Choosing the Number of PMM Donors

Why Donor-Pool Size Matters

In predictive mean matching (PMM), an imputed value is taken from an observed outcome whose prediction score is close to the score for the record with a missing value. The number of candidate donors controls how many observed outcomes are eligible for selection. A very small pool can concentrate selection on the closest score matches; a larger pool can offer more outcome variety, but may include cases whose scores are less similar. Choosing \(k\) is therefore a balance, not a search for a universally best number.

The choice should be evaluated in the setting where PMM is used: with the specified imputation model, predictors, donor eligibility rules, and missingness pattern. A suitable pool size cannot repair a poor prediction model, inadequate overlap, or implausible assumptions about the missing data.

Write the ordered distances as \(d_{(1)} \le d_{(2)} \le \cdots\), and let \(y_{(j)}\) be the observed outcome belonging to the record at rank \(j\). The pool size is an integer from 1 to the number of eligible observed cases. With \(k=1\), the closest eligible donor is selected. With a larger \(k\), selection can reach farther down the ordered list. These statements concern distances on the prediction scale; they do not imply that outcomes themselves are ordered by their values.

Assumptions and What the Choice Can Control

The comparison of pool sizes is meaningful only if the matching procedure is being used appropriately. The prediction model should use relevant, adequately specified predictors, and its scores should rank cases in a way that makes nearby scores informative about plausible donor outcomes. Donors must be eligible under the analysis design, and there must be observed cases with scores sufficiently comparable to those of records needing imputation. In addition, the missing-data assumptions and imputation model must support the intended analysis; selecting \(k\) alone does not establish those assumptions.

As in other imputation procedures, the missingness mechanism and the variables included in the imputation model matter. The standard missing-data terminology is covered in the earlier material on Missing Data. Here, the focus is narrower: how the size of the PMM pool changes the set from which a donor can be drawn, conditional on the chosen model and eligible donors.

Two useful quantities make the trade-off concrete. The mean distance in a pool is \(\bar d_k = k^{-1}\sum_{j=1}^{k}d_{(j)}\). If selection is uniform among the \(k\) donors, the conditional expected imputed outcome is \(\bar y_k = k^{-1}\sum_{j=1}^{k}y_{(j)}\), and its conditional variance is \(k^{-1}\sum_{j=1}^{k}(y_{(j)}-\bar y_k)^2\). That variance describes variation from selecting a donor within this fixed pool; it is not a complete measure of uncertainty in the imputation or analysis.

What Changes as \(k\) Changes?

Because the distances are ordered, the average distance cannot decrease when the pool grows. To see why, the average for \(k+1\) donors is the average of the first \(k\) distances and the next, larger-or-equal distance:

The inequality holds because \(d_{(k+1)}\) is at least as large as each of the first \(k\) distances, and therefore at least as large as their average. Thus, a larger pool weakly increases the average matching distance. The effect on the set of possible imputed outcomes is less predictable: a new donor may have an outcome similar to those already included, or it may have a very different one. Pool size does not guarantee a particular direction or amount of change in outcome variance.

Using Validation to Compare Candidate Sizes

A practical comparison can temporarily hide some observed outcomes, treat those records as if their outcomes were missing, and use the remaining observed cases as donors. Because the hidden outcomes are known for this check, one can compare how well candidate pool sizes recover them. The masking and donor selection should reflect the intended analysis as closely as possible, including the prediction model, score calculation, and eligibility rules.

This check evaluates predictive performance for the masked cases; it does not prove that imputations are unbiased or that the missing-data assumptions hold. It is also useful to inspect matching distances and the range or distribution of imputed values across candidate sizes. These diagnostics complement, rather than replace, substantive justification.

Choosing and Reporting a Pool Size

There is no pool size that is best for every data set. A small pool emphasizes the closest prediction-score matches, but it may allow very few distinct outcomes to be selected. A larger pool can broaden the possible draws, while also admitting more distant matches. The consequences depend on the local distribution of scores and outcomes, the number of eligible observed cases, and the purpose of the imputation.

A defensible choice compares a modest set of plausible values of \(k\) while holding the other modeling decisions fixed. Examine the resulting matching distances, imputed outcome distributions, and, when feasible, masked-observation performance. Then assess whether the substantive conclusions are sensitive to reasonable alternatives. If candidate sizes yield very different imputed values or conclusions, investigate the model and the sparse regions of the score distribution rather than treating one setting as automatically correct.

PMM can be contrasted with a parametric regression imputation that generates outcomes from a fitted outcome model rather than selecting observed donor outcomes. PMM’s donor selection retains observed outcome values, while the pool size governs which of those values can be drawn. Neither approach is universally preferable: the choice depends on the model, data structure, and inferential goal. Choosing \(k\) is a tuning decision within PMM, not a substitute for comparing appropriate imputation approaches.

Diagnostics, Limitations, and a Common Mistake

Warning signs include a large increase in average match distance as \(k\) grows, imputed values that change sharply across plausible pool sizes, or masked-observation errors that worsen consistently for larger pools. Also check whether donors are genuinely eligible and whether the score model distinguishes cases relevant to the outcome. These checks can flag poor performance, but none can establish that the missingness assumptions are true.

A masked-observation check has limitations. The observed cases used for validation may not represent cases with missing outcomes, especially when missingness depends on unobserved values. The check is also sensitive to how cases are masked and to random donor selection. Consequently, validation results should inform the choice, not be treated as a proof of inferential validity. Pool-size comparisons should be interpreted alongside the intended analysis and the model’s assumptions.

Common mistake: “A larger \(k\) always produces more variable imputations, so choose the largest pool to preserve uncertainty.” This reasoning is not generally correct. New donors may have outcomes close to the existing pool, and the conditional outcome variance can rise, fall, or stay similar. Conversely, \(k=1\) removes random donor selection within a fixed pool but does not remove all imputation uncertainty, such as uncertainty associated with model estimation or other random draws in the imputation procedure. Compare the actual distances and outcomes; do not infer their behavior from \(k\) alone.