Predictive Scores From Linear Imputation Models

Why PMM Needs Predictive Scores

Predictive mean matching (PMM) uses an imputation model to put observed cases and cases with missing outcomes on a common predictive scale. The model’s fitted value for a case is its predictive score. A missing-outcome case can then be compared with observed-outcome cases using their scores, so that cases with similar model-predicted outcomes can be considered as donor candidates.

A score is not the outcome itself. For an observed case, the outcome may be above or below its fitted value; for a missing case, the fitted value is not treated as the imputed value. The score summarizes how the specified model’s covariates predict the outcome. PMM uses that summary to identify candidate donors, whose observed outcomes remain the source of imputations.

The score has the same units as \(Y\), because it is a predicted value of \(Y\). The coefficients have units that depend on their predictors: for example, a slope for age is measured in outcome units per year. The score is determined by the fitted coefficients and the case’s covariates; the observed outcome does not enter its calculation once the model has been fitted.

Model and Assumptions

For a linear imputation model, write the conditional mean as

Here \(X_i\) includes the intercept if the model has one. The fitted model estimates \(\beta\) from cases whose outcomes are observed. Under ordinary least squares (OLS), if \(X_{\mathrm{obs}}\) is the design matrix for those cases and \(y_{\mathrm{obs}}\) their outcome vector, then, when \(X_{\mathrm{obs}}^\top X_{\mathrm{obs}}\) is invertible,

The score for any case is then calculated using this same \(\hat{\beta}\). The key structural assumption for interpreting the score as a useful prediction is that the chosen linear predictor represents the conditional mean adequately over the covariate patterns where scores will be used. This need not mean every predictor has a straight-line relationship in its raw form: transformations, indicator variables, and interactions can be included in \(X_i\) when justified.

A score can be computed even if the model is not a good description of the conditional mean. But then its ranking may be unhelpful: cases that should be similar in expected outcome may receive dissimilar scores, or cases with different expected outcomes may receive similar scores. The model also requires usable predictor values for each case being scored. In a chained-equations iteration, those values may include values imputed in an earlier step of the iteration.

For the score calculation itself, normal errors are not required. OLS gives the displayed coefficient estimate when the design matrix has full column rank; normality is not needed to multiply a design row by that estimate. A correctly specified conditional mean supports interpreting the fitted value as an estimate of \(E(Y_i\mid X_i)\). Broader assumptions about missingness and imputation-model adequacy matter to the validity of an analysis, but they do not change the arithmetic definition of the score.

Calculating Scores and Reading Their Meaning

The calculation has two parts: first fit the model from observed outcomes, then multiply each case’s design row by the fitted coefficient vector. For an observed case, this produces a fitted value alongside its actual outcome. For a missing case, it produces a prediction based on covariates alone. Candidate-donor comparisons use those fitted values, not the observed cases’ residuals or the missing case’s unknown outcome.

Fitted Values From an Estimated Model

In practice, the coefficients are usually estimated from observed outcomes rather than supplied in advance. The following small calculation shows how the fitted model creates scores. The example is deliberately simple so the OLS arithmetic can be checked directly.

How Model Structure Changes Scores

The design row \(X_i\) is more than a list of raw measurements. It encodes the model that was chosen. A categorical predictor must be represented using an appropriate indicator coding; nonlinear relationships may require transformations or basis terms; and an interaction requires a product term. Omitting a term that matters can distort scores even if the calculation is performed correctly.

This example also illustrates an important property of score construction: different covariate patterns can produce the same score. A score compresses the design row into one number, so it does not preserve every distinction among covariates. That compression is intentional for this matching scale, but it makes model specification important. If a predictor is essential for finding scientifically comparable donors, a score model that ignores it may treat unlike cases as close.

A Short Justification and Practical Checks

The OLS formula explains why observed and missing cases are scored with the same operation. Let \(X_{\mathrm{all}}\) contain the design rows for all cases that can be scored. Once \(\hat{\beta}\) has been estimated on observed outcomes, the vector of scores is \(X_{\mathrm{all}}\hat{\beta}\). For a particular row \(X_i\), the corresponding entry is exactly \(X_i^\top\hat{\beta}\). The outcome values for observed cases were used to estimate \(\hat{\beta}\), but an individual case’s outcome is not substituted into its own score.

When the OLS design matrix has full column rank, the fitted values for observed cases are the projection of their outcome vector onto the columns of the design matrix. The residual vector is \(y_{\mathrm{obs}}-X_{\mathrm{obs}}\hat{\beta}\), which is distinct from the fitted-value vector \(X_{\mathrm{obs}}\hat{\beta}\). This distinction is central to PMM: score closeness compares fitted values, whereas the observed outcome is retained for possible donation.

Useful checks include verifying that the same coefficient vector and coding are used for observed and missing cases, confirming that all required predictor values are available, and inspecting whether the resulting scores are plausible across important covariate groups. Strongly unusual scores, abrupt gaps, or many identical scores can signal coding errors, sparse design patterns, or a model that cannot distinguish relevant cases. These checks do not establish that the imputation model is correct, but they can reveal errors before score comparisons are used.

Alternatives, Limitations, and Common Pitfalls

A linear fitted-value score is one way to place cases on a predictive scale. Other imputation approaches may use a different outcome model or may directly draw a value from an assumed conditional distribution. PMM’s donor-based approach differs from deterministic regression prediction: rather than assigning the fitted conditional mean as the missing value, PMM uses the score to locate observed cases and can use an observed outcome as the imputation. This preserves the observed outcome’s scale and values, but it does not guarantee a suitable donor if observed cases with relevant covariate patterns are scarce.

The score is only as informative as the model that creates it. A linear model can be inadequate when the conditional mean has important curvature, interactions, or subgroup differences not represented in the design. Extreme extrapolation is also a concern: a missing case may receive a score beyond the range of scores for observed cases. A score cannot manufacture observed donors in that region. Conversely, different cases may share a score because the model maps several covariate patterns to the same fitted value.

A common mistake is to say, “The missing case’s predicted score is 49, so 49 is its imputed value.” That confuses the matching score with the donated outcome. The correction is: 49 is the model’s fitted prediction used to compare cases; the imputed value, when PMM proceeds, is an observed outcome from a selected donor. Another mistake is to score observed cases using their actual outcomes and missing cases using fitted values. That compares unlike quantities and defeats the purpose of a common predictive scale. Both groups must receive fitted-value scores from the same model.