Why Check Restrictions for Compatibility?
An available-case restriction supplies a value for an outcome mean that is not observed in a dropout pattern: it sets that mean equal to a mean among people who were observed at the visit. The restriction is an identifying assumption, not a consequence of the observed data. Even so, the assumptions an analysis imposes should be internally coherent. If two stated restrictions assign different values to the same missing-outcome mean, no data completion can satisfy both.
Compatibility deserves particular attention when restrictions are applied at several visits or both within covariate strata and marginally. A marginal mean and a covariate-adjusted mean use different weights. Consequently, a restriction that appears reasonable at each level on its own can conflict with another restriction when both are imposed on the same target.
For the calculations below, consider a monotone three-visit design. The pattern \(111\) is observed at all three visits, \(110\) is observed at visits 1 and 2, and \(100\) is observed only at visit 1. Thus \(Y_2\) is missing in pattern \(100\), and \(Y_3\) is missing in patterns \(110\) and \(100\). The pattern labels indicate observation, not outcome values.
The restriction is defined only where the relevant available-case mean can be formed. In a stratified analysis, this requires observed cases at visit \(t\) in each stratum being used. For an empirical calculation, the means and covariate weights below are calculated from the stated hypothetical counts. In a population model, the corresponding quantities are conditional expectations and pattern-specific covariate distributions.
How Marginal and Stratified Restrictions Can Conflict
Suppose the strata of \(X\) are \(A\) and \(B\). Write \(m_x\) for the observed mean at a visit among available cases with \(X=x\). Under a stratified restriction, the missing-outcome mean in pattern \(r\), within stratum \(x\), is set to \(m_x\). Its marginal mean is then an average using the covariate distribution of pattern \(r\), not the distribution of the available cases.
The marginal available-case restriction instead uses the covariate distribution among people observed at that visit. Denote those weights by \(P(X=x\mid \text{available at }t)\). The two restrictions agree only when their weighted means agree. This gives an immediate algebraic diagnostic.
Subtracting the right side from the left shows that agreement requires \(\sum_x [P(X=x\mid R=r)-P(X=x\mid \text{available at }t)]m_x=0\). This holds, for example, if the two covariate distributions are the same, or if the available-case means are constant across strata. Otherwise it may or may not hold by coincidence. Covariate imbalance alone does not prove incompatibility; the weighted means must actually be compared.
Worked Example: Finding Incompatible Implied Means
Consider hypothetical data with covariate strata \(A\) and \(B\). The counts at each pattern and the observed visit-specific means are as follows. Entries for a visit that a pattern does not observe are not available.
| Pattern | Count in A | Count in B | Observed mean at visit 2 | Observed mean at visit 3 |
|---|---|---|---|---|
| 111 | 8 | 2 | A: 8; B: 18 | A: 10; B: 30 |
| 110 | 2 | 8 | A: 12; B: 22 | Not observed |
| 100 | 1 | 9 | Not observed | Not observed |
First check visit 2, where patterns \(111\) and \(110\) are available. In stratum \(A\), the available-case mean is \((8\times8+2\times12)/(8+2)=88/10=8.8\). In stratum \(B\), it is \((2\times18+8\times22)/(2+8)=212/10=21.2\). The available-case mean marginal over strata is \((88+212)/(10+10)=15\).
The stratified restriction for the missing \(Y_2\) mean in pattern \(100\) assigns means 8.8 in \(A\) and 21.2 in \(B\). Pattern \(100\) has 1 person in \(A\) and 9 in \(B\), so this implies the marginal mean \((1/10)\times8.8+(9/10)\times21.2=0.88+19.08=19.96\). The marginal restriction assigns 15 instead. These cannot both be the marginal \(Y_2\) mean in pattern \(100\): the two restrictions are incompatible for these data and weights.
Now check visit 3. Only pattern \(111\) is available, with means 10 in \(A\) and 30 in \(B\). Its overall available-case mean is \((8\times10+2\times30)/(8+2)=140/10=14\). The stratified restriction for pattern \(110\) assigns means 10 and 30 in the two strata. Since that pattern has 2 people in \(A\) and 8 in \(B\), it implies \((2/10)\times10+(8/10)\times30=2+24=26\) for its marginal mean. The marginal restriction says 14, so the visit-3 restrictions conflict for pattern \(110\). For pattern \(100\), the same stratified means imply \((1/10)\times10+(9/10)\times30=1+27=28\), again distinct from the marginal restriction of 14.
The arithmetic identifies the source of conflict: pattern \(100\) is concentrated in stratum \(B\), while visit-2 available cases are evenly split across strata; at visit 3, the available cases are concentrated in \(A\), unlike either missing pattern. Because the stratum-specific means differ, changing the weights changes the resulting marginal mean.
A Compatibility Check Across Visits and Strata
A useful audit treats each missing pattern-by-visit mean as a target and records every restriction that assigns it a value. The first question is whether those assignments refer to the same target. If they do, their values must agree. The second question is whether the available-case means and covariate weights exist in every stratum required by the restriction. Finally, check any known support limits: an assigned mean must be possible for the outcome being modeled.
Write down the visit \(t\), missing pattern \(r\), and, for conditional restrictions, stratum \(x\). Keep \(Y_2\) and \(Y_3\) as separate targets.
At each visit and stratum, combine only patterns observed at that visit, using their observed counts as weights.
