Propagating Imputation-Model Uncertainty at Fixed Delta

Why Parameter Uncertainty Still Matters at Fixed Delta

A delta adjustment specifies how the distribution of missing outcomes departs from a reference imputation model. Fixing delta means holding that sensitivity parameter at a chosen value during an analysis. It does not mean that the parameters of the reference imputation model should be treated as known. If those parameters are uncertain, their uncertainty should generally be carried into the completed data sets and the final inference.

For example, a reference model might predict a missing clinical measurement from treatment group and baseline measurements. Even if the analyst fixes the delta adjustment, the fitted regression coefficients are estimated from finite observed data. Different proper parameter draws can therefore give different predictions for missing outcomes. Predictive draws also vary because individual outcomes are not exactly their conditional means. Multiple imputation should retain these sources of variation rather than repeatedly imputing from one set of fitted coefficients.

For an additive adjustment, use the convention that a positive delta adds that amount to the reference-model imputation for a missing outcome. Thus, if \(Y_{\mathrm{mis}}^{(m,0)}\) is a reference-model draw in imputation \(m\), the adjusted draw is \(Y_{\mathrm{mis}}^{(m,\delta)}=Y_{\mathrm{mis}}^{(m,0)}+\delta\). This tutorial concerns uncertainty when delta is fixed; it does not assign a probability distribution to delta or estimate delta from the observed outcomes.

What Is Drawn, and What Is Held Fixed?

Let \(\theta\) denote the unknown parameters of the reference imputation model, and let \(X_{\mathrm{mis}}\) denote the covariates for records with missing outcomes. In a proper Bayesian implementation, each imputation draws \(\theta^{(m)}\) from its posterior distribution given the observed data, then draws missing outcomes from the predictive distribution conditional on \(\theta^{(m)}\) and the covariates. A valid frequentist proper-imputation procedure can use an appropriate sampling or bootstrap approximation to parameter uncertainty instead. The key is that imputations must reflect uncertainty in the fitted model, not just uncertainty in individual outcomes conditional on one fixed fitted model.

For an additive delta adjustment, the operation is applied to the predictive draws after they are generated from the reference model. If the imputation-model draw gives a conditional mean \(\mu^{(m)}(X_i)\), and the predictive residual draw is \(\varepsilon_i^{(m)}\), then the adjusted imputation is

The model parameters vary across \(m\), the predictive residuals vary across \(m\) and records, and delta stays the same. Parameter variation and residual variation are not interchangeable: the former reflects uncertainty about the fitted conditional distribution, while the latter reflects outcome variation around that distribution.

This procedure relies on a suitable reference imputation model, proper parameter and predictive draws, and a delta adjustment that is coherent with the outcome scale and its constraints. It also relies on the imputation and analysis procedures being sufficiently compatible for the intended inference. The fixed delta is a sensitivity assumption, not something the observed data can establish. As in the earlier material on checking the reference model, poor prediction or implausible adjusted values remain problems even if uncertainty is propagated correctly.

Why Parameter Draws Affect Between-Imputation Variation

The role of parameter uncertainty is visible for a simple target: the completed-data mean. Suppose \(n\) outcomes are to be included, \(n_{\mathrm{obs}}\) are observed, and \(n_{\mathrm{mis}}\) are missing, so \(n=n_{\mathrm{obs}}+n_{\mathrm{mis}}\). In a simple model with a common conditional mean \(\theta\), each missing value has the form \(\theta+\varepsilon_i+\delta\). The completed-data mean in imputation \(m\) is then

The fixed observed-data term and fixed delta term do not vary between imputations. The contribution of the parameter draw to the completed-data mean is \((n_{\mathrm{mis}}/n)\theta^{(m)}\). If the predictive residual term is set aside just to isolate this contribution, then

This establishes why varying the model parameters matters: their uncertainty feeds into uncertainty in the completed-data estimate, with greater influence when a larger fraction of outcomes is missing. In a real imputation, predictive residual variation also contributes, and the two sources may interact in more complicated models. Across imputations, variation in the analysis estimates reflects the combined between-imputation variation; it is not, in general, a separately identified estimate of parameter uncertainty alone.

