Why Draw Parameters Before Computing PMM Scores?
Predictive mean matching (PMM) uses a regression model to construct prediction scores, then uses those scores to find observed cases that can supply values for missing outcomes. The fitted regression is uncertain: another sample from the same process could produce different coefficients. If each imputation uses only one fixed coefficient estimate, that source of uncertainty does not enter its scores. Drawing regression parameters before calculating the scores allows the relative positions of recipients and observed cases to vary across imputations.
The distinction is important because a parameter draw can change the ranking of candidate donors, not merely shift every score by the same amount. That change is possible when cases differ across multiple predictors and the relative sizes of the regression coefficients are uncertain. The later step of selecting a donor from a ranked set is separate: this tutorial concerns how parameter uncertainty changes the scores and their rankings.
Consider an incomplete outcome \(Y\) and predictors collected in a design vector \(z\), which includes an intercept when the model has one. The regression score at coefficients \(\beta\) is \(z^\mathsf{T}\beta\). In a chained-equations update, the regression is fitted among cases with observed \(Y\), using the current values of the predictors. For a missing \(Y\), the score is calculated from that case’s predictors; observed cases also receive scores from their predictors. The score is a model-based quantity used for matching, not an imputed outcome itself.
A Standard Regression-Parameter Draw
One common construction uses the normal linear model for the observed outcomes. Let \(X_o\) be the design matrix for the \(n_o\) cases with observed outcomes, \(y_o\) their outcome vector, and \(p\) the number of regression coefficients, including the intercept if present. Assume
- the design matrix has full column rank and \(n_o>p\);
- the observed outcomes satisfy \(y_o=X_o\beta+\varepsilon\);
- the errors are independent and normally distributed with mean zero and common variance \(\sigma^2\); and
- the standard reference prior \(p(\beta,\sigma^2)\propto 1/\sigma^2\) is used for this illustrative posterior draw.
Under these assumptions, calculate \(\hat\beta=(X_o^\mathsf{T}X_o)^{-1}X_o^\mathsf{T}y_o\), the residual sum of squares \(SSE=(y_o-X_o\hat\beta)^\mathsf{T}(y_o-X_o\hat\beta)\), and \(s^2=SSE/(n_o-p)\). Draw \(\sigma^2\) from its posterior distribution and then draw \(\beta\) conditional on that variance:
Equivalently, draw \(U\sim\chi^2_{n_o-p}\), set \(\sigma^2=SSE/U\), and then draw \(\beta\) from the displayed conditional normal distribution. This is one standard Bayesian construction, not a claim that every PMM implementation uses precisely the same prior or variance-drawing convention. The essential point for this tutorial is that scores are computed from a random coefficient vector representing parameter uncertainty, rather than only from \(\hat\beta\).
Worked Scores and Their Uncertainty
Worked Example: A Coefficient Draw Changes the Scores
Suppose a hypothetical imputation model has an intercept and one binary predictor \(x\). Its design vector is \(z=(1,x)^\mathsf{T}\). For one parameter draw, let \(\beta=(1,2)^\mathsf{T}\). Calculate scores for a missing case with \(x=1\) and an observed case with \(x=0\).
The design vectors are \(z_m=(1,1)^\mathsf{T}\) and \(z_o=(1,0)^\mathsf{T}\). Their scores under this draw are
Thus the score difference is \(3-1=2\), in the outcome’s prediction units. Now suppose another imputation draws \(\beta=(1,4)^\mathsf{T}\). The scores become \(5\) and \(1\), so the difference is \(4\). The intercept contributes equally to both scores; it is the drawn slope that changes their separation. These calculations show how the score scale can vary across imputations even when the predictor values do not.
Worked Example: Score-Difference Uncertainty
For a second hypothetical calculation, suppose a coefficient draw is described by a normal distribution with mean \((2,1)^\mathsf{T}\) and covariance matrix \(\begin{pmatrix}0.25&0\\0&0.36\end{pmatrix}\). Compare cases with design vectors \(z_1=(1,1)^\mathsf{T}\) and \(z_2=(1,0)^\mathsf{T}\). This distribution is stipulated for the calculation; it need not be treated as a fitted clinical model.
The score difference is the linear contrast of the coefficients:
The contrast has mean \(c^\mathsf{T}(2,1)^\mathsf{T}=1\). Its variance is \(c^\mathsf{T}\operatorname{Cov}(\beta)c=0.36\), so its standard deviation is \(\sqrt{0.36}=0.600\) outcome units. In this example the intercept cancels exactly, and the uncertainty in the difference comes from the slope. This is uncertainty in the model-based score contrast; it is not the residual standard deviation of an individual outcome.
How Parameter Draws Change Rankings
A score ranking compares distances between predictions. Since those predictions are linear combinations of the coefficients, changing their relative values can change which observed case is closest to a missing case. This is especially clear with more than one predictor: different cases can be separated from the recipient in different predictor directions, and uncertainty about the coefficient vector changes the importance of those directions.
Worked Example: Rankings Reverse Across Imputations
Consider a hypothetical model with an intercept and two predictors, \(x_1\) and \(x_2\). A missing case has predictors \((0,0)\). Two observed cases, A and B, have predictors \((1,0)\) and \((0,1)\), respectively. For this illustration, compare absolute differences between their predicted scores and the missing case’s score. The intercept cancels from each difference.
