Neighboring-Case Restriction for Adjacent Dropout Patterns

Why Borrow From an Adjacent Pattern?

In a longitudinal study, a participant who drops out after visit $j$ has an unobserved outcome at visit $j+1$. A participant who remains through visit $j+1$ has that outcome observed, as well as the same earlier history. The neighboring-case restriction uses this adjacent longer-history pattern as a source for the missing outcome distribution. Its central claim is conditional: among participants with the same specified history through visit $j$, the outcome distribution at visit $j+1$ is the same in the two patterns.

This restriction is a modeling choice about the unobserved data, not a consequence of observing adjacent patterns. It is useful to state it at the distributional level because it specifies more than equality of means: it also equates conditional probabilities, quantiles, and other features of the outcome distribution, when those are defined.

Definition: Let $D=k$ denote the dropout pattern in which outcomes through visit $k$ are observed and outcomes after visit $k$ are missing. For adjacent patterns $D=j$ and $D=j+1$, let $H_j$ be a specified history observed through visit $j$, and let $Y_{j+1}$ be the next outcome. The neighboring-case restriction assigns the missing conditional distribution in pattern $D=j$ the distribution observed in pattern $D=j+1$: $$ F_{Y_{j+1}\mid H_j,D=j}(y\mid h) = F_{Y_{j+1}\mid H_j,D=j+1}(y\mid h), $$ for histories $h$ in the relevant common support. Equivalently, where conditional densities or probability mass functions exist, those conditional distributions are equal.

Here $H_j$ must be specified before applying the restriction. It may include the measured outcomes through visit $j$ and any baseline covariates selected for conditioning. It cannot include $Y_{j+1}$, which is the outcome being assigned, or later outcomes. The same definition applies to a prespecified subset of the observed history, but changing that subset changes the assumption.

The equality is about the conditional distribution of $Y_{j+1}$, not the distribution of the entire future outcome sequence. It also does not say that the two dropout patterns have the same distribution of histories $H_j$. Those histories can be distributed differently, even when the conditional distributions agree at each history value.

Conditions for Using the Restriction

The equality has a substantive interpretation only when the two patterns can be compared at the same histories. For a history value $h$, there must be observed participants in the longer pattern $D=j+1$, or a defensible model-based way to estimate their conditional outcome distribution. This is a positivity or overlap requirement: the histories encountered in the shorter pattern should be represented in the longer pattern. If a history occurs only in $D=j$, its missing-outcome distribution cannot be learned directly from the neighboring cases without additional extrapolation assumptions.

The outcome and history must also be defined consistently in both patterns: the same visit, measurement scale, and conditioning variables must be used. A fitted conditional distribution should respect the outcome’s support. For example, a model for a binary outcome must yield probabilities between zero and one, and a model for a count cannot assign probability to negative counts.

The substantive assumption is that, after conditioning on $H_j$, pattern membership does not distinguish the next-outcome distributions for these two patterns. That assumption is not established by a large sample or by good covariate overlap. It may be plausible when the specified history captures important predictors of both dropout and the next outcome, but unmeasured differences can still make the distributions unequal.

Worked Example: Borrowing a Conditional Binary Distribution

Suppose a hypothetical study has a binary outcome $Y_3$, coded 1 for improvement and 0 otherwise. Participants with pattern $D=2$ have observed visits 1 and 2 but missing $Y_3$; participants with pattern $D=3$ also have observed $Y_3$. Let the specified history be $H_2$, summarized here by two categories. The counts below are hypothetical.

History $H_2$	Pattern	Improved at visit 3	Did not improve	Total
Stable	$D=3$	36	24	60
Worsening	$D=3$	12	28	40

Assume both history categories occur in pattern $D=2$, and use the neighboring-case restriction. Among longer-pattern participants with stable history, the estimated conditional probability of improvement is $$ \widehat{P}(Y_3=1\mid H_2=\text{Stable},D=3) = \frac{36}{60}=0.600. $$ The restriction assigns that same conditional probability to participants with $H_2=\text{Stable}$ and $D=2$. For worsening history, the corresponding calculation is $$ \widehat{P}(Y_3=1\mid H_2=\text{Worsening},D=3) = \frac{12}{40}=0.300. $$ Thus a participant in pattern $D=2$ with worsening history receives an assigned probability of improvement of 0.300, not the stable-history probability of 0.600. This assignment is conditional on history and relies on overlap in both categories; it is not a claim that the marginal improvement rates are equal across dropout patterns.

What the Equality Implies

A short justification shows how the restriction supplies the missing conditional distribution. For any history $h$ in the common support and any outcome set $A$, the restriction gives $$ P(Y_{j+1}\in A\mid H_j=h,D=j) = P(Y_{j+1}\in A\mid H_j=h,D=j+1). $$ The right-hand side is estimable from participants in the longer pattern, subject to sampling uncertainty and the chosen estimation method. Therefore, under the restriction, it can be used for the unobserved conditional probability on the left. Taking $A=(-\infty,y]$ for every $y$ yields equality of the conditional cumulative distribution functions; taking $A=\{1\}$ for a binary outcome yields equality of the conditional probabilities of the value 1.

For a numerical outcome with finite conditional means, equality of the distributions also implies equality of conditional means. This follows because the mean is determined by the distribution. But equality of conditional means alone does not establish equality of distributions: two distributions can have the same mean and different variances, tail probabilities, or shapes. The distributional restriction is therefore stronger than a mean-only restriction.

