Estimands And Notation For Multiple-Imputation Pooling

Why Pooling Needs a Shared Target

Multiple imputation produces \(M\) completed datasets from the same incomplete data. An analysis is then applied to each completed dataset. The results can be combined only if the analyses address the same estimand: the same precisely defined population quantity, for the same population, under the same model and on the same measurement scale. Otherwise, the resulting numbers do not represent repeated estimates of one target.

For example, a treatment-effect coefficient adjusted for age is not automatically the same target as an unadjusted treatment-effect coefficient. Nor are a regression coefficient on the log-odds scale and its exponentiated odds ratio interchangeable as pooling inputs. A common estimand is not just a common variable name; it includes the quantity being estimated and the scale on which it is expressed.

The \(Q_i\) values will generally differ because the imputed values differ. The \(U_i\) values can also differ because the completed datasets differ. A variance is not a standard error: