Conditional Neighboring-Case Restrictions Within History

Why Condition on Shared History?

Adjacent dropout patterns can contain different mixes of patients even when they share some observed history. If the outcome distribution varies across that history, borrowing the neighbor’s overall outcome distribution also borrows the neighbor’s history mix. A conditional neighboring-case restriction instead borrows the neighbor’s outcome distribution within each shared history, then combines those conditional distributions using the target pattern’s history distribution.

This distinction is important for interpretation: the restriction makes a statement about outcomes conditional on history, not about the distribution of history itself. The resulting marginal outcome distribution for the target pattern is therefore standardized to that pattern’s own history mix.

Definition and Identification Formula

Consider two adjacent patterns in a pattern-mixture model. Let \(G=A\) denote the target pattern and \(G=B\) its neighboring pattern. Let \(H\) be history observed in both patterns, and let \(Y\) be an outcome observed in pattern \(B\) but missing in pattern \(A\). For example, with three scheduled visits, \(A\) might denote dropout after visit 1, \(B\) dropout after visit 2, \(H=Y_1\), and \(Y=Y_2\). The pattern labels are just names for the two patterns being considered.

Definition: A conditional neighboring-case restriction within history assumes that the conditional distribution of the missing outcome in target pattern \(A\) equals the observed conditional distribution in neighboring pattern \(B\), at each shared history value: \(F_{Y\mid H,G=A}(y\mid h)=F_{Y\mid H,G=B}(y\mid h)\). The restriction applies only to history values represented in the target pattern and supported by the neighboring pattern.

For discrete \(H\), this restriction identifies the target pattern’s marginal distribution of \(Y\) by reweighting the neighbor’s conditional distributions with the target pattern’s history probabilities:

F_{Y\mid G=A}(y) = \sum_h F_{Y\mid H,G=B}(y\mid h)\,P(H=h\mid G=A).

For continuous \(H\), replace the sum by an integral over the target history distribution. The corresponding target-pattern mean, when it exists, is

E(Y\mid G=A) = \begin{cases} \sum_h E(Y\mid H=h,G=B)\,P(H=h\mid G=A), & \text{if } H \text{ is discrete},\\ \int E(Y\mid H=h,G=B)\,dF_{H\mid G=A}(h), & \text{if } H \text{ is continuous}. \end{cases}

The distributional formula is stronger than borrowing only a conditional mean: it specifies the conditional distribution of \(Y\) at each history value, and therefore determines the standardized marginal distribution as well. The history probabilities in the formula belong to pattern \(A\), not pattern \(B\).

Assumptions and Support

The identifying statement is an assumption about the unobserved outcomes in pattern \(A\). Its application requires a well-defined outcome and a history \(H\) that is observed in both patterns in the same way. It also requires support overlap: every history value that occurs with positive probability in \(A\) must have a corresponding conditional outcome distribution in \(B\). For discrete history, this means \(P(H=h\mid G=B)>0\) whenever \(P(H=h\mid G=A)>0\). For continuous history, the target history distribution must lie within the region where the neighbor’s conditional distribution is supported and can be estimated.

If a target history value has no neighboring cases, the restriction does not supply the missing conditional distribution there. Extrapolating a fitted conditional model into that region would require an additional modeling assumption; it is not justified by neighboring-case equality alone. Even with overlap, sparse histories can make conditional estimates unstable.

The restriction does not assert that pattern membership is random, nor that the two patterns have the same distribution of \(H\). It asserts equality of the outcome distributions conditional on \(H\). Because \(Y\) is unobserved in pattern \(A\), the key equality cannot be checked directly from the observed data. The observed history frequencies and the neighbor’s conditional distributions can be examined, but they do not verify the restriction for the missing outcomes.

Worked Example: Binary Outcome With Different History Weights

Suppose \(H\) is a binary baseline risk category, with values 0 and 1, and \(Y\) is a binary event at the next visit. In hypothetical data, the observed neighboring pattern has event probabilities \(0.20\) at \(H=0\) and \(0.80\) at \(H=1\). Its history proportions are \(P(H=0\mid G=B)=0.75\) and \(P(H=1\mid G=B)=0.25\). The target pattern has proportions \(P(H=0\mid G=A)=0.25\) and \(P(H=1\mid G=A)=0.75\). Assume both histories occur in both patterns and impose the conditional restriction.

