Why Match Predictions but Impute Observed Outcomes?
A regression model can predict the expected value of an outcome for a case with a missing value. But replacing every missing value with that prediction is not the same as drawing a plausible value of the outcome: the prediction is typically an average, and may not even be a possible observed value. Predictive mean matching (PMM) uses regression predictions to find comparable observed cases, then imputes by drawing one of those cases’ actual outcomes.
This distinction is the central idea of PMM. The regression prediction guides the search for a donor; it is not itself the imputed value. PMM is used within an imputation model for a variable that has observed and missing values, with predictors available for both groups.
In a basic formulation, the score for case \(k\) is \(\hat m_k=x_k^\mathsf{T}\hat\beta\), where \(x_k\) is that case’s predictor vector and \(\hat\beta\) is estimated from cases with observed outcomes. A simple matching distance is \(d_{ik}=|\hat m_i-\hat m_k|\). Donors are observed cases with small distances. Implementations can differ in how scores and donor sets are constructed; the defining feature is the draw of an observed outcome after matching on a fitted score.
Conditions That Make the Donor Idea Plausible
PMM does not make missingness assumptions disappear. For inference under a missing-at-random assumption, the observedness of \(Y\) must be conditionally independent of its unobserved value given the variables used to explain missingness. In notation, if \(R\) indicates whether \(Y\) is observed, this assumption is \(P(R=1\mid Y,X)=P(R=1\mid X)\). The predictors should include relevant information about both the outcome and its missingness. A positive chance of observing \(Y\) across the predictor patterns of interest is also needed; otherwise, there may be no comparable observed cases to donate.
The working regression model must provide useful scores: cases close in score should tend to have outcome distributions that are relevant to one another. This does not require that the regression perfectly predict each individual outcome. It does require enough information and overlap for observed cases to serve as plausible donors for missing cases. Matching on a score cannot repair omitted predictors, severe lack of overlap, or an inappropriate relationship between predictors and outcome.
A PMM donor is an observed case, not a synthetic value from the regression equation. Consequently, the imputed value is one of the observed outcomes. That property can be useful, but it does not establish that the value is appropriate for the recipient case, or that the donor pool represents the recipient’s conditional outcome distribution. Model uncertainty and donor selection also matter for multiple imputation; a single deterministic fit and a fixed donor would not represent all relevant uncertainty.
Worked Example: A Continuous Outcome and Two Nearby Donors
Suppose a hypothetical data set has a continuous outcome, annual clinic visits, and a fitted regression model using age and a health-status measure. For one case with a missing outcome, the fitted score is 53.0 visits. Among the observed cases, four candidate scores and outcomes are:
| Observed case | Fitted score (visits) | Observed outcome (visits) |
|---|---|---|
| A | 48 | 44 |
| B | 51 | 59 |
| C | 56 | 52 |
| D | 62 | 67 |
Assume, for illustration, that the two closest observed scores form the donor set. The missing case’s score is 53.0 visits. The absolute distances are \( |53-48|=5\), \( |53-51|=2\), \( |53-56|=3\), and \( |53-62|=9\) visits. Cases B and C are therefore the two closest donors.
A random draw from these two donors gives either B’s observed outcome, 59 visits, or C’s observed outcome, 52 visits. If the draw selects C, the imputation is 52 visits. It is not 53 visits: 53 is the fitted score used for matching. This example assumes that the regression scores meaningfully order comparable cases and that the observed donors are relevant to the missing case; the arithmetic alone cannot establish those conditions.
Why Not Substitute the Fitted Value?
A fitted conditional mean summarizes the center of an outcome distribution. It does not generally reproduce the spread of individual outcomes around that center. Directly inserting a fitted value for every missing observation can make the completed data set too concentrated and can understate uncertainty in later analyses. PMM instead uses the fitted value to locate observed cases and draws an outcome that includes observed case-to-case variation.
A simple model illustrates the distinction. Suppose \(Y=m(X)+\epsilon\), with \(E(\epsilon\mid X)=0\) and \(\operatorname{Var}(\epsilon\mid X)=\sigma^2\). If \(m(X)\) were known, direct mean substitution would always insert \(m(X)\), with no variation across repeated substitutions at the same \(X\). Actual outcomes still vary around that mean:
A donor draw can retain some of the outcome variation because the imputed value is an observed \(Y_j\), not just a fitted mean. This is a justification for the donor-based idea, not a claim that any donor draw has exactly the correct conditional distribution. That depends on whether the donor pool is suitable and on the imputation procedure’s treatment of uncertainty.
Worked Example: A Count Outcome With a Fractional Prediction
In a hypothetical count-variable imputation, a fitted count model gives a missing case an expected count score of 1.95 visits. Two close observed scores are 1.8 and 2.1 visits, and their corresponding observed outcomes are 1 and 4 visits. The absolute score distances are:
Suppose these two cases are the donor set and the procedure draws each with probability \(1/2\). The imputation is 1 visit with probability \(1/2\) or 4 visits with probability \(1/2\). The expected value across these possible donor draws is \((1+4)/2=2.5\) visits, but 2.5 is not the imputed value in either draw. Nor is the fitted score 1.95 the imputation. The donor’s observed count is used, so the result remains an integer count in this example.
