Why Restrict the Donor Pool?
Predictive mean matching (PMM) imputes a missing value by selecting an observed value from a donor whose predicted score is close to the score for the incomplete case. Ordinarily, the donor search is made over an available set of observed cases. But sometimes two cases that are close on the prediction scale are not eligible to donate to one another. For example, an outcome may have different meanings, permitted ranges, or clinical pathways across genuine eligibility groups. A close score alone does not make a donor appropriate.
In this setting, a defensible stratum is a substantively meaningful group that is known for both the incomplete case and potential donors, and within which observed values are eligible to be exchanged for imputation. Restricting donors to that group can prevent cross-group matches. The tradeoff is that each case has fewer candidates: nearest eligible donors may be farther away, and a stratum may not contain enough observed values to support the requested donor search.
The score and the donor search are defined for the current PMM imputation step, including its model and parameter draw. The stratum restriction changes the candidate set; it does not, by itself, change the prediction model, make the model adequate, or establish that the missingness mechanism is ignorable. PMM and its prediction-scale matching are described in the earlier material on Predictive Mean Matching.
When Is a Stratum Justified?
A restriction is appropriate when cross-stratum values are not reasonable substitutes for the incomplete case, not merely because a variable happens to divide the data into groups. A group might encode an eligibility rule, a treatment-specific measurement protocol, or a category that changes the meaning or allowable support of the outcome. The clinical or scientific reason should be stated before examining which restriction produces convenient imputations.
The stratum must also be available for the incomplete case. If group membership is unknown precisely when \(Y\) is missing, a rule is needed to determine or impute that membership; it is not valid to silently assign the case to whichever stratum has donors. In addition, each relevant stratum needs observed \(Y\) values. If a stratum has no observed donors, within-stratum PMM cannot produce an imputation from observed values for its incomplete cases.
A useful conceptual condition is that the observed values within a stratum are plausible donors for incomplete cases in that same stratum, given the prediction information used by PMM. Restriction is not a substitute for relevant predictors, adequate overlap, or a suitable imputation model. Nor does a meaningful eligibility label automatically guarantee that all cases within a stratum are comparable: important differences may remain within it.
What Restriction Guarantees—and What It Does Not
Suppose the unrestricted candidate set for case \(i\) is \(C_i\), and the eligible within-stratum set is \(C_i(S_i)\). By definition, \(C_i(S_i)\) is a subset of \(C_i\). If both sets are nonempty, the nearest eligible donor cannot be closer on the score scale than the nearest unrestricted donor. This follows because minimizing over a subset cannot produce a value smaller than minimizing over the full set:
This simple inequality describes the main tradeoff precisely: restricting donors can enforce eligibility, but it cannot improve the best available score distance. If the nearest unrestricted match belongs to a different stratum, the restricted match may be substantially farther away. A small score distance is useful only to the extent that the score captures relevant similarities; it does not prove that two outcomes are exchangeable.
If the procedure selects several nearest donors and samples among them, all selected candidates should obey the restriction. Sampling from a within-stratum set preserves the restriction; selecting a cross-stratum case because it is among the globally nearest scores does not. The number of requested donors must be feasible in the restricted pool. Ties and donor reuse can also matter, especially in small strata.
Worked Example: Preventing a Cross-Eligibility Match
Hypothetical data. Suppose a clinical service imputes a measurement for patients in one of two documented eligibility groups, A and B. The groups have different protocols, so the team has justified that donors must come from the same group. The displayed scores are from the same PMM model and parameter draw. The observed donor values are in measurement units.
| Case | Stratum | Score \(\eta\) | Observed \(Y\) |
|---|---|---|---|
| A1 | A | 10.0 | 24 |
| A2 | A | 10.8 | 27 |
| A3 | A | 12.0 | 31 |
| B1 | B | 9.9 | 40 |
| B2 | B | 10.4 | 44 |
| B3 | B | 11.1 | 47 |
| Target | A | 10.5 | Missing |
For the target, the score distance to B2 is \(|10.5-10.4|=0.1\), while the distance to A2 is \(|10.5-10.8|=0.3\) and the distance to A1 is \(|10.5-10.0|=0.5\). With \(k=2\), unrestricted selection would choose B2 and A2. But B2 is ineligible as a donor under the stated protocol.
