Why the Same Missing Outcome Can Have Different Restrictions
Suppose a study records an outcome at three visits, but some participants have missing outcomes at one or more visits. For participants with a missing third-visit outcome, the observed data do not directly reveal the distribution of that outcome in their response pattern. A restriction supplies the missing distribution by relating it to outcomes observed in other patterns. The choice of reference pattern matters: complete cases, all cases with the outcome observed, and a nearby response pattern need not have the same outcome distribution.
A fair comparison holds the conditioning set fixed. Otherwise, differences between restrictions can arise from adjusting for different information rather than from choosing a different reference group. Here the common conditioning variable is a baseline covariate \(X\), observed for everyone. The examples use three visits and allow intermittent missingness; a pattern such as \(101\) means the first and third outcomes are observed and the second is missing.
Three Restrictions Under One Conditioning Set
For the target pattern \(100\), the outcome \(Y_3\) is missing. The following definitions make the comparison precise. In each case, the conditioning variable is \(X\), and the restriction applies separately at each value \(x\) with relevant observed data.
Available-case restriction: Set \(P(Y_3\mid R=100,X=x)\) equal to the distribution of \(Y_3\) among all participants for whom \(Y_3\) is observed, conditional on \(X=x\). This distribution is a mixture over the patterns with \(R_3=1\).
Neighboring-case restriction: Set \(P(Y_3\mid R=100,X=x)=P(Y_3\mid R=101,X=x)\). Here \(101\) is the selected neighboring pattern: it differs from \(100\) only in the second response indicator and has \(Y_3\) observed.
These are alternative identifying assumptions, not three calculations that must agree. The available-case restriction needs a mixture because several response patterns observe \(Y_3\); the other two use one reference pattern each. All require that the reference pattern or patterns have observations at the relevant \(X\) values. This is a positivity condition. The restrictions also presume that \(Y_3\) is measured comparably across patterns and that the chosen conditional equality is appropriate for the scientific setting. Neither equality is established merely because its right-hand side can be calculated from observed data.
The available-case mixture follows from the law of total probability. Conditional on \(X=x\) and on \(Y_3\) being observed, the observed-outcome distribution averages the pattern-specific distributions using the pattern proportions in that group:
Thus, its weights are not generally the proportions of patterns in the whole study. They are the proportions among participants with \(Y_3\) observed, within the same \(X\) stratum.
Worked Distribution Comparison
Worked Example: Three Implied Distributions in One Covariate Stratum
Hypothetical data. Work within one \(X\) stratum, so conditioning on \(X=x\) is implicit. Let \(Y_3\) take values 0, 10, and 20 points. The observed pattern-specific counts are:
| Response pattern | Count at 0 | Count at 10 | Count at 20 | Total |
|---|---|---|---|---|
| 111 | 4 | 10 | 6 | 20 |
| 101 | 1 | 4 | 5 | 10 |
| 011 | 5 | 4 | 1 | 10 |
| 001 | 2 | 2 | 1 | 5 |
Each listed pattern has \(R_3=1\), so its \(Y_3\) values are observed. The target pattern \(100\) has \(Y_3\) missing. The empirical distributions for the complete-case reference \(111\) and neighboring reference \(101\) are, respectively,
For the available-case distribution, pool all 45 participants whose \(Y_3\) is observed. The category totals are \(12\) at 0, \(20\) at 10, and \(13\) at 20. Therefore,
The same result follows from pattern weights: \(20/45,10/45,10/45,5/45\) for patterns \(111,101,011,001\), respectively. For example, the probability assigned to 0 is \( (20/45)(4/20)+(10/45)(1/10)+(10/45)(5/10)+(5/45)(2/5)=12/45 \).
Consequently, the restrictions assign the target pattern three different distributions: complete cases give \((0.20,0.50,0.30)\), available cases give approximately \((0.2667,0.4444,0.2889)\), and neighboring cases give \((0.10,0.40,0.50)\). Their implied means are also different. The complete-case mean is \(0(0.20)+10(0.50)+20(0.30)=11.0\) points; the available-case mean is \(0(12/45)+10(20/45)+20(13/45)=10.22\) points; and the neighboring-case mean is \(0(0.10)+10(0.40)+20(0.50)=14.0\) points.
These are empirical reference distributions and the implications of applying each restriction. They are not direct observations of \(Y_3\) in pattern \(100\).
Keep Conditioning Fixed, Then Standardize
When \(X\) varies, restrictions must first be applied within \(X\) strata. A marginal distribution for the target pattern is then obtained by averaging those conditional distributions over the distribution of \(X\) in that target pattern. Using the \(X\) distribution of the complete cases or of all available cases instead changes the population being described.
Worked Example: Comparing Restrictions Across Two Covariate Strata
Hypothetical data. Suppose the target pattern \(100\) contains 8 participants with \(X=A\) and 12 with \(X=B\), so its covariate weights are \(0.40\) and \(0.60\). Suppose \(Y_3\) takes values 0, 10, and 20. The complete-case distributions are \((0.20,0.50,0.30)\) for \(A\) and \((0.30,0.40,0.30)\) for \(B\). The neighboring-case distributions are \((0.10,0.40,0.50)\) for \(A\) and \((0.20,0.30,0.50)\) for \(B\).