First calculate the neighbor’s unconditional event probability:

P(Y=1\mid G=B) =(0.20)(0.75)+(0.80)(0.25) =0.15+0.20 =0.35.

Under the conditional restriction, the event probabilities borrowed within histories remain \(0.20\) and \(0.80\). To obtain the target marginal probability, weight them by the target history proportions:

P(Y=1\mid G=A) =(0.20)(0.25)+(0.80)(0.75) =0.05+0.60 =0.65.

Thus, the restricted target-pattern event probability is \(0.65\), or 65%, while the neighbor’s unconditional event probability is 35%. The 30-percentage-point difference arises from reweighting the same conditional probabilities by different history mixes. The calculation assumes conditional equality and adequate support; it does not show that equality is true.

Derivation: Why Target Weights Appear

The standardization formula follows from the law of total probability. For discrete \(H\), partition the event \(Y\le y\) in pattern \(A\) according to the possible history values:

\begin{aligned} P(Y\le y\mid G=A) &=\sum_h P(Y\le y\mid H=h,G=A)P(H=h\mid G=A)\\ &=\sum_h P(Y\le y\mid H=h,G=B)P(H=h\mid G=A). \end{aligned}

The second line uses the conditional neighboring-case restriction. Notice that the derivation does not replace the target weights with the neighbor’s weights: the conditioning event on the left is \(G=A\), so the history distribution in the decomposition must also be conditional on \(G=A\). The same reasoning applied to expectations gives the mean formula, provided the relevant expectations exist.

Worked Example: Standardizing a Continuous Outcome

Suppose \(Y\) is a hypothetical symptom score measured in points, and \(H\) has two strata. In neighboring pattern \(B\), the conditional means are 10 points for \(H=0\) and 20 points for \(H=1\). The target pattern has history proportions 0.60 and 0.40, respectively; the neighbor’s proportions are 0.80 and 0.20. Assume both history values occur in both patterns and the conditional distributions of \(Y\), not just their means, are equal across patterns within each history.

The conditional restriction gives target-pattern conditional means of 10 and 20 points. Its standardized mean is

E(Y\mid G=A) =(10)(0.60)+(20)(0.40) =6+8 =14\text{ points}.

Unconditional borrowing would instead use the neighbor’s history weights:

E(Y\mid G=B) =(10)(0.80)+(20)(0.20) =8+4 =12\text{ points}.

The two marginal means differ by 2 points. The conditional restriction also determines a target marginal distribution by mixing the neighbor’s within-stratum distributions with weights 0.60 and 0.40. The mean calculation alone does not describe that full distribution.

Conditional and Unconditional Borrowing

Unconditional neighboring-pattern borrowing sets the entire target marginal distribution equal to the neighbor’s marginal distribution: \(F_{Y\mid G=A}(y)=F_{Y\mid G=B}(y)\). Conditional neighboring-case borrowing instead uses the neighbor’s conditional outcome distributions and standardizes them to the target history distribution. These are different restrictions: the first borrows both outcome behavior and the neighbor’s history mix, while the second borrows outcome behavior within history and retains the target’s mix.

Feature	Conditional neighboring-case restriction	Unconditional neighboring-pattern borrowing
Equality imposed	Conditional outcome distributions given shared history	Marginal outcome distributions
History weights in target marginal	Target pattern’s history weights	Neighbor’s history weights are inherited through its marginal
Key support concern	Target histories need neighboring conditional support	Does not require stratum-by-stratum borrowing, though the marginal restriction remains an assumption
Interpretation	Borrow within history, then standardize	Borrow the neighbor’s overall outcome distribution

The two approaches will generally give different marginal distributions when history weights differ and conditional outcome distributions vary across history. However, this is not a necessary difference. Matching history distributions or history-invariant conditional outcome distributions are sufficient conditions for agreement, but they are not necessary. Weighted conditional distributions can also agree by coincidence.