Restricting the search to stratum A leaves A1, A2, and A3 as candidates. The two smallest distances are \(0.3\) for A2 and \(0.5\) for A1, so the eligible donor set is \(\{A2,A1\}\). If PMM samples uniformly from these two selected donors, the imputed value is 27 or 24, each with probability \(1/2\). The restriction prevents an ineligible donor value of 44 from entering the draw; it does not claim that the selected donor is equally close to the target as B2.
Sparsity Is the Cost of a Smaller Candidate Set
A stratum restriction can leave too few observed cases for the intended donor search. If a procedure requests \(k\) distinct donors but only \(m\) eligible observed cases are available, then \(k\leq m\) is necessary to select \(k\) distinct donors. When \(k>m\), silently drawing fewer donors or allowing ineligible donors changes the specified procedure. If reuse is permitted, repeated selections do not create additional independent observed support; they repeat values from the same small pool.
Even when enough donors exist numerically, sparse score support can be a problem. The closest eligible donor may be far away, or several incomplete cases may repeatedly draw from the same one or two observed cases. A restricted search can therefore preserve the eligibility rule while still producing a weakly supported imputation. Increasing the number of donors does not manufacture information, and relaxing a justified restriction merely to fill the donor set can violate the reason the restriction was imposed.
Worked Example: A Sparse Eligibility Stratum
Hypothetical data. Suppose stratum C is a genuine eligibility group, but only two observed cases have \(Y\). Their scores and observed outcome values are \((6.1, 18)\) and \((7.0, 22)\). An incomplete case in C has score \(6.8\). The requested donor count is \(k=3\), with distinct donors.
The two eligible score distances are \(|6.8-6.1|=0.7\) and \(|6.8-7.0|=0.2\). There are only \(m=2\) eligible observed cases, so \(k=3>m=2\): three distinct within-stratum donors cannot be selected. If the procedure instead used the two available eligible candidates, the nearer donor would be the case with \(Y=22\); if it samples from both nearest candidates, the donor values could only be 18 and 22.
The calculation shows both the available choices and the limitation: even the nearest eligible donor is \(0.2\) score units away, and no third distinct donor exists. A cross-stratum case could not be added without abandoning the justified restriction. The analyst should report the sparse support and revisit the imputation strategy with the scientific team, rather than conceal the shortage by treating an ineligible case as valid. If there were no observed cases in C, there would be no within-stratum donor value at all.
Distinguishing Eligibility From Arbitrary Grouping
Stratification is not automatically beneficial whenever subgroup labels are present. An administrative label, clinic, or demographic category should not be used as a hard donor barrier unless there is a substantive reason that cross-group donors are not suitable. An unnecessary restriction may discard close matches, increase donor distances, and create sparse cells. Conversely, if the group genuinely changes eligibility or the interpretation of \(Y\), pooling donors can create a close prediction-score match that is inappropriate on the outcome scale.
Worked Example: A Restriction Without an Eligibility Rationale
Hypothetical data. Suppose two clinics record the same measurement using the same protocol, and there is no clinical or measurement reason to prohibit cross-clinic donors. A target at clinic X has score \(5.0\). The observed donors are X1 with score \(4.0\) and \(Y=12\), X2 with score \(5.2\) and \(Y=18\), Y1 with score \(5.0\) and \(Y=19\), and Y2 with score \(5.1\) and \(Y=20\). Consider selecting one nearest donor.
With all donors eligible, the distances are \(1.0\), \(0.2\), \(0\), and \(0.1\), respectively, so Y1 is nearest and its value is 19. If clinic is imposed as a stratum without justification, only X1 and X2 remain; X2 is nearest, at distance \(0.2\), and its value is 18. This arithmetic does not establish that 19 is a better imputation than 18. It shows that the restriction changed both the nearest match and the donor value, despite no stated reason that clinic X and clinic Y donors were ineligible to exchange.
The decision must come from the data-generating and measurement context, not from which donor value seems preferable after the fact. If clinics actually used different measurement protocols, that would be a different justification and the restricted comparison would need to be evaluated under that context.
