What Can Observed Data Check?
A restriction in a pattern-mixture model specifies how distributions in one response pattern relate to distributions in another. Its purpose is often to describe outcomes that are missing for one pattern, by borrowing information from a pattern in which those outcomes are observed. The central diagnostic question is therefore not simply whether two pattern-specific summaries look similar. It is: which part of the restriction has implications for quantities the data actually reveal?
A restriction about an unobserved outcome is generally not directly verifiable from the pattern in which that outcome is missing. However, observed data can reveal whether the proposed source distribution is available for the histories where it is needed, and whether the restriction also entails equalities among variables observed in both patterns. These checks help expose unsupported borrowing, contradictions with observable implications, and fragile estimates. They do not turn assumptions about missing outcomes into established facts.
Map the Restriction to Observed Variables
Let \(Y_1,\ldots,Y_T\) be repeated outcomes, \(X\) baseline covariates, and \(R\) a response pattern. For a pattern \(r\), write \(O(r)\) for the set of outcome visits observed in that pattern. The observed data reveal the pattern and the values of \(X\) and \(Y_j\) for \(j\in O(r)\); they do not reveal the other outcomes for that individual.
For example, with three visits, a monotone pattern \(R=100\) observes \(Y_1\) but not \(Y_2\) or \(Y_3\), whereas \(R=110\) observes \(Y_1,Y_2\) but not \(Y_3\). A restriction that borrows the distribution of \(Y_2\) from \(R=110\) for individuals in \(R=100\) requires information about \(Y_2\) conditional on histories such as \(Y_1\) and \(X\). The target pattern supplies those histories, but not \(Y_2\). The source pattern supplies observed \(Y_2\), but its history distribution may not cover every target history.
For each proposed restriction, make three distinctions before calculating a summary:
- Target quantity: What conditional distribution or mean is needed for the pattern with missing outcomes?
- Source quantity: In which pattern is the outcome observed, and which conditioning variables are also observed there?
- Observable implication: Does the restriction constrain any distribution or summary that is observed in both patterns, or does it concern only an outcome missing in the target pattern?
A restriction can be compatible even when it is not testable from the observed data. Conversely, a restriction may entail an equality for a shared, observed variable that is contradicted by the population observed-data distribution. In finite samples, estimated differences are subject to sampling variation, so a discrepancy in sample summaries is a diagnostic signal, not by itself proof of incompatibility.
Support and Estimability of Borrowed Distributions
A conditional source distribution can be estimated nonparametrically for a target history only if the source pattern contains observations with that history, or with histories sufficiently close under a justified smoothing or model. For discrete histories, a basic support requirement is that every target history of interest also occurs in the source pattern. For continuous histories, the corresponding requirement is adequate overlap of their distributions; a few observations at the edge of the source range do not provide strong support.
This is a question of conditional estimability, not just total sample size. A large source pattern may still have no observations in a particular target stratum. A model can produce a prediction beyond observed support, but then that prediction relies on the model's extrapolation rather than direct information from the source pattern. Sparse cells also matter: a conditional distribution may technically be estimable but too imprecise to serve as a stable borrowed distribution.
Worked Example: A Target History Missing From the Source Pattern
Suppose a three-visit study has 20 individuals in pattern \(R=100\) and 10 in pattern \(R=110\). The candidate restriction assigns the conditional distribution of \(Y_2\) in \(R=110\) to \(R=100\), given whether \(Y_1\) is low or high. The observed counts are:
| Pattern | Low \(Y_1\) | High \(Y_1\) |
|---|---|---|
| \(R=100\), where \(Y_2\) is missing | 12 | 8 |
| \(R=110\), where \(Y_2\) is observed | 10 | 0 |
The target pattern has both histories: 12 low and 8 high. The source pattern has observations only for low \(Y_1\). Thus the source conditional distribution \(Y_2\mid Y_1=\text{low},R=110\) can be summarized, but the data contain no observations from which to estimate \(Y_2\mid Y_1=\text{high},R=110\).
The restriction requires that latter distribution for the 8 high-\(Y_1\) individuals in \(R=100\). Its nonparametric source estimate is therefore unavailable. The zero source count is not evidence that the conditional distribution of \(Y_2\) is degenerate or that its mean is zero; there are no observed \(Y_2\) values in that stratum. A parametric model might predict the missing conditional distribution, but the high-history prediction would be extrapolation. The compatibility check identifies a data-support limitation, not a direct contradiction of the restriction.
Checking Implications That Are Observable
Some restrictions constrain conditional distributions of outcomes that are observed in both patterns. Those implications can be compared using pattern-specific summaries, provided that the outcome and conditioning variables are observed in each group being compared. The comparison must use the conditioning set specified by the restriction. An unadjusted, marginal comparison does not in general check a conditional equality.
For a distributional equality, compare the relevant conditional distributions, not only their means. Equality of means is a necessary implication of equality of distributions when the means exist, but it is not sufficient: distributions can have the same mean and different variances or shapes. For a restriction that concerns only conditional means, comparing those conditional means addresses its stated implication, but does not establish other aspects of the distributions.
Worked Example: Comparing a Shared, Observed Conditional Outcome
Suppose a candidate restriction states that the distribution of \(Y_2\) is the same in patterns \(R=111\) and \(R=110\) conditional on a binary \(Y_1\) history. Consider the \(Y_1=0\) stratum. In a hypothetical data set, \(Y_2\) is observed for 30 individuals in \(R=111\), of whom 18 have \(Y_2=1\); it is observed for 20 individuals in \(R=110\), of whom 9 have \(Y_2=1\). The conditional outcome is observed in both patterns, so the equality has an observable implication in this stratum.
