PMM Compared With Parametric Regression Imputation

Two Ways to Generate a Missing Value

For a continuous variable with missing values, predictive mean matching (PMM) and normal linear regression imputation can use the same predictors but generate values in different ways. PMM uses a fitted model to find observed cases with similar predicted values, then takes an observed outcome from a selected donor. Normal regression imputation predicts the missing outcome from a fitted regression and adds a random residual drawn from a specified distribution. One method selects from observed values; the other generates from a model-based distribution.

The distinction matters most when the missing case lies near the edge of the observed data, when observed outcomes are skewed or bounded, or when the regression model’s distributional assumptions are doubtful. Neither approach is automatically reliable merely because it produces a value. Credibility depends on the imputation model, the available information for the missing case, and whether comparable observed cases exist.

A deterministic regression prediction, such as substituting only the fitted conditional mean, is not the stochastic regression procedure described here. It omits residual variation and, if used repeatedly, assigns the same predicted value to cases with the same predictors. A proper multiple-imputation procedure incorporates uncertainty rather than treating the fitted mean as the observed outcome.

Models and Assumptions Behind the Comparison

Let \(Y\) be the variable with missing values and let \(X\) denote the predictors observed for a case. A linear predictive model estimates a conditional mean, often written \(\widehat{m}(X)=X^\mathsf{T}\widehat{\beta}\). In a stochastic regression draw, the imputed value has the form

where \(\beta^{*}\) and \(\sigma^{2*}\) are draws reflecting uncertainty about the regression parameters and residual variance. The displayed normal distribution describes the residual conditional on the predictors and parameters. The procedure relies on an adequately specified conditional mean and, for this version, an approximately normal residual distribution with the modeled variance structure. A basic linear model usually assumes a common residual variance; if variance changes substantially with predictors, that assumption is also questionable.

In PMM, a predictive score for the missing case and each observed case is calculated from a fitted model. Donors are selected by closeness of their scores, and an observed \(Y\) value is copied. Parameter draws may also be used in constructing scores. PMM avoids requiring the outcome itself to follow a normal distribution, but it still relies on a useful predictive model: cases close in score should be reasonable substitutes for one another with respect to the outcome distribution.

Both methods need observed cases that can inform the missing case. The imputation model should include predictors that are important for the outcome and for explaining missingness; its conditional relationships should be sufficiently well represented by the observed cases. Under a missing-at-random analysis, the observed predictors must provide the information needed to model the outcome among cases with missing values. Neither method repairs a lack of overlap: if no observed cases resemble a missing case, PMM must borrow a poor match and regression must extrapolate its fitted relationship.

A Shared Numerical Comparison

Why Donor Support and Extrapolation Differ

The support property of PMM follows directly from its donor rule. If the eligible donor outcomes are \(y_1,\ldots,y_r\), then every imputed value must be one of those values:

This is useful when a normal model might generate an implausible value outside the range of observed outcomes. But “within observed support” has a limited meaning: it guarantees that PMM copies an observed value, not that the donor is truly comparable or that the full population support has been observed. A sample may fail to contain rare but valid values, and a poorly matched donor may be inappropriate even though its value is observed.

By contrast, if a regression draw has conditional distribution \(N(m,\sigma^2)\) with \(\sigma>0\), its distribution has positive probability beyond any finite observed maximum \(y_{\max}\):

where \(\Phi\) is the standard normal cumulative distribution function. That can be sensible if the outcome is genuinely unbounded and the normal model is adequate. It is a problem if the outcome has a strict bound, such as a score that cannot exceed a fixed maximum, unless the model or draw is adapted to respect that bound.

A further distinction is the range of predictors. PMM’s copied outcome stays among donor values even when the missing case’s predictor values lie beyond those of observed cases. That does not mean PMM has solved extrapolation: the nearest donor may be far away on the prediction scale and represent a very different case. Regression can also extrapolate by extending its fitted mean function beyond the observed predictor range; its generated value can then move still further because of the residual draw.

Conditional-Distribution Assumptions in Practice

PMM is often useful when a normal residual model is implausible, but it is not distribution-free in every relevant sense. Its donor pool is a local empirical approximation to the conditional outcome distribution. For that approximation to be useful, the score must rank cases in a way that brings together cases with comparable outcomes, and there must be enough suitable donors near the missing case. If distinct groups receive similar scores but have different outcome distributions, matching by score alone can select a misleading donor.

Normal regression imputation makes a stronger shape assumption for the conditional residual distribution. When the conditional distribution is close to normal and the linear mean and variance structure are adequate, it can generate a range of plausible values, including values not observed in the finite sample. If conditional outcomes are skewed, heavy-tailed, multimodal, or bounded, a normal draw may underrepresent tails, create impossible values, or spread probability in the wrong places. PMM may preserve some features of observed values, including skewness or unusual values, but only when relevant cases are available and the donor matching is suitable.

Comparison, Diagnostics, and Limitations

When reviewing either approach, check whether donor distances are small enough to make the donors credible, whether there are enough candidate donors, and whether important predictors or strata are missing from the score model. For regression, inspect whether the fitted mean form and residual spread are plausible and whether generated values respect the outcome’s scientific limits. In both methods, compare imputed and observed distributions in relevant predictor groups; a close overall mean alone does not establish that conditional imputations are credible.

The key limitation of PMM is that observed support is not a substitute for conditional comparability. The key limitation of normal regression imputation is that a convenient parametric distribution can misrepresent the conditional outcome. Both approaches also depend on observed data covering the cases that need imputation and on a suitable set of predictors. These limitations are reasons to examine the model and imputations, not rules that one method should always be preferred.