Why the Wording of a Restriction Matters
A pattern-mixture restriction is not fully described by saying that a missing outcome is “borrowed from a neighboring pattern.” That phrase leaves important questions unanswered: which pattern supplies the borrowed information, which outcome visit is restricted, what observed history and covariates are held fixed, and whether the claim concerns an entire distribution or just its mean. Different answers produce different assumptions, even when they use the same donor pattern.
The goal here is disciplined reporting. A concise statement should let another analyst reconstruct the restriction without guessing what “similar,” “same,” or “borrowed” means. It should also avoid implying that an identifying assumption has been established by the observed data.
For illustration, consider outcomes \(Y_1,Y_2,Y_3\), a baseline covariate \(X\), and patterns \(S=(1,1,0)\) and \(S=(1,1,1)\). The first pattern has \(Y_1,Y_2\) observed and \(Y_3\) missing; the second has all three outcomes observed. Write \(H_2=(Y_1,Y_2)\) for the observed history through visit 2. In a restriction for visit 3, one might condition on \(H_2\) and \(X\), but that choice must be stated rather than left implicit.
Four Elements of a Reportable Assumption
A useful report names four elements explicitly. First, identify the target pattern and the borrowed pattern, preferably by their observed/missing indicators or a clear verbal description. Second, identify the outcome visit being restricted. Third, state the conditioning variables, including the relevant observed history and any covariates. Fourth, state whether the equality is distributional or applies only to a specified functional such as a conditional mean.
Using the illustrative patterns above, a distributional restriction at visit 3 is
Here \(\mathcal{L}\) denotes a conditional probability distribution. The assumption says that, at each relevant history \(h\) and covariate value \(x\), the full conditional distribution of the missing visit-3 outcome in the target pattern matches the conditional distribution among subjects in the borrowed pattern. By contrast, a conditional mean restriction is
This second statement equates conditional expectations only. It does not require equal variances, equal tail probabilities, or equal distributions. In either expression, the target-pattern conditional quantity involves an outcome that is unobserved for that pattern. It is therefore an identifying assumption, not a direct comparison of two fully observed outcome distributions.
Worked Examples: From Informal Language to Explicit Restrictions
Worked Example: Reporting a Visit-3 Distributional Restriction
Suppose the target pattern is \(S=(1,1,0)\), the borrowed pattern is \(S=(1,1,1)\), and the proposed restriction concerns visit 3. The analyst wants to condition on both earlier outcomes and baseline age \(X\). Write the assumption precisely and say what it asserts.
The conditioning history is \(H_2=(Y_1,Y_2)\), and the stated covariate is age \(X\). The restriction is:
In words: among subjects with the same values of \(Y_1,Y_2\), and age, the conditional distribution of the visit-3 outcome for the target pattern is assumed to equal the corresponding conditional distribution for the complete pattern. This includes the conditional mean and, when they exist, the variance and other distributional features.
The statement is meaningful only over histories and ages for which the borrowed pattern provides relevant information. It does not say that the two patterns have the same distributions of \(Y_1,Y_2\) or age. It says that the conditional distribution of \(Y_3\) is equal after conditioning on those quantities.
Worked Example: A Mean Restriction Is Not a Distributional Restriction
At a fixed history and covariate value, suppose the borrowed pattern’s visit-3 outcome has conditional mean 12 points and conditional standard deviation 3 points. A proposed restriction for the target pattern equates its conditional mean to the borrowed-pattern mean. What can be reported, and what cannot be concluded?
The reportable assumption is
Thus the assumed target-pattern conditional mean is 12 points at this \(h,x\). The borrowed-pattern standard deviation of 3 points is not part of this equality. For example, a target conditional distribution with mean 12 and standard deviation 5 points would satisfy the stated mean restriction but would not satisfy a distributional-equality restriction.
A distributional assumption would instead require equality of the two conditional laws. If that stronger restriction held and the conditional variance existed, the target conditional standard deviation would also be 3 points. That conclusion does not follow from the mean restriction alone.
Worked Example: Why Conditioning Must Be Named
Consider two age strata in a hypothetical dataset. For a target pattern, the assumed conditional means at visit 3 are 10 points for younger subjects and 20 points for older subjects. The borrowed pattern has the same two conditional means. In the target pattern, 80% of subjects are younger; in the borrowed pattern, 20% are younger. Calculate the pattern-specific marginal means and compare them.
The target-pattern marginal mean, averaging over its age composition, is
The borrowed-pattern marginal mean is
The conditional means agree in both age strata, but the marginal means differ by \(18-12=6\) points because the age compositions differ. Therefore, reporting only that “the means are equal” is ambiguous and may be misleading. The precise assumption is equality of conditional means given age, not equality of marginal means across patterns.
