive patterns. The probability of belonging to pattern \(k\) is \(\pi_k\), and the mean outcome within that pattern is \(\mu_{k,t}\). The law of total expectation therefore gives the overall mean as a weighted sum of pattern means. The derivation also makes clear why an assigned mean must be supplied for every pattern missing visit \(t\).

Assumptions, Diagnostics, and Alternatives

A boundary assignment is interpretable only after its assumptions are made explicit. At minimum, state the monotone pattern definition, the donor pattern, the outcome being transferred, and whether the equality concerns a mean or a fuller distribution. For a distributional assignment, equal means alone are insufficient: the assumed distributions must match in the features the analysis uses. Also check that the pattern proportions and donor summaries refer to the intended analysis population and that the donor pattern has enough observations for a stable estimate.

Observable comparisons can reveal problems but cannot verify equality for outcomes missing in the recipient pattern. For example, patterns 1 and 2 both observe \(Y_1\); a large difference in their observed \(Y_1\) distributions warns that a broad marginal borrowing assumption may be implausible. Conversely, similar observed \(Y_1\) distributions do not prove that their missing \(Y_2\) distributions are equal. The distinction is between checking observable implications and testing the missing-outcome assumption itself.

When an adjacent donor is absent or too sparse, the main alternatives are to use a farther pattern that observes the outcome, to state a model-based extrapolation, or to present sensitivity analyses under multiple plausible assignments. Each changes the assumption being made. A farther donor may be less comparable; a model requires its own specification and support; sensitivity analysis shows how conclusions vary but does not make an unsupported value data-identified. Complete-case analysis avoids assigning outcomes to incomplete patterns, but targets a different population summary unless further assumptions connect completers to the full population.

• Check donor support: confirm that the donor pattern actually observed the visit being assigned.
• Check pattern size: a mathematically available donor can still yield an unstable estimate when its sample is small.
• Check observable compatibility: compare outcomes observed in both patterns, while treating this as a diagnostic rather than proof.
• Check the full assignment: ensure every missing pattern-visit mean needed in the mixture has a specified source.

Common Boundary Mistakes

A common error is to say, “Pattern 2 is adjacent to pattern 1, so it can supply all of pattern 1's missing outcomes.” Adjacency does not make unobserved outcomes available: pattern 2 supplies \(Y_2\), not \(Y_3\), in a three-visit study. The correction is to check donor observation visit by visit and add a separate, named assumption wherever the chain stops.

A second error is to assume that completers automatically count as a donor for the latest dropout pattern. They may be an appropriate adjacent donor, but only under a restriction that explicitly permits that comparison. Another error is to call an assigned value “estimated from the data” when the pattern-specific outcome is entirely unobserved. The observed data may estimate pattern weights and donor summaries; the equality or extrapolation that fills the gap remains an assumption.