From Regression Scores to an Imputed Value
Predictive mean matching (PMM) uses a regression model to locate observed cases that are similar to a case with a missing outcome. It does not usually fill in the missing value with the predicted mean itself. Instead, it randomly selects an observed outcome from a small pool of cases whose regression scores are close to the score for the incomplete case. A hand calculation makes each part of this procedure visible: calculate scores, calculate distances, rank eligible donors, form the pool, and draw one observed outcome.
The calculation below focuses on one missing outcome and five possible donors. For clarity, the fitted regression coefficients are treated as fixed in the numerical example. In an actual MICE iteration, the model may use a parameter draw to represent estimation uncertainty, as established in the earlier material on parameter draws and uncertainty in PMM. The donor-selection arithmetic is the same once the score coefficients for that iteration are specified.
The score is a ranking device, not the imputed value. A donor’s observed outcome can differ from its score because the regression is not an exact description of every observed outcome. That distinction is central: PMM uses the model to guide selection but takes the imputed value from the observed outcome column.
Assumptions and Conditions for the Hand Calculation
The arithmetic assumes that the specified regression score can meaningfully rank cases for this missing outcome. In practice, this calls for a sensible imputation model: relevant predictors should be included, the relationship used to construct the score should represent the important conditional structure, and the score should distinguish cases with materially different expected outcomes. A linear score need not make the outcome itself normally distributed; PMM returns observed donor values rather than drawing a new outcome from a normal residual model.
The donor pool must also be appropriate. Candidate donors need observed outcomes, and eligibility restrictions—such as justified strata—must be applied before ranking. The earlier material on restricting PMM donors within justified strata concerns that eligibility decision; this calculation takes the eligible pool as given. There must be enough eligible observed cases with scores near the missing case’s score for a plausible match. If there is little or no overlap, choosing the closest available cases does not make them genuinely comparable.
For valid inference from imputed data, PMM also relies on assumptions about the missingness and imputation models. In particular, the predictors and observed information must support a defensible missing-at-random assumption and adequate overlap for the cases being imputed. PMM does not establish those assumptions by matching scores. A numerically successful donor draw cannot repair an inappropriate model, unmeasured predictors related to missingness, or a region with no comparable observed cases.
Worked Five-Donor Calculation
Worked Example: Ranking Five Donors and Drawing One Outcome
Suppose a hypothetical dataset has five eligible cases with observed outcomes \(Y\) and one predictor \(X\). The fitted regression for this illustration is \( \widehat{Y}=10.3+1.9X \). The values are chosen to allow a full hand calculation; they are not real study data.
| Case | \(X\) | Observed \(Y\) |
|---|---|---|
| A | 1 | 12 |
| B | 2 | 15 |
| C | 3 | 14 |
| D | 4 | 20 |
| E | 5 | 19 |
First, the stated regression is consistent with the five observations if it is the ordinary least-squares fit with an intercept. The means are \(\bar{x}=3\) and \(\bar{y}=16\). The slope is
and the intercept is \(16-1.9(3)=10.3\). The fitted scores for the five observed cases are therefore \(12.2, 14.1, 16.0, 17.9,\) and \(19.8\), respectively.
Now suppose a sixth case has \(X=3.4\), but its outcome is missing. Its score is
Calculate each absolute score distance. For example, case C has distance \(|16.0-16.76|=0.76\), while case D has distance \(|17.9-16.76|=1.14\).
| Case | Score | Distance from 16.76 | Rank |
|---|---|---|---|
| A | 12.20 | 4.56 | 5 |
| B | 14.10 | 2.66 | 3 |
| C | 16.00 | 0.76 | 1 |
| D | 17.90 | 1.14 | 2 |
| E | 19.80 | 3.04 | 4 |
For a donor-pool size of \(k=3\), the pool is C, D, and B, in that order of closeness. If the draw selects each pool member with equal probability, each has probability \(1/3\) of supplying the imputation. Suppose the random draw selects D. The missing outcome is then imputed as D’s observed value, \(Y=20\), not as the missing case’s score of 16.76. The value 20 is one possible stochastic imputation; a different valid draw could give 14 or 15.
Why the Donor Draw Is Stochastic
Once the pool is fixed, equal-probability random selection gives each of its \(k\) observed outcomes the same chance of being used. Consequently, the conditional mean of the imputation draw, given that pool, is the average of the pool outcomes. In the five-donor example, the pool outcomes are 14, 20, and 15, so
This expectation is not itself the imputed value in a single draw. The realized value is one of the observed outcomes. Random selection allows repeated imputations to reflect uncertainty about which nearby observed case is an appropriate donor, rather than repeatedly assigning the same deterministic value.
The ranking step has a simple geometric justification: under the specified score, absolute distance measures how close a donor’s model-based expected outcome is to the missing case’s model-based expected outcome. Selecting the \(k\) smallest distances therefore selects the closest cases on that one-dimensional prediction scale. It does not prove that they are close on every predictor, nor that their outcomes are interchangeable. Those are model and context questions, not consequences of the distance calculation.
Worked Example: How Pool Size Changes the Draw
Use the same hypothetical scores and missing-case score \(16.76\), but set \(k=2\). The two smallest distances remain C at \(0.76\) and D at \(1.14\), so the pool outcomes are 14 and 20. An equal-probability draw gives each outcome probability \(1/2\).