The model score can be fractional even when the outcome is a count; that is not itself a problem because the score is used for matching. The example assumes the two observed cases are valid candidates, not that closeness of their scores proves they have the right outcome distributions for the missing case.
What a Donor Draw Does—and Does Not—Guarantee
When several suitable donors are available, random selection avoids always assigning the same donor outcome to every missing case with a similar score. For example, if two selected donors have outcomes 14 and 16 visits and are drawn with equal probability, the possible imputed values are 14 and 16; the mean of the donor outcomes is 15 visits. Repeated draws can therefore reflect some outcome variation that direct insertion of a mean would suppress.
However, an observed value is not automatically a good value for every recipient. A donor with a nearby score can still differ on important predictors, particularly when the score compresses information from many predictors into one number. If a missing case has no observed cases with comparable scores, the donor-based approach has limited ability to supply relevant information. It cannot create empirical evidence for predictor patterns absent from the observed cases.
PMM also does not guarantee a correctly specified imputation distribution merely because every imputation is an observed value. A donor pool that is too broad, unrepresentative, or poorly matched can give plausible-looking values while misrepresenting relationships in the data. Further choices about score construction, parameter uncertainty, and donor selection affect performance; they should be assessed rather than inferred from the name of the method.
Worked Example: Donor Outcomes Retain Variation
Consider a hypothetical continuous outcome measured in visits. For a missing case, the fitted score is 15.2 visits. Three observed cases have scores and outcomes \((12,10)\), \((15,17)\), and \((18,16)\), with both quantities measured in visits. Assume the two closest cases are used as donors.
The score distances are \( |15.2-12|=3.2\), \( |15.2-15|=0.2\), and \( |15.2-18|=2.8\) visits. The two nearest donors are thus the second and third cases, with observed outcomes 17 and 16 visits. If either donor is drawn with probability \(1/2\), the expected donor outcome is \((17+16)/2=16.5\) visits. The variance across these two possible donor outcomes is \(\{(17-16.5)^2+(16-16.5)^2\}/2=0.25\) visits squared.
The fitted score, 15.2 visits, is not substituted. Instead, one imputation is 17 or 16 visits, depending on the donor draw. The calculation shows how donor sampling can vary imputations; it does not show that the two donors are adequate. In particular, this small hypothetical pool would require substantive scrutiny before being treated as representative.
Diagnostics, Alternatives, and Common Mistakes
PMM should be checked as part of the imputation model, not accepted solely because its values look reasonable. Useful checks include comparing score distributions for missing cases and observed cases, examining whether missing cases have nearby observed scores, and checking whether selected donor outcomes are sensible in context. Compare observed and imputed distributions, including their ranges, concentrations, and unusual values. Also assess whether important relationships with predictors are retained in completed data. These checks can reveal a model that is extrapolating or a donor pool that does not represent the cases receiving imputations.
The main alternatives answer the missing-value problem differently. Direct regression prediction inserts a fitted mean or other predicted value; it is simple, but deterministic mean substitution omits residual outcome variation. Parametric stochastic regression imputation draws from a specified outcome model; it can represent a modeled conditional distribution, but depends on the adequacy of that distributional model. PMM uses a regression score to find observed cases and draws an observed outcome; it avoids requiring the imputed value to be a newly generated outcome from a parametric distribution, but it depends on available and suitable donors. None of these choices removes the need to justify predictors and missingness assumptions.
A common mistake is to say, “The regression predicts 15.2, so PMM imputes 15.2.” That is direct prediction, not the defining PMM step. The correction is: use 15.2 as a score, find observed cases with nearby scores, then draw one case’s observed outcome. Another mistake is to conclude that because the donor value was observed, it must be appropriate. Observed status alone says nothing about whether the case is a good match.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A missing case has a fitted score of 20. An observed donor has a score of 19.5 and an outcome of 24. What value is used if that donor is drawn?
- Why can direct substitution of a fitted conditional mean understate variability?
- A missing count has a fitted score of 2.4. Its selected donor has observed outcome 3. Is 2.4 or 3 the PMM imputation?
- Name one condition needed for observed cases to serve as useful donors for missing cases.
- Does drawing an observed outcome by itself establish that the imputation is valid? Explain briefly.
Solutions. 1. The imputation is 24, the donor’s observed outcome; 19.5 is its matching score. 2. A conditional mean contains no case-specific residual variation, whereas outcomes can vary around that mean. 3. The imputation is 3, the donor’s observed count. 4. For example, observed cases must have relevant, nearby scores, with adequate overlap for the missing case’s predictor pattern. 5. No. Donor outcomes can be inappropriate if matching is poor, the pool is unrepresentative, or key assumptions about missingness and predictors fail.