Calculate marginal and stratified assignments separately. Aggregate stratified assignments using the covariate distribution of the target pattern.
If two assignments to the same target differ, the joint restrictions are incompatible. Differences between means for different visits are not, by themselves, a contradiction.
Confirm that every required stratum has available cases and that assigned means satisfy any genuine outcome-range or other imposed constraints.
This audit is an algebraic check of the restrictions, not a test of their truth. Missing outcomes cannot be used to verify directly whether their means equal available-case means. The observed data can reveal that a set of assumptions is contradictory, but compatibility does not establish that the assumptions are scientifically credible.
Worked Example: A Compatible Marginal and Stratified Pair
Suppose at one visit the observed means are still 10 in stratum \(A\) and 30 in stratum \(B\), and the available cases are 8 in \(A\) and 2 in \(B\). Suppose a missing pattern has the same covariate proportions: 8 in \(A\) and 2 in \(B\). The available-case marginal mean is \((8\times10+2\times30)/10=14\).
Under the stratified restriction, the missing pattern has mean 10 in \(A\) and 30 in \(B\). Aggregating with its own proportions gives \((8/10)\times10+(2/10)\times30=8+6=14\). The marginal restriction also assigns 14. In this case both restrictions can hold simultaneously for the pattern’s marginal mean, despite different stratum means, because the available cases and the missing pattern have the same covariate composition.
This agreement is specific to the weights and means shown. It does not establish compatibility for another visit, another pattern, or another conditioning set; those targets require their own calculations.
Cross-Visit Differences Are Not Automatically Contradictions
Available-case restrictions are visit-specific unless an additional model states otherwise. In particular, setting a pattern’s missing \(Y_2\) mean to an available-case mean at visit 2 does not require its missing \(Y_3\) mean to have that same numerical value. The outcomes refer to different visits, and a difference may represent a genuine change over time.
Worked Example: Checking Visit-Specific Assignments
Return to the visit-2 stratified assignments in the first example. For pattern \(100\), they assign \(E(Y_2\mid R=100,X=A)=8.8\) and \(E(Y_2\mid R=100,X=B)=21.2\). At visit 3, the stratified restriction assigns \(E(Y_3\mid R=100,X=A)=10\) and \(E(Y_3\mid R=100,X=B)=30\). These are four assignments to two different outcomes at two visits.
There is no contradiction merely because 8.8 differs from 10 or 21.2 differs from 30. The restrictions do not assert \(Y_2=Y_3\), or equality of their means. But if a separate assumption required, for example, \(E(Y_2\mid R=100,X=A)=E(Y_3\mid R=100,X=A)\), it would conflict with these assignments because \(8.8\ne10\). Such an equality must be stated as an additional restriction; it is not part of the available-case rule by itself.
Diagnostics, Alternatives, and Limitations
Several warning signs deserve attention. A marginal and a stratified calculation may use different covariate weights; confusing those weights can conceal a real contradiction. A stratum with no available cases makes its available-case mean undefined, so a stratified restriction cannot be calculated there without changing the analysis or adding assumptions. Sparse strata can also make empirical means unstable, even when they are defined. Finally, for a bounded outcome, a restriction that implies a mean outside the outcome’s possible range is infeasible, regardless of whether the weighting arithmetic is internally consistent.
Marginal and stratified restrictions are often alternative specifications rather than requirements that must be imposed together. A marginal restriction gives a single overall mean and leaves the within-stratum missing means less directly specified. A stratified restriction uses covariate-specific available-case means and then averages them according to the missing pattern’s covariate distribution. A complete-case restriction is another distinct identifying assumption; it should not be treated as interchangeable with an available-case restriction. Which specification is defensible depends on the substantive assumptions, not solely on which is easiest to calculate.
The central limitation is that compatibility is necessary for a joint set of restrictions to be usable, but it is not sufficient for identification to be convincing. The restrictions concern unobserved outcomes, so their empirical plausibility cannot be established just by checking observed means. Nor does mean compatibility guarantee compatibility of full outcome distributions or of additional assumptions on associations across visits. This tutorial’s checks concern the mean restrictions stated here.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- At a visit, available-case means are 5 in stratum \(A\) and 15 in stratum \(B\). Available cases are 50% in each stratum, while a missing pattern is 20% in \(A\) and 80% in \(B\). What marginal means do the marginal and stratified restrictions imply?
- In Question 1, state one change to the weights that would make the two restrictions agree without changing the stratum-specific means.
- Why is a restricted \(Y_2\) mean of 9 and a restricted \(Y_3\) mean of 12 not, by itself, evidence of incompatibility?
- What should an analyst conclude if a required covariate stratum has no cases observed at the visit?
Solutions. (1) The marginal restriction uses the available-case weights: \(0.5(5)+0.5(15)=10\). The stratified restriction uses the missing pattern’s weights: \(0.2(5)+0.8(15)=1+12=13\). They are incompatible if both assign the same pattern’s marginal mean. (2) Give the missing pattern the available cases’ 50%/50% covariate distribution; then its stratified aggregate is \(0.5(5)+0.5(15)=10\), matching the marginal restriction. (3) \(Y_2\) and \(Y_3\) are different visit-specific targets; equality across visits is not implied by available-case restrictions. (4) The stratum-specific available-case mean is undefined there, so the stated stratified restriction cannot be applied without an additional assumption or a changed specification.