For the available-case restriction, suppose the conditional pattern weights for \(111,101,011,001\) are \((0.50,0.25,0.15,0.10)\) at \(A\) and \((0.20,0.40,0.30,0.10)\) at \(B\). The corresponding available-case distributions, calculated by mixing the observed pattern distributions with those weights, are \((0.24,0.45,0.31)\) at \(A\) and \((0.31,0.38,0.31)\) at \(B\). Each set of weights sums to 1.
Standardize each restriction to the target pattern’s \(X\) distribution. For the complete-case restriction, the marginal probabilities are \(0.40(0.20,0.50,0.30)+0.60(0.30,0.40,0.30)=(0.26,0.44,0.30)\). For the neighboring-case restriction, they are \(0.40(0.10,0.40,0.50)+0.60(0.20,0.30,0.50)=(0.16,0.34,0.50)\). For the available-case restriction, they are \(0.40(0.24,0.45,0.31)+0.60(0.31,0.38,0.31)=(0.282,0.408,0.310)\).
The corresponding means are \(10.4\), \(13.4\), and \(10.28\) points, respectively. For example, the available-case mean is \(0(0.282)+10(0.408)+20(0.310)=10.28\) points. The calculations compare restrictions under the same conditioning set and the same target-pattern covariate mix.
From a Distribution to Implied Missing Values
Worked Example: Expected Category Counts for the Target Pattern
Return to the first example and suppose there are 15 participants in target pattern \(100\), all in the same \(X\) stratum. Under the complete-case restriction, the implied expected counts at 0, 10, and 20 points are \(15(0.20)=3\), \(15(0.50)=7.5\), and \(15(0.30)=4.5\). These sum to 15. Under the available-case restriction, they are \(15(12/45)=4.00\), \(15(20/45)=6.67\), and \(15(13/45)=4.33\), summing to 15 apart from rounding. Under the neighboring-case restriction, they are \(15(0.10)=1.5\), \(15(0.40)=6.0\), and \(15(0.50)=7.5\), again summing to 15.
These are expected or fractionally allocated counts implied by the distributions, not claims that fractional people were observed. The arithmetic makes the practical consequences of the assumptions visible: the neighboring-case restriction assigns more missing outcomes to the highest category than either alternative in this example.
Diagnostics, Alternatives, and Limits
Several checks help prevent a mechanical comparison from being misleading. First, verify that each reference distribution is conditional on the same \(X\) and that the final marginalization uses the target pattern’s \(X\) distribution. Second, check that pattern weights are nonnegative and sum to 1, and that category probabilities in each implied distribution also sum to 1. Third, inspect pattern counts within each stratum. Zero counts make a reference distribution unavailable; very small counts make its empirical estimate unstable. Combining strata may improve numerical stability, but it changes the conditioning set and requires a defensible modeling choice.
The alternatives answer different substantive questions. Complete-case restriction uses only fully observed records and is simple to describe, but can be imprecise when complete cases are few. Available-case restriction uses every pattern that observes \(Y_3\), but its mixture can reflect the composition of those patterns rather than the composition of the target pattern. Neighboring-case restriction uses one selected nearby pattern and may be attractive when that pattern is scientifically relevant, but the result can depend strongly on which neighbor is chosen. No option is automatically preferable in every study.
A frequent mistake is to say, “\(Y_3\) is observed in the available cases, so their pooled distribution is the distribution for participants in pattern \(100\).” The first clause describes observed data; the second asserts an unverified equality. The correction is to state the available-case restriction explicitly and explain that it identifies the missing distribution only if the assumption is accepted. Similarly, the observed differences between the three reference distributions do not test which restriction is true: the target outcomes are missing, so their distribution cannot be checked directly from these data alone.
Finally, matching a distribution conditional on \(X\) is not the same as matching it conditional on additional outcomes such as \(Y_1\). Adding an observed history to the conditioning set can change all three reference distributions and may create sparse or empty strata. The conditioning variables must be chosen deliberately and kept common for a meaningful comparison. As with other pattern-mixture restrictions, conclusions about the missing outcomes depend on untestable assumptions; results should identify the restriction used and, when relevant, show how conclusions vary across plausible alternatives.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For target pattern \(100\), which patterns contribute to an available-case mixture for \(Y_3\)?
- In the first worked example, what is the available-case probability of \(Y_3=20\), and what conditional pattern proportions produce it?
- Why must the stratum-specific distributions in the second worked example be averaged using the target pattern’s \(X\) proportions?
- If a target pattern has no participants in a particular \(X\) stratum but the neighboring pattern does, what issue arises for standardization?
Solutions. (1) The patterns with \(R_3=1\) contribute: \(111,101,011,\) and \(001\). (2) The probability is \(13/45\approx0.2889\). The pattern proportions are \(20/45,10/45,10/45,\) and \(5/45\); mixing their probabilities of 20 points, \(6/20,5/10,1/10,\) and \(1/5\), gives \(13/45\). (3) The target-pattern proportions define the population for which the missing distribution is sought. Using other weights would describe a different covariate mix. (4) There is no target-pattern mass in that stratum to standardize, but the target distribution there is not required for this target’s marginal distribution. If the target population or desired estimand includes that stratum, the missing target distribution cannot be obtained there without additional assumptions or extrapolation.