The possible imputations are thus 14 and 20, and the expected value of the draw is
With \(k=3\), the pool additionally includes B, whose outcome is 15, and the expected draw becomes \(49/3\approx16.33\). Increasing \(k\) changes both the set of possible donors and the distribution of possible imputations. It may make the draw less dependent on one very close case, but it can also admit cases with less similar scores. The choice of \(k\) is therefore a modeling decision, not a cosmetic setting.
Worked Example: A Tie at the Donor-Pool Boundary
Consider five hypothetical eligible donors with scores \(8, 10, 12, 14,\) and \(16\), and corresponding observed outcomes \(40, 50, 70, 90,\) and \(100\). The missing case has score 11. The distances are \(3, 1, 1, 3,\) and \(5\). If \(k=3\), the two closest cases are unambiguously the donors with scores 10 and 12. The third position is tied between the cases with scores 8 and 14.
Suppose the stated tie rule selects either boundary-tied donor with probability \(1/2\). Then the pool always contains outcomes 50 and 70, and contains outcome 40 or 90 with equal probability. Conditional on a particular formed pool, each of its three donors is selected with probability \(1/3\). Averaging over the random tie resolution, the donor probabilities are \(1/3\) for outcomes 50 and 70, and \(1/6\) each for outcomes 40 and 90. The expected imputation is
A different documented tie rule could produce a different donor distribution. The important point is to handle ties explicitly and consistently; silently choosing whichever tied case appears first in a data table can make the result depend on arbitrary row order.
Checking the Calculation and Recognizing Failure
Confirm that the same fitted coefficients and predictor coding are used for observed and missing cases. Check that no required predictor is itself unavailable without an appropriate treatment.
Include only cases with observed outcomes that are valid donors for the missing case. Apply any justified restrictions before calculating the nearest eligible scores.
Recalculate the absolute differences and confirm that the selected pool contains the smallest eligible distances. Record how ties are resolved.
Every selected donor must have an observed outcome, and the imputation must equal that outcome. A prediction score or average of donor outcomes is not a realized donor draw.
Warning signs include a missing-case score far outside the range of eligible donor scores, a pool whose distances are all large, or a pool composed of cases that differ importantly on predictors despite similar scores. A small donor pool can also lead to frequent reuse of a few observed outcomes; a larger pool may reduce that concentration while admitting less similar matches. These checks do not identify one universally correct pool size. They reveal when the chosen ranking and donor set may be doing poor work.
A concise comparison helps clarify what PMM does and does not do. The alternatives below differ in how they produce an imputed value; none removes the need for a defensible imputation model and appropriate missing-data assumptions.
| Approach | What supplies the imputed value? | Key distinction |
|---|---|---|
| PMM | An observed outcome from a nearby-score donor | Uses regression scores for matching and a stochastic donor draw for the value |
| Deterministic nearest-donor assignment | The outcome of a fixed closest donor | Can repeatedly give the same case the same value, omitting donor-selection variation |
| Direct regression-based imputation | A value generated from the outcome model | Does not require the imputed value to equal an observed outcome |
Common Mistakes and Limitations
- Using the prediction as the imputation. The score 16.76 in the main example locates the missing case; it is not the PMM imputation. The imputed value must be the observed outcome of the selected donor.
- Ranking by outcome distance. The missing outcome is unknown, so its distance from observed outcomes cannot be calculated. Rank donors by the specified prediction-score distance instead.
- Ignoring eligibility or ties. A nearby case is not automatically an eligible donor, and a tie at the pool boundary needs an explicit rule.
- Treating the nearest available donor as a good match. The nearest score may still be far away. Ranking always identifies a nearest case when candidates exist; it does not establish adequate overlap.
- Assuming donor values remove all model dependence. The score model determines who can donate. If it ranks cases poorly, using observed outcomes does not fix that failure.
PMM’s attraction is that a single imputed value comes from the observed outcome distribution and therefore remains within observed support. Its limitation is that this support may not represent plausible values for a poorly matched case, especially in sparse regions or when the candidate set is restricted. The hand calculation should therefore be read as a transparent audit of one donor draw, not as evidence by itself that the chosen model, assumptions, or donor pool are adequate.
Check Your Understanding
Answer each question using the definitions and calculations in this tutorial. Short solutions follow the questions.
- A missing case has score 9.5. Three eligible donors have scores 8, 10, and 13, with observed outcomes 6, 11, and 15. For \(k=2\), identify the donor pool and the possible imputed values.
- In the five-donor example, why is 16.76 not a valid realized PMM donor imputation?
- Two eligible donors tie for the last place in a pool. Name one defensible way to handle the tie and one problem with choosing by row order without documenting the rule.
- All eligible donor scores lie between 10 and 12, while a missing case has score 25. What does selecting the nearest donors establish, and what does it fail to establish?
Solutions. (1) The distances are \(|8-9.5|=1.5\), \(|10-9.5|=0.5\), and \(|13-9.5|=3.5\). The pool is the donors with scores 10 and 8, so the possible imputations are their observed outcomes, 11 and 6. (2) The value 16.76 is the missing case’s prediction score, not an observed outcome from a donor; PMM draws an observed donor value. (3) One can randomly select among tied boundary candidates using a documented rule. Choosing by row order can make donor selection depend on arbitrary data ordering. (4) Ranking establishes which eligible scores are closest to 25, but it does not establish that any is a good match or that adequate overlap exists.