Why PMM Imputations Stay Within Observed Support

What “Within Observed Support” Guarantees

Predictive mean matching (PMM) is often described as keeping imputations in a realistic range. That description is useful only when “range” is stated precisely. The mechanical guarantee is that a PMM imputation is copied from an observed outcome selected as a donor. It therefore belongs to the set of observed donor outcomes, and it cannot be smaller than the smallest or larger than the largest outcome among all observed cases. This is a guarantee about the source of the imputed value—not a guarantee that the value is appropriate for the particular person whose outcome is missing.

Let \(Y\) be the outcome being imputed, and let \(O\) denote the indices of cases with observed \(Y\). Let \(M\) denote the indices of cases with missing \(Y\). The empirical observed support is the set of distinct observed outcome values:

For a missing case \(j\in M\), PMM forms a donor candidate set \(C_j\) from observed cases, often using closeness of predicted scores. It selects a donor index \(d(j)\in C_j\) and assigns the donor’s observed outcome to the recipient. Donor selection may be random among a set of close candidates; the selected donor can depend on the score model and the random draw.

The guarantee requires a nonempty donor set and a selected donor whose outcome is observed. It does not require the prediction model to be correct, the nearest donors to be good matches, or the donor pool to represent every kind of recipient. Those properties matter for whether the imputation is useful, but not for the range inequality itself. If donors are further restricted to a subgroup, the imputed value is in the support of that subgroup’s eligible observed donors, which is itself a subset of \(\mathcal{S}_{\mathrm{obs}}\).

Why the Bound Holds

The support result follows directly from the donor assignment rule. Since \(d(j)\in C_j\) and \(C_j\subseteq O\), the selected donor is one of the cases whose outcome was observed. Every observed outcome is at least the observed minimum and at most the observed maximum. Substituting the donor outcome for the imputation gives the claimed inequalities.

Because \(Y_j^{\mathrm{imp}}=Y_{d(j)}\), it follows that

The set statement is stronger than the interval statement. For example, if observed outcomes are 2, 5, and 9, a donor-based imputation must be one of 2, 5, or 9—not merely any number between 2 and 9. In a continuous-valued dataset, exact repeats may be uncommon, but PMM still copies an observed value rather than generating an arbitrary value between the observed extremes.

Range Is Not Conditional Plausibility

Observed support is a marginal, sample-level description: it records which outcomes occurred among observed cases. Conditional plausibility asks a different question: is a donor’s outcome credible for a recipient with this recipient’s covariates? A value can satisfy the first criterion and fail the second. For example, if outcome distributions differ sharply by a covariate, a donor from a dissimilar covariate pattern may supply an outcome that is globally observed but unusual for the recipient.

PMM relies on the predictive scores and donor eligibility rules to find observed cases that are useful analogues. The support proof does not verify those choices. Poorly specified predictive models, insufficient overlap between observed donors and missing cases, or a candidate pool that contains dissimilar cases can undermine conditional plausibility while leaving every imputation inside the observed range. A valid-looking number is not, by itself, evidence of a good match.

A second limitation is that the empirical range is not necessarily the population range or a known scientific boundary. If a finite observed sample does not include rare large values, its maximum can be below values that occur in the population. PMM cannot donate a value that was never observed, so it cannot extrapolate beyond the donor outcomes available to it. Conversely, being inside the observed range does not make a value physically or scientifically valid in every conditional context. Known constraints and the design of the donor pool still matter.

Assumptions, Checks, and Alternatives

For the mechanical bound, the necessary conditions are modest: the observed outcomes are correctly identified, the candidate pool contains only cases with observed outcomes, and the imputation copies the selected donor’s outcome on the scale being reported. If donor matching is performed on a transformed scale, the returned value must still be the corresponding observed donor outcome on the outcome scale for this direct support argument to apply. An implementation that instead uses a model-generated value, alters the donor value, or accidentally admits cases with missing outcomes is not following the donor-copying rule used in the proof.

The inferential value of PMM requires more than these mechanical conditions. Donors should be sufficiently comparable to recipients for the analysis, and the model and donor rules should provide useful matches across the covariate patterns of interest. The support theorem does not establish these conditions, nor does it establish validity of the missing-data assumptions used for inference. Those are separate questions in multiple imputation; the earlier material on Missing Data should be consulted for the standard missing-data framework.

Practical checks can separate implementation problems from limitations of the guarantee:

• Check that every imputed value is an observed donor outcome. A value outside \([L_{\mathrm{obs}},U_{\mathrm{obs}}]\), or not present among observed values when exact copying is intended, warrants checking the implementation and data scale.
• Inspect donor identities or donor frequencies, if available. Confirm that donors have observed outcomes and belong to the defined candidate pool.
• Compare observed and imputed outcomes across important covariate groups. Overall range checks can pass even when a subgroup’s imputed values look unlike its comparable observed cases.
• Review whether missing cases have plausible observed donors near their predictive scores. If donor candidates are consistently dissimilar, the range guarantee cannot repair the lack of suitable matches.

These checks have limits. A completed dataset can pass every range check and still have poor conditional imputations. Likewise, an observed outcome’s presence in the donor set does not prove that its donor was a good analogue. Diagnostics should therefore distinguish the easy-to-verify support property from the harder question of donor suitability.

The main contrast is with parametric regression imputation. A regression-based draw can generate values outside the observed sample range, even when its model is appropriate; that is not automatically an error because the sample range is not the population support. PMM avoids such out-of-range generated values by copying observed outcomes, but it cannot create values absent from its donors. Deterministic clipping of a model-generated value to the observed minimum or maximum is another alternative, but it changes the generated value without making it a donor observation or ensuring a good conditional match. No range rule alone resolves the choice: the data, model, donor pool, and inferential goal matter.