Worked Example: Agreement by Coincidence

Let \(H\) have three values. In hypothetical data, the neighboring pattern has history weights \((1/3,1/3,1/3)\), while the target pattern has weights \((5/12,1/6,5/12)\). Suppose the neighbor’s conditional probabilities of a binary event at the three histories are \(0\), \(0.5\), and \(1\), respectively. Assume all three histories are supported in both patterns and impose the conditional restriction.

The neighbor’s unconditional event probability is

P(Y=1\mid G=B) =(1/3)(0)+(1/3)(0.5)+(1/3)(1) =0+\frac{1}{6}+\frac{1}{3} =\frac{1}{2}.

The conditionally standardized target probability is

P(Y=1\mid G=A) =(5/12)(0)+(1/6)(0.5)+(5/12)(1) =0+\frac{1}{12}+\frac{5}{12} =\frac{1}{2}.

The history weights differ and the conditional event probabilities vary, yet the two marginal event probabilities both equal \(0.5\). The weighted sums happen to agree. Therefore, observing agreement of these marginals would not establish matching weights or history-invariant conditional distributions.

Diagnostics, Limitations, and Common Pitfalls

Useful diagnostics concern whether the restriction can be applied coherently and estimated stably, rather than whether it is empirically true for the unobserved target outcomes. Tabulate or graph the history distributions in both patterns, inspect the overlap, and examine the number of neighboring observations available across the target history range. For continuous histories, check whether the target values require substantial extrapolation from the neighbor. If conditional distributions are estimated with a model, assess whether the chosen form represents the observed neighbor data adequately; poor fit in pattern \(B\) undermines the borrowed conditional distributions.

The restriction remains an identifying assumption, not a fact established by these diagnostics. It may be implausible if the patterns differ in prognostic features omitted from \(H\), or if the conditional outcome behavior genuinely changes with pattern even after conditioning on the shared history. A richer history may address an omitted prognostic factor only when that history is observed comparably and support remains adequate. Adding variables can also create sparse strata and unstable estimates.

Common mistake: “The neighbor’s event rate is 35%, so the target event rate under the neighboring-case restriction is also 35%.” This uses the neighbor’s marginal history weights and silently changes a conditional restriction into unconditional borrowing. The correction is to apply the borrowed conditional event probabilities to the target pattern’s history weights. In the first worked example, that produces 65%, not 35%.

A second mistake is to claim that different history distributions and varying conditional outcomes guarantee different marginal results. They usually lead to different results, but weighted averages may coincide, as the third example demonstrates. The correct comparison is to calculate both weighted distributions (or the particular summary of interest), not to infer inequality from those features alone.

Key takeaway: A conditional neighboring-case restriction borrows the neighbor’s outcome distribution at each shared history value. To recover the target pattern’s marginal distribution, standardize those conditional distributions using the target pattern’s history distribution, and check that the required histories have neighboring support.

Check Your Understanding

Answer each question using the definitions and formulas above. Solutions follow.

What equality does the conditional neighboring-case restriction impose, and what does it leave unrestricted?
A target pattern has history proportions 0.70 and 0.30. The neighbor’s conditional event probabilities are 0.10 and 0.60 at the two histories. What is the standardized target event probability?
What support condition is needed to use the restriction for a discrete history?
Name two sufficient conditions for the conditional-standardized and unconditional neighboring distributions to agree, and explain why they are not necessary.
Can observed data directly verify equality of the conditional outcome distributions across patterns when the outcome is missing in the target pattern? What can be checked instead?

Solutions. (1) It equates \(F_{Y\mid H,G=A}(y\mid h)\) and \(F_{Y\mid H,G=B}(y\mid h)\) at supported histories; it does not equate the patterns’ history distributions. (2) The result is \((0.10)(0.70)+(0.60)(0.30)=0.07+0.18=0.25\), or 25%. (3) Every history with positive probability in the target pattern must have positive probability in the neighbor. (4) Matching history distributions and history-invariant conditional outcome distributions are each sufficient; neither is necessary because different weighted sums can coincide by cancellation. (5) No, because the target outcome is unobserved. One can check observed history overlap, stratum counts, and the neighbor’s conditional fit, but those checks do not verify the missing-outcome equality.