Diagnostics and Practical Comparisons
Before using restricted PMM, record the eligibility definition and verify that stratum membership is known for both observed and incomplete cases. Then examine, for each stratum, the number of observed donors, the number of incomplete cases, the score distances to selected donors, and the observed range and distribution of \(Y\). These checks can reveal an empty stratum, a donor count that is impossible, unusually large nearest distances, or repeated reliance on a few observed values. Also verify that every selected donor has the same stratum label as its recipient.
Large distances are a warning that the selected donors may be poor score matches, not an automatic rule for dropping cases. Compare distances and donor availability across strata, and check whether the score model gives useful ranking within each one. Look for strata in which observed values cover only a narrow part of the cases’ score range. A restriction cannot repair lack of overlap within the eligible group, and it cannot ensure that important predictors have been included.
| Approach | Candidate rule | Main advantage | Main limitation |
|---|---|---|---|
| Pooled PMM | Search among observed cases across groups | Larger donor pool and potentially closer scores | Can select donors who are not eligible or comparable |
| Within-stratum PMM | Search only among observed cases in the recipient’s justified stratum | Prevents cross-stratum donor assignments | Fewer candidates, potentially larger distances or no donors |
| Separate model by stratum | Fit prediction models within groups, then match within groups | Allows prediction relationships to differ by group | Can be unstable when strata have little observed data |
Pooled PMM and within-stratum PMM differ in donor eligibility, while a separate model by stratum changes the prediction-model fitting as well. These are distinct decisions. Restricting the donor pool does not require that separate models be fitted; fitting separate models does not remove the need to check whether each group has adequate observed support. The appropriate choice depends on the substantive eligibility rule and the information available within groups.
Common Mistakes and Limitations
- Restricting by convenience. A group label alone does not justify excluding donors. State why cross-group values are unsuitable before applying the restriction.
- Assuming restriction improves all matches. The nearest within-stratum donor can only be as close as, or farther than, the nearest unrestricted donor on the same score scale.
- Ignoring an empty or sparse stratum. A restriction cannot provide an observed donor when none exists, and repeated reuse of a tiny pool does not create new information.
- Treating a close score as proof of comparability. Prediction distance is a ranking device; it cannot establish that the donor and recipient are interchangeable on all relevant features.
- Using an unknown group assignment as if it were known. Donor eligibility cannot be checked without a defensible stratum for the incomplete case.
The central limitation is that the restriction enforces a chosen boundary; it does not validate that boundary, guarantee adequate support, or establish the broader assumptions required for valid multiple imputation. When eligibility and available observations conflict, the conflict is a substantive modeling problem to address transparently, not a reason to make the donor rule appear successful by changing it silently.
Check Your Understanding
Answer each question using the donor eligibility rule, not just the prediction scores.
- A missing case belongs to a justified stratum with four observed donors. The requested number of distinct donors is five. Can the specified within-stratum donor set be formed? Explain.
- For one target, the nearest unrestricted score distance is \(0.08\), while the nearest within-stratum distance is \(0.42\). Is this inconsistent with the mathematics of restriction? What does it suggest?
- A hospital identifier is available for every case, but there is no protocol or clinical reason to exclude cross-hospital donors. Is hospital automatically a justified stratum?
- A missing case has an unknown eligibility-group label, and the groups have different donor rules. Can the analyst simply use the group with more observed donors?
- Why does repeatedly reusing the only two eligible observed donors not amount to having a large donor pool?
Solutions. (1) No. Five distinct donors require at least five eligible observed cases, but only four are available. The donor count or imputation strategy must be addressed without silently adding ineligible cases. (2) No. The restricted candidate set is a subset of the unrestricted set, so its minimum distance can be larger. The increase suggests weaker score proximity within the eligible group and should prompt a support check. (3) No. The identifier alone does not justify the restriction; a substantive reason is needed. (4) No. Group assignment must be determined or handled using a defensible model or rule; choosing the group with more donors assigns eligibility for convenience. (5) Reuse repeats the same observed values. It may enable draws under a reuse-permitting procedure, but it does not add distinct observed outcomes or broaden the information available within that stratum.