Pooling Estimates at a Fixed Delta

Suppose \(M\) completed data sets are analyzed using the same analysis model. Let \(Q_m\) be the estimate of the target parameter from completed data set \(m\), and let \(U_m\) be its estimated sampling variance. Rubin’s pooling quantities are the average estimate \(\bar{Q}\), the average within-imputation variance \(\bar{U}\), the between-imputation variance \(B\), and the total variance \(T\):

The within-imputation variance \(\bar{U}\) represents the estimated sampling variance within completed data sets. The between-imputation variance \(B\) measures how much the estimates vary across imputations. When imputations are proper, that between-imputation variation carries uncertainty due to missing outcomes, including uncertainty arising from imputation-model parameter draws and predictive draws. The factor \(1+1/M\) accounts for the finite number of imputations in the total-variance estimate. A common large-complete-data-degrees-of-freedom approximation for scalar inference uses

This degrees-of-freedom formula is an approximation; settings with finite complete-data degrees of freedom can require a small-sample adjustment. If \(B=0\), the displayed expression is not used by direct division, and inference requires the appropriate limiting or software-handled case. Increasing \(M\) can reduce Monte Carlo noise in pooled quantities, but it cannot repair an improper imputation model.

What the Pooled Uncertainty Does—and Does Not—Represent

At a fixed delta, \(\bar{Q}\), \(B\), and \(T\) describe inference conditional on the chosen sensitivity assumption and the imputation and analysis procedures. Rubin’s rules do not add uncertainty about which delta value is scientifically appropriate. A fixed-delta confidence interval therefore does not cover uncertainty across alternative delta values unless that uncertainty is separately incorporated through a specified model or analysis.

For a simple additive shift applied to every missing outcome, with the same missing records in every imputation, the completed-data mean changes by the constant \(n_{\mathrm{mis}}\delta/n\). If the same delta is added in every imputation, it shifts every \(Q_m\) for that mean by the same amount; subtracting \(\bar{Q}\) removes that common shift, so \(B\) is unchanged. This special result should not be generalized automatically. A nonlinear estimand, a bounded outcome transformation, or an analysis whose variance depends on the shifted values can change \(U_m\) or the variation in \(Q_m\) when delta changes.

A useful way to separate the questions is: first, conditional on the chosen delta, did the imputation procedure represent uncertainty in the reference model and outcomes? Second, across plausible delta values, how sensitive are the substantive conclusions to the untestable departure assumption? The first question is answered within the multiple-imputation analysis at each fixed delta; the second requires comparing analyses under different sensitivity assumptions.

Diagnostics, Alternatives, and Common Pitfalls

Several checks help identify whether uncertainty is being propagated as intended:

• Inspect parameter draws. Confirm that model parameters vary across imputations as expected. Identical parameter estimates in every imputation can indicate plug-in imputation, although some parameters may legitimately have little variability.
• Check imputed values and predictions. Compare their ranges and distributions with the outcome scale, covariate patterns, and the chosen delta. Implausible values may indicate a reference-model or adjustment problem rather than a pooling problem.
• Check Monte Carlo stability. Repeat the analysis with more imputations if pooled estimates or standard errors appear unstable. More imputations improve numerical stability, not the validity of the reference model or delta assumption.
• Compare the correct quantities. \(B\) reflects between-imputation variation in the analysis estimates; it is not a direct diagnostic of the variance of one regression coefficient in the imputation model.

A plug-in alternative estimates the reference-model parameters once, then imputes missing outcomes repeatedly conditional on those fixed estimates. That approach can preserve predictive residual variation but omits parameter-draw variation, and can consequently understate uncertainty. Drawing model parameters but using conditional means without predictive residual draws also omits outcome variation. Conversely, drawing parameters and residuals but then adding a random delta answers a different question: it no longer conditions on delta being fixed.

Correct propagation of uncertainty cannot compensate for a misspecified reference model, incompatible imputation and analysis models, too few imputations for stable computation, or a clinically implausible delta. Nor does it make the missing-outcome distribution identifiable from observed data alone. The interpretation remains conditional on the assumptions used to generate the missing values.