From a Regression Fit to an Imputed Outcome
Predictive mean matching (PMM) combines a regression model with donor-based imputation. The regression does not directly supply the missing outcome. Instead, it gives each case a prediction score; the score helps locate observed cases whose outcomes can serve as donors. PMM then fills a missing outcome by randomly selecting one of those donors’ actual observed values.
This sequence matters. A regression prediction can be fractional, fall outside the observed range, or fail to preserve features of the observed outcome distribution. A donor value is an outcome that was actually observed. Using it can keep an imputation within the observed support, although it does not by itself guarantee a suitable model, a suitable donor, or valid inference.
For the examples below, the regression score is the fitted value \(\hat{Y}_i = X_i^\mathsf{T}\hat{\beta}\), where \(\hat{\beta}\) is estimated from cases with observed outcomes. The same fitted model is used to score observed and missing cases. This makes the scores comparable. In a MICE iteration, predictors that are themselves incomplete must have current values available to calculate the scores; those values may be their current imputations in the iterative procedure.
This is a basic illustration of the score-to-donor mechanics. Implementations may add parameter uncertainty to the fitted model, and the choice of donor-pool size and sampling details affects the procedure. Those choices do not change the central distinction: scores guide donor selection, while the imputed value is an observed outcome.
The Ordered Steps and Their Conditions
A PMM update requires more than carrying out calculations in sequence. The outcome model must be fitted to the right cases, scores must be available on a common scale, and the selected donor must be a defensible stand-in for the missing case. The main assumptions and checks are summarized in the steps below.
Use cases in \(O\), where \(Y_i\) is observed, to estimate the specified relationship between \(Y\) and \(X\). The model form and predictors should represent important features of the outcome conditional on the predictors. For the fitted regression to be useful, its estimation also requires adequate information in the observed cases and a defensible model for the score relationship.
For every \(i\) in \(O\) and \(M\), calculate \(\hat{Y}_i\) using the same fitted model and the predictor values available in the current update. A missing case whose score lies far beyond the observed scores has no close score-based analogue; matching cannot create information in such a region.
For each \(i\) in \(M\), compare its score with the scores of cases in \(O\), then form a donor set from observed cases with nearby scores. The donors must have observed outcomes, and score proximity should correspond to similarity relevant to the outcome. A close score is a selection aid, not proof that two cases are interchangeable.
Randomly select an observed case from the donor set and assign its \(Y\) value to the missing outcome. The draw is from observed outcomes, not from the regression predictions. Repeat the update for each missing case, following the implementation’s donor-sampling rule.
The score model is a working model for locating donors; its role is not to claim that \(Y_i\) equals its fitted value. Useful conditions include a reasonably specified predictor relationship, enough observed cases to estimate it, and overlap between missing-case scores and observed-case scores. The predictors must also be available for the cases being scored. If a model is poorly specified or the observed cases do not cover the missing cases’ predictor region, a numerical match can still be a poor match in context.
Worked Example: Fit, Score, and Draw
Suppose five hypothetical cases have fully observed predictor \(x\) and outcome \(y\). A sixth case has \(x=2.5\) but a missing outcome. The observed data are \((x,y)=(0,2),(1,3),(2,5),(3,4),(4,6)\). Use a simple linear regression with an intercept to illustrate one PMM update. Assume the donor set consists of the two observed cases with scores closest to the missing case’s score.
1. Fit the regression. The observed means are \(\bar{x}=2\) and \(\bar{y}=4\). The least-squares slope is
The intercept is \(\hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x}=4-(0.9)(2)=2.2\). Thus the fitted score is \(\hat{y}=2.2+0.9x\). This model is estimable here because there are five observed outcomes and the predictor varies across the observed cases; the linear form is an illustrative working model, not a claim that the data prove linearity.
2. Score the observed and missing cases. The observed scores, in order of \(x\), are \(2.2, 3.1, 4.0, 4.9,\) and \(5.8\). For the case with \(x=2.5\),
3. Identify nearby donors. The scores \(4.0\) and \(4.9\) are each \(0.45\) from \(4.45\), so the two nearest observed cases have \(x=2\) and \(x=3\). Their observed outcomes are 5 and 4, respectively.
4. Draw an outcome. If the random draw selects the case with \(x=2\), the imputation is \(y=5\). If it selects the case with \(x=3\), the imputation is \(y=4\). The regression score \(4.45\) is used to locate donors; it is not itself assigned as the imputation.
Why the Draw Is an Observed Outcome
PMM separates the job of ranking cases from the job of supplying an imputed value. The regression score summarizes the fitted outcome level associated with the predictors. If two cases have similar scores, they may be plausible candidates to supply an outcome for one another, subject to the model and context. But the score is not a recorded response. PMM selects a donor’s response instead.