The sample proportions are
Their difference, in the order \(R=111\) minus \(R=110\), is
Thus the observed conditional proportions differ by 0.150, or 15 percentage points. This is a visible discrepancy with the proposed equality of conditional distributions, but these sample values alone do not establish that the population distributions differ: sample proportions vary across samples. A formal comparison would need an inferential procedure and its assumptions. Also, this check concerns the stated \(Y_1=0\) stratum only; the restriction must be checked over every relevant history. If the restriction instead concerned an outcome missing in one of these patterns, this direct comparison would not be available.
Deriving a Standardized Borrowed Mean
When the source conditional mean is estimable over the target histories, the restriction can imply a target-pattern mean by averaging the source conditional means over the target pattern's observed history distribution. This calculation makes clear why checking overlap comes before borrowing.
To justify the formula, condition the target-pattern mean on \(H\), then apply the restriction inside the conditional expectation:
The target history weights are used because the desired mean is for pattern \(r\). Simply averaging source-pattern outcomes would instead weight histories according to \(P(H\mid R=s)\), which generally describes a different population.
Worked Example: Borrowing and Reweighting Over Target Histories
Suppose \(R=100\) is the target pattern, \(R=110\) is the source, and a candidate restriction equates the conditional mean of \(Y_2\) across these patterns given whether \(Y_1\) is low or high. In a hypothetical sample, the target pattern has 12 low-\(Y_1\) and 8 high-\(Y_1\) individuals. The source has 10 low-\(Y_1\) individuals with mean \(Y_2=4.0\), and 10 high-\(Y_1\) individuals with mean \(Y_2=8.0\). Both patterns contain both histories, so the basic discrete support check passes.
The target history proportions are
Under the conditional-mean restriction, the estimated mean for the missing \(Y_2\) values in \(R=100\) is
This is a restriction-based estimate, not an observed mean of \(Y_2\) in \(R=100\). It uses the source conditional means and the target pattern's history proportions. The support check is necessary but not enough to ensure a precise estimate: only 10 source observations inform each conditional mean here, and uncertainty in both source means and target proportions should be considered in inference.
Diagnostics, Alternatives, and Limits
A useful compatibility assessment records both what was checked and what remained assumption-dependent. For each target history, report its target-pattern count, the corresponding source-pattern count, and the source summary used. For continuous histories, inspect the distributions and ranges rather than relying only on a statement that the overall pattern sizes are large. For shared observed variables, compare conditional summaries using the restriction's actual conditioning variables. Large discrepancies, empty cells, unstable estimates, and strong dependence on model specification are reasons to investigate further.
| Approach | What it can establish | Main limitation |
|---|---|---|
| Pattern-specific conditional summaries | Whether observed implications show notable discrepancies and whether strata have support | Sparse samples create uncertainty; they cannot reveal outcomes missing in a pattern |
| Pooling or aggregating patterns | May provide more observations for a summary when the pooling is justified | Can conceal differences or fail to preserve the required conditional distribution |
| Parametric conditional model | Can produce estimates for sparse or continuous histories under a specified model | Unsupported histories require extrapolation; conclusions depend on model assumptions |
| Sensitivity analysis across restrictions | Shows how conclusions vary under alternative assumptions about missing outcomes | Does not identify which assumption is correct from the observed data alone |
These checks have several limits. First, passing support checks does not validate the equality of conditional distributions for outcomes unobserved in the target pattern. Second, a small empirical difference in a shared observed summary is not proof of equality, just as a large difference is not automatically decisive without accounting for uncertainty and the restriction's full conditioning structure. Third, coarse history categories can hide variation within categories; overly fine categories can leave too few observations to estimate distributions. Finally, a model-based estimate in a source-empty region is not the same as a data-supported conditional estimate.
A common mistake is to see a large source sample and conclude that every borrowed conditional distribution is estimable. That reasoning ignores the histories on which the restriction conditions. The correction is to check support history by history, then distinguish direct estimation from model-based extrapolation. Another mistake is to treat a restriction about an unobserved target outcome as if a favorable observed-data comparison confirmed it. Such a comparison can assess only the implications that are themselves observable.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A target pattern has 15 individuals in a history stratum, but the source pattern has none. What can be concluded about nonparametric estimation of the source conditional outcome distribution in that stratum?
- A restriction equates conditional means of an outcome observed in both patterns, given a binary history. In one stratum, the means are 3.2 and 3.7. What is the observed difference, and what does it establish?
- A target pattern has history proportions 0.25 and 0.75. Source conditional means are 10 and 6. Under the stated conditional-mean restriction, calculate the standardized target mean.
- Why can equal marginal outcome means fail to check a restriction that equates conditional means given history?
Solutions. (1) The conditional distribution cannot be estimated nonparametrically in that stratum because the source has no observed outcomes there. A model may extrapolate, but that adds assumptions. (2) In the stated order, the difference is \(3.2-3.7=-0.5\). It is a sample discrepancy of 0.5 outcome units, not by itself proof of a population violation; uncertainty and the full restriction matter. (3) The standardized mean is \((0.25)(10)+(0.75)(6)=2.5+4.5=7.0\). (4) Marginal means average over each pattern's own history distribution. Different history weights can produce different marginal means even when conditional means are equal, or mask conditional differences. The restriction must be assessed using the specified conditional histories.