Justification: What a Conditional Restriction Implies After Averaging
A conditional restriction can imply a marginal equality only when the averaging distributions are also aligned. This follows from the law of total expectation. For a covariate or history \(Z\), the pattern-specific mean can be written as
If the conditional means for target pattern \(s\) and borrowed pattern \(r\) are equal at every relevant \(z\), then their marginal means are integrals of the same conditional-mean function—but generally with different distributions of \(Z\). Hence the marginal means need not agree. If, in addition, \(F(z\mid S=s)=F(z\mid S=r)\), the integrals are equal. The calculation in the third example illustrates the failure of equality when the distributions of the conditioning variable differ.
This derivation explains why the conditioning set belongs in the assumption statement. Omitting it can silently change a conditional restriction into a marginal claim. It also clarifies that equality of conditional means does not determine a full conditional distribution.
Checking the Statement Before Reporting It
A precise statement can still be poorly supported or internally unclear. Before using one, check each element in order:
Give their response indicators or unambiguous descriptions, and distinguish the target from the borrowed pattern.
State the visit and outcome scale. Do not let “post-dropout outcomes” stand in for a particular visit when visits may have different restrictions.
State the observed history and covariates. Check that the donor-pattern data contain subjects with relevant histories and covariate values.
Use “conditional distribution” for equality of the full law, or “conditional mean” for equality of expectations. Do not substitute one phrase for the other.
Poor overlap in histories or covariates is a warning: the borrowed conditional quantity may require extrapolation rather than direct support from comparable donor-pattern subjects. Sparse donor data can also make the corresponding conditional distribution or mean unstable to estimate. These are diagnostics of practical fragility, not tests that can verify the unobserved target-pattern outcome.
Observed data can reveal contradictions with observable implications of a proposed restriction or reveal that the needed donor histories are absent. But because \(Y_t\) is missing in the target pattern at the restricted visit, the equality involving that target conditional outcome cannot generally be checked directly. A report should therefore describe the restriction as assumed, not as confirmed by a nonsignificant test or proven by agreement in the observed data.
Comparing Restriction Statements and Common Pitfalls
| Statement | What it specifies | What remains unclear or does not follow |
|---|---|---|
| “Borrow from the complete pattern.” | Suggests a donor pattern. | Does not identify visit, conditioning set, or equality type. |
| “The conditional mean at visit 3 is equal given \(Y_1,Y_2,X\).” | Names a visit, conditioning variables, and mean-level equality. | Still needs target and donor patterns to be identified. |
| “The conditional distributions at visit 3 are equal given \(Y_1,Y_2,X\), for patterns \((1,1,0)\) and \((1,1,1)\).” | Identifies both patterns, visit, conditioning set, and distributional equality. | Does not establish that the assumption is true; support and plausibility still require consideration. |
The main alternative to a precisely stated restriction is not a different mathematical equality but an underspecified verbal rule. A separate modeling choice is whether to impose a distributional restriction or a weaker mean restriction. The distributional version constrains all features of the conditional law; the mean version leaves conditional variance, shape, and tails unrestricted. Neither is automatically preferable: the choice depends on the intended identifying assumption and the analysis, and it must be communicated explicitly.
- Vague borrowing: “Use the neighboring pattern” is not enough. Name the pattern and the outcome visit.
- Hidden conditioning: “Equal means” does not say whether equality is conditional on history, covariates, or neither.
- Overclaiming: Equal conditional means do not imply equal distributions. Equal conditional distributions do not imply identical pattern compositions of the conditioning variables.
- Calling an assumption a finding: Agreement among observed outcomes cannot directly establish equality involving a missing target-pattern outcome.
Check Your Understanding
For each question, identify what an assumption statement must say. Brief solutions follow the questions.
- For target pattern \((1,0,0)\) and donor pattern \((1,1,0)\), an analyst restricts \(Y_2\). What information is still needed to report the restriction fully?
- Write a conditional mean restriction for \(Y_3\) in target pattern \((1,1,0)\) borrowed from \((1,1,1)\), conditional on \(Y_1,Y_2\), but not on \(X\).
- If two patterns have equal conditional means given \(X\), must their marginal means be equal? State the condition that would ensure equality by averaging.
- A report says, “The target and donor distributions are the same on average.” Why is this wording inadequate?
Solutions. (1) The analyst must specify the conditioning variables or history and whether the restriction equates conditional distributions or only conditional means; the target, donor, and visit are already named. (2) Write \(E(Y_3\mid Y_1,Y_2,S=(1,1,0))=E(Y_3\mid Y_1,Y_2,S=(1,1,1))\), for relevant histories. This is a conditional-mean, not distributional, equality. (3) No. Equal conditional means given \(X\) can yield different marginal means if the patterns have different distributions of \(X\). Equal distributions of \(X\) across the two patterns, together with the conditional-mean equality over their support, ensures equal marginal means. (4) “On average” does not identify the outcome visit, patterns, conditioning set, or whether “average” means a marginal mean or a conditional mean. It also does not assert equality of full distributions. A complete statement must name each of these elements explicitly.