When a Missing Case Lies Beyond the Observed Data
Predictive mean matching (PMM) assigns a missing outcome by selecting an observed case as a donor and copying that case’s outcome. This can make PMM useful when a newly generated outcome would be implausible, such as a negative count or a value beyond an instrument’s range. But the same feature creates a limitation: when a missing case lies in a sparse region of covariate space, PMM still has to borrow an outcome from somewhere in the observed data. The selected donor may be far away in the covariates, in its predictive score, or both.
A distant donor is not automatically inappropriate. In a well-supported setting, the covariate model and observed data may still identify relevant donors. The concern is whether the available observed cases provide credible information about the outcome for this particular missing case. A fitted score can be calculated for a case with unusual covariates, but a calculated score does not create observed outcomes near that case.
Assumptions Behind Using a Donor
PMM is a procedure, not a guarantee that an available donor is informative. For the selected donor’s outcome to represent a missing case, the observed data need to support the prediction in the region where that case lies. In particular, the prediction model should capture outcome-relevant relationships well enough for its score to help identify comparable cases; observed cases should exist with relevant covariates or predictive scores; and their outcomes should be plausible substitutes for the missing outcome. If the missingness process is treated as missing at random conditional on the modeled covariates, that assumption is also needed for the usual missing-data interpretation. Missing at random does not, by itself, supply information in a region with no observed cases.
The critical distinction is between computing a prediction and having observed support for it. A model can assign a score to a case outside the covariate range of the observed cases. PMM can then find the closest observed scores, but “closest available” need not mean “close enough to justify borrowing.” Model uncertainty or a parameter draw can alter score distances or donor rankings; it cannot add an observed outcome in the empty region.
Worked Example: A Case Beyond the Observed Covariate Range
Suppose a hypothetical dataset has observed cases at \(X=0,1,2\), with outcomes \(Y=2,4,5\). A missing case has \(X=8\). For illustration, suppose the fitted prediction score is \(2+1.5X\). What donor does nearest-score PMM select, and what does this tell us?
The observed scores are \(2+1.5(0)=2\), \(2+1.5(1)=3.5\), and \(2+1.5(2)=5\). The missing case’s score is \(2+1.5(8)=14\). The absolute score distances are therefore \(12\), \(10.5\), and \(9\), respectively. The closest observed score belongs to the case at \(X=2\), so the donor’s observed outcome is \(Y=5\).
The score model assigns a value of 14 to the missing case, but there is no observed case at or near \(X=8\): the largest observed \(X\) is 2. The imputation is \(5\), not 14, because PMM copies the donor’s observed outcome. The available donor is the closest among these three cases, yet its covariate value is still six units below the missing case’s. The example illustrates a data-support limitation, not proof that the imputation is wrong: its relevance depends on whether the relationship at the edge of the observed range is informative about the distant case.
Why the Donor Constraint Limits Extrapolation
For a fixed donor pool \(D\), PMM assigns the missing case an outcome from the observed values in that pool. This gives a simple support result that does not depend on the prediction model being linear or correctly specified.
The same bound applies to the expected imputed value over donor selection: it is a weighted average of donor outcomes, with nonnegative selection probabilities that sum to one. For example, if two donors have outcomes 30 and 70 and each is selected with probability one-half, the expected imputed outcome is \((0.5)(30)+(0.5)(70)=50\). It cannot exceed 70 or fall below 30. The realized imputation is one of the donor values, not necessarily that average.
This support restriction is local to the eligible donor pool. If donors are selected from the full observed data, an imputation may be outside the observed outcome range of a particular covariate neighborhood, while remaining within the range of all eligible donors. Conversely, a narrow donor pool can impose a narrow outcome range even when a missing case’s true outcome could lie outside it. PMM therefore preserves observed values; it does not establish that the observed values span the outcomes needed in sparse regions.
Worked Example: Similar Scores but Sparse Covariate Support
Consider two hypothetical observed donors with covariates \((X_1,X_2)=(9,1)\) and \((1,9)\), and outcomes 30 and 70. A missing case has covariates \((10,10)\). Suppose the score is \(X_1+X_2\), and these are the only eligible donors.
Each observed donor has score \(9+1=10\) or \(1+9=10\). The missing case has score \(10+10=20\), so each donor’s absolute score distance is \(10\). If PMM selects randomly between the tied donors, the imputation is 30 or 70; with equal selection probabilities, its expected value is \((30+70)/2=50\).
The two donors are also both far from the missing case in covariate space: each has Euclidean distance \(\sqrt{(10-9)^2+(10-1)^2}=\sqrt{82}\), approximately 9.055. The score cannot distinguish them, and neither donor has covariates near \((10,10)\). Their outcomes differ by 40 units, so the selection entails substantial uncertainty about which observed outcome is a useful substitute. The score match is not evidence that either donor is locally representative.