Worked Example: Why Conditional Equality Does Not Mean Marginal Equality

Consider a hypothetical setting with two history categories, $H=A$ and $H=B$. Suppose the conditional probability of a positive outcome under the restriction is 0.8 for $H=A$ and 0.2 for $H=B$, in both adjacent patterns. Pattern $D=j$ has 80% of participants with history $A$, while pattern $D=j+1$ has 30% with history $A$. The conditional distributions agree by history, but the history proportions differ.

The implied marginal positive-outcome probability for the shorter pattern is $$ 0.80(0.80)+0.20(0.20)=0.64+0.04=0.68. $$ For the longer pattern it is $$ 0.30(0.80)+0.70(0.20)=0.24+0.14=0.38. $$ The marginal probabilities, 0.68 and 0.38, differ even though the conditional probabilities are identical at each history. The difference arises from the patterns’ different history distributions. This example demonstrates why the restriction must be stated conditional on the specified history.

Using the Assigned Distribution

Once the conditional distribution is assigned, it can be combined with the history distribution in the shorter pattern to describe its missing next outcomes. For instance, if the shorter pattern has history proportions $q(h)$, its implied marginal probability for an outcome set $A$ is $$ P(Y_{j+1}\in A\mid D=j) = \sum_h P(Y_{j+1}\in A\mid H_j=h,D=j)\,P(H_j=h\mid D=j), $$ for discrete histories. The neighboring-case restriction replaces the first factor with the corresponding conditional probability estimated in $D=j+1$. For continuous histories, the analogous expression integrates over the history distribution. This averaging uses the shorter pattern’s history distribution; it does not substitute the longer pattern’s history proportions.

Worked Example: Implied Mean for a Shorter Pattern

In a hypothetical study, $Y_3$ is a score in points and $H_2$ has two categories. Among longer-pattern participants, the estimated conditional mean scores are 70 points for history $A$ and 50 points for history $B$. Suppose 25% of the shorter-pattern participants have history $A$ and 75% have history $B$. Under the neighboring-case restriction, the shorter-pattern conditional means are assigned as 70 and 50 points, respectively. Its implied mean is $$ (0.25)(70\text{ points})+(0.75)(50\text{ points}) = 17.5\text{ points}+37.5\text{ points} = 55.0\text{ points}. $$ The 25% and 75% weights come from the shorter pattern, because the target is its missing-outcome mean. Using the longer pattern’s history proportions instead would answer a different question. This calculation uses only conditional means; it does not by itself specify the full outcome distribution unless the distributional restriction is also imposed.

Diagnostics, Alternatives, and Limitations

Some aspects of the setup can be checked in the observed data. Inspect history support and pattern-specific counts; very small cells or histories absent from the longer pattern signal unstable estimation or unsupported extrapolation. Compare observed conditional distributions across the adjacent patterns only where the outcome is observed in both. Such comparisons can reveal discrepancies that challenge the restriction, but they cannot verify it for the shorter pattern’s missing outcomes. A lack of detected difference is not proof of equality.

The restriction differs from available-case approaches, which use the observed cases for a visit according to their own pooling or conditioning rule, and from a complete-case restriction, which borrows information from participants observed through the end of follow-up. Here the source is specifically the adjacent pattern with one additional observed visit, conditional on a shared history. It can therefore use a closer observed pattern than complete cases alone, but its plausibility is not guaranteed by that proximity.

The conditional distribution may be estimated with stratum-specific empirical proportions or distributions, or with a regression model when histories are numerous or continuous. A model can stabilize estimates but introduces functional-form assumptions; poor fit, sparse data, or extrapolation beyond observed histories can distort the assigned distribution. Uncertainty in the estimated neighboring distribution must also be reflected in any later inference that uses it.

Most importantly, the restriction is an identifying assumption for the missing distribution, not a testable fact about outcomes that were never observed in pattern $D=j$. Different restrictions can yield different completed-data conclusions while fitting the same observed data. Sensitivity analysis is appropriate when substantive knowledge does not make this particular equality compelling.

Common pitfall: “The patterns have the same conditional outcome distribution, so they must have the same overall outcome distribution.” This is wrong when the patterns have different distributions of the conditioning history. The correction is to keep the conditioning explicit and, when forming a marginal distribution for the shorter pattern, average its assigned conditional distribution over the shorter pattern’s own history distribution.

Key takeaway: The neighboring-case restriction assigns the next unobserved outcome distribution in a shorter-history pattern from its adjacent longer-history pattern, conditional on a specified shared history. It requires overlap and a substantive equality assumption; observed data alone cannot confirm the missing part of that equality.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

Write the neighboring-case restriction for $Y_{j+1}$, patterns $D=j$ and $D=j+1$, and history $H_j$.
Why is overlap in $H_j$ needed to apply the restriction using observed longer-pattern cases?
In a binary-outcome example, the longer pattern has 18 positive outcomes among 30 participants in a particular history stratum. What conditional probability is assigned to shorter-pattern participants in that stratum?
Can conditional distribution equality imply marginal distribution equality when the patterns have different history distributions? Explain.
Does a good observed-data fit prove the neighboring-case restriction? Why or why not?

Solutions. (1) $F_{Y_{j+1}\mid H_j,D=j}(y\mid h)=F_{Y_{j+1}\mid H_j,D=j+1}(y\mid h)$ on common support. (2) Without longer-pattern participants at a history value, their conditional distribution there cannot be estimated directly; applying the restriction would require extrapolation. (3) The assigned probability is $18/30=0.600$, or 60.0%. (4) No. Different history distributions can produce different marginal distributions despite equality within each history, as shown by averaging the conditional distributions with different pattern-specific weights. (5) No. The missing outcomes in the shorter pattern are unobserved, so observed-data fit cannot verify their conditional distribution.