Suppose four parameter draws give the following slope pairs \((\beta_1,\beta_2)\). For each draw, the score differences are \(d_A=|\beta_1|\) and \(d_B=|\beta_2|\).
| Draw | \((\beta_1,\beta_2)\) | \(d_A\) | \(d_B\) | Smaller difference |
|---|---|---|---|---|
| 1 | (2.0, 0.5) | 2.0 | 0.5 | B |
| 2 | (0.5, 2.0) | 0.5 | 2.0 | A |
| 3 | (1.5, 1.0) | 1.5 | 1.0 | B |
| 4 | (0.8, 1.2) | 0.8 | 1.2 | A |
For example, in draw 1 the missing case’s score is the intercept, while A’s score differs from it by \(2.0\) and B’s by \(0.5\), so B is closer. In draw 2 the differences are \(0.5\) and \(2.0\), so A is closer. In this set of four illustrative draws, each case ranks closer twice. These four hypothetical draws do not estimate a real probability; they demonstrate that coefficient uncertainty can reverse a ranking. Actual PMM uses its specified parameter-drawing procedure and its observed data.
Why This Matters in Multiple Imputation
Multiple imputation is intended to represent uncertainty about missing values, not merely to fill gaps repeatedly with near-identical predictions. A fixed-coefficient PMM implementation can still vary through later donor selection, but its regression-based scores do not reflect coefficient uncertainty. If that uncertainty is appreciable, a fixed plug-in fit can make the matching step too dependent on one estimated regression.
A common standard implementation draws the regression parameters, computes scores for the missing and observed cases, and proceeds to the separate donor-selection step. It is useful to keep the sources of randomness distinct:
- Parameter uncertainty: a new regression coefficient vector changes the model scores and potentially their ranking.
- Donor-selection randomness: a later random choice among eligible candidates can vary even when the score ranking is fixed.
Drawing coefficients does not guarantee good imputations. The regression model still needs a defensible set of predictors and an adequate form for the conditional relationship. The standard normal-model draw also relies on its stated assumptions. Strong departures from linearity, severe heteroskedasticity, dependence not represented in the model, or poorly identified coefficients can make the drawn scores unreliable. Parameter uncertainty represents uncertainty under the model; it does not automatically correct model misspecification or make missingness ignorable.
Diagnostics, Alternatives, and Common Pitfalls
A practical check is to inspect coefficient draws and the resulting score behavior over repeated imputations. Coefficients that barely vary despite sparse information, extreme draws caused by a nearly rank-deficient design, or scores that are implausibly sensitive to small coefficient changes warrant investigation. Also check that each update uses the intended observed-outcome cases and that its design matrix is estimable. In chained equations, predictor values for incomplete cases may themselves change between iterations, so changes in scores can reflect both updated predictors and new parameter draws.
| Approach | What varies across imputations? | Main implication |
|---|---|---|
| PMM with a parameter draw | Regression parameters, and in some implementations residual variance; donor selection occurs separately | Model-score uncertainty can alter candidate rankings |
| PMM with fixed fitted coefficients | Potentially only the later donor choice | Score rankings omit coefficient uncertainty |
| Parametric regression imputation | Depends on its specified parameter and residual draws | It draws outcomes from a model rather than obtaining the missing value from a matched observed case |
These approaches are not interchangeable in every respect. In particular, using a parameter draw in PMM does not make the final imputed value a direct draw from the linear regression’s outcome distribution: the value still comes from an observed donor. Conversely, a fixed-coefficient score can still produce different donors if the later selection is randomized, but that randomness is not a substitute for parameter uncertainty.
- Pitfall: using only the fitted coefficients. “The regression has been estimated, so its predictions are known” ignores sampling uncertainty. Use the required parameter-drawing procedure when the implementation calls for it.
- Pitfall: drawing coefficients independently for each case. This can make comparisons reflect unrelated model draws. Use a shared draw to score the recipient and the observed cases for that imputation.
- Pitfall: assuming every ranking must change. Some coefficient changes shift all scores similarly or leave the relevant ordering intact. Rankings change when the changes alter relative score distances; they need not change in every draw.
- Pitfall: treating more parameter variation as proof of validity. Wide variation can signal weak information or an unstable model. Check estimability and model adequacy rather than interpreting variability alone as a virtue.
Use cases with observed outcome values and the current predictor values, checking that the design is estimable.
Use the implementation’s specified procedure to represent uncertainty in the regression coefficients.
Apply the same coefficient draw to the missing case and observed cases being compared.
Check for unstable coefficients or implausible sensitivity before interpreting changes in donor rankings.
Check Your Understanding
Answer each question, then compare with the short solutions below.
- Why does a PMM procedure draw regression parameters before computing prediction scores?
- For \(z_1=(1,1)^\mathsf{T}\), \(z_2=(1,0)^\mathsf{T}\), and \(\beta=(3,2)^\mathsf{T}\), calculate both scores and their difference.
- In the two-predictor ranking example, why can changing the coefficients reverse which observed case has the smaller score difference?
- Why should one coefficient draw be shared across the cases being scored within an imputation?
- Does parameter uncertainty alone guarantee valid PMM imputations? State one reason it may not.
Solutions. (1) A draw carries regression-parameter uncertainty into the scores, so relative score distances and rankings can vary across imputations. (2) The scores are \(z_1^\mathsf{T}\beta=3+2=5\) and \(z_2^\mathsf{T}\beta=3\), so their difference is \(5-3=2\) outcome units. (3) The two observed cases differ from the recipient along different predictors; changing the relative coefficient sizes changes the relative contributions of those directions. (4) A shared draw makes all scores comparable under one coherent regression model. (5) No. For example, a misspecified conditional model can produce unreliable scores despite correctly drawing parameters under that model.