Diagnosing Sparse-Region Problems
Sparse support should be assessed for the cases that need imputation, not only by checking whether a model fitted successfully. Useful diagnostics include the distance from each missing case’s score to its nearest observed score, how many observed cases are comparably close, the range of covariates among candidate donors, and whether donor outcomes vary substantially. These checks help distinguish a modest gap from a case for which every available donor is remote.
| Diagnostic pattern | What it may indicate | What it does not prove |
|---|---|---|
| Large nearest-score distance for a missing case | The case is unlike observed cases on the prediction scale. | That the imputation is biased by a known amount. |
| Few observed cases near the missing case’s covariates | Limited local support, even if score distances are similar. | That matching on the score is necessarily invalid. |
| Very different outcomes among the nearest candidates | Donor choice may lead to materially different imputations. | Which candidate outcome is closest to the unobserved truth. |
| Missing covariates beyond observed covariate ranges | The imputation may rely on extending relationships beyond observed data. | That a particular extension is correct or incorrect. |
These diagnostics are clues about support, not automatic thresholds for accepting or rejecting an analysis. Distances depend on how covariates are represented and scaled, and a single scalar score can conceal differences in the underlying covariates. Examine the data in context, document which missing cases lack nearby observed cases, and assess whether conclusions depend on imputations for those cases. A plausible overall distribution can conceal severe problems for a small but influential sparse subgroup.
Identify cases with unusual covariates or scores relative to cases whose outcomes are observed.
Check how far the best available donors are from the missing cases and whether candidate outcomes are similar or widely dispersed.
State where support is weak and avoid presenting donor-based imputations there as if they were directly informed by comparable observed cases.
What PMM Can and Cannot Do
A model-based approach can, depending on its specification, generate values outside the observed outcome range by extending a modeled relationship. PMM instead returns an observed donor outcome, making its output less prone to certain implausible generated values but unable to invent an outcome beyond the donor values. This is a conceptual contrast, not a rule that either approach is preferable in every sparse region. The model-based result also depends on assumptions, and a numerical prediction beyond observed support is not validated merely because it can be calculated.
When covariates are sparse, neither method creates empirical information that the data do not contain. A defensible analysis should identify this limitation and consider whether substantive knowledge, the chosen covariates, or the target question supports the intended inference. Changes to donor eligibility require justification: they cannot be treated as a mechanical cure for missing local support. In particular, a donor set that excludes relevant observed cases may introduce a different mismatch.
Worked Example: An Observed-Outcome Ceiling
In a hypothetical application, observed outcomes among all eligible donors range from 12 to 18 units. A missing case lies in a covariate region with no observed cases nearby. Suppose its fitted score is 27, while the nearest eligible donor scores 19 and has an outcome of 18 units.
The score distance to that donor is \(|27-19|=8\). If this donor is selected, PMM imputes 18 units. More generally, because all eligible donor outcomes are between 12 and 18 units, every PMM imputation must satisfy \(12\leq Y_{\mathrm{imp}}\leq18\). In particular, PMM cannot impute 27 units, even if a score of 27 seems to suggest a larger outcome.
The score is not itself an outcome, so the difference between 27 and 18 is not an estimate of imputation bias. The key conclusion is narrower: PMM’s donor mechanism cannot produce an imputed value above 18 units from this donor pool. If outcomes above 18 are plausible for the missing case, the available donor values do not resolve that possibility.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A missing case has a score of 25. The closest observed donor has a score of 17 and an outcome of 9. What does the score distance tell you, and what does it not tell you?
- All eligible donor outcomes lie between 4 and 11 units. Can PMM impute 12 units for this case? Explain.
- Two equally likely donors have outcomes 20 and 60. Calculate the expected imputed outcome, and state whether that expectation is itself necessarily a possible imputed value.
- Why does a missing-at-random assumption not guarantee that a sparse-region PMM imputation is well supported?
Solutions. 1. The absolute score distance is \(|25-17|=8\), indicating a gap on the prediction scale. It does not show that the donor is representative, quantify bias, or establish how the missing outcome relates to the score. 2. No. PMM must copy an eligible donor value, so \(4\leq Y_{\mathrm{imp}}\leq11\). 3. The expectation is \((0.5)(20)+(0.5)(60)=40\). The realized imputation is 20 or 60, so 40 is not necessarily a possible draw. 4. Missing at random conditional on modeled covariates concerns the missingness mechanism; it does not create observed cases with comparable covariates. A case in a region with little or no observed support can therefore still lack an informative donor.