This gives a direct support property: every imputation produced by this basic procedure equals an outcome observed in the donor pool. For example, if all eligible donor outcomes are whole numbers, the imputed value is also a whole number. This property is useful, but limited. It does not show that the conditional distribution is correct, ensure that the donor is substantively comparable, or solve a lack of observed cases in an important region.
Worked Updates and Practical Checks
Worked Example: Two Missing Outcomes in One Update
Consider a hypothetical update with observed cases having scores and outcomes \((10,12),(12,15),(15,18),(18,20)\). Two cases have missing outcomes, with scores 11 and 17. For illustration, use each missing case’s two closest observed scores as its donor set, then select one donor uniformly at random.
For score 11, the distances to observed scores 10, 12, 15, and 18 are \(1,1,4,7\). The donor outcomes are therefore 12 and 15. If the random draw selects the donor with score 12, the first imputation is 15; if it selects the donor with score 10, it is 12.
For score 17, the distances are \(7,5,2,1\). Its donor outcomes are 18 and 20. A draw selecting the donor with score 18 imputes 20; a draw selecting the donor with score 15 imputes 18. Each missing case is matched using its own score. The example assumes scores were calculated from the same fitted model and that these observed outcomes are valid donor candidates; neither assumption can be checked from the distance arithmetic alone.
Several diagnostics help distinguish a functioning update from one that merely returns a number. First check that the outcome model was fitted to observed outcomes and that scores for observed and missing cases use the same predictors and model. Then examine whether missing-case scores fall within or near the observed-score distribution. Finally, check that the selected donors make sense for the study: a shared score may hide important differences in predictors if the model omitted relevant structure.
- Score extrapolation: if a missing case’s score is well outside the observed range, its “nearest” donor may still be far away. Treat the resulting imputation as weakly supported.
- Misspecified score model: systematic residual patterns or an inappropriate functional form can make scores a poor guide to outcome similarity.
- Unavailable predictors: a case cannot be scored as intended unless its predictor values are available in that update. Verify how incomplete predictors are represented in the current MICE iteration.
- Unsuitable donors: inspect donor outcomes and relevant case characteristics, not only the distance. Proximity on a score is not a substitute for subject-matter judgment.
- Random variation: different random draws can produce different imputations. A single completed data set does not reveal all uncertainty associated with missing values.
Worked Example: Diagnosing a Poorly Supported Match
Suppose a hypothetical observed-case score range is 8 to 22, while a missing case has score 31. The closest observed score is 22, so its score distance is \(31-22=9\). The nearest donor may still be used by a mechanical rule, but the distance signals that this case is outside the observed score range.
The arithmetic identifies the problem; it does not fix it. Before accepting the imputation, check whether the score model is appropriate, whether predictor values for the missing case are correct, and whether any observed cases cover that case’s relevant characteristics. If no observed donor is plausibly comparable, the imputation is an extrapolation in donor form. The donor outcome remains observed, but its suitability for this missing case is uncertain.
Alternatives, Limitations, and a Common Mistake
A parametric regression imputation can use the model to generate a value directly, potentially including values not seen among observed outcomes. PMM instead uses the model’s scores to choose observed outcomes. Mean imputation is simpler still, but assigns the same fixed value to many missing cases and does not represent their variation. These approaches answer different practical needs; none is automatically reliable without appropriate modeling and uncertainty treatment.
PMM’s attraction is also a limitation. It cannot produce an outcome beyond the observed donor support, which may be unsuitable when the missing cases genuinely lie outside that support. Its performance depends on observed cases being adequate donors and on the prediction score ordering cases usefully. The full missing-data analysis must also address assumptions about the missingness process and uncertainty; the donor step alone does not establish those conditions. The earlier material on Overlaps Missing Data covers the broader missing-data framework.
A reliable PMM update therefore requires attention to the entire chain: fit on observed outcomes, score both groups consistently, identify donors with adequate score proximity and substantive relevance, and draw an observed outcome. A plausible final value is not enough if one of the preceding steps was unsupported.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- In PMM, what role does the regression prediction play, and what value is actually assigned to a missing outcome?
- A missing case has score 14. Observed cases have scores 12, 15, and 19, with outcomes 8, 11, and 17. If the two closest cases form the donor set, what outcomes may be imputed?
- Why should observed and missing cases be scored using the same fitted model?
- A missing case’s score is far above every observed score. What does that suggest, and why does choosing the nearest donor not remove the concern?
Solutions. 1. The regression prediction is a score used to locate donors; PMM assigns a randomly selected donor’s observed outcome. 2. The score distances are \(2,1,5\), so the donor outcomes are 8 and 11; either may be imputed under the stated rule. 3. A common fitted model puts the scores on a comparable scale, so their proximity has a consistent interpretation. 4. The case lacks close score overlap with observed cases. The nearest donor can still be far away, so the match may be unsupported extrapolation rather than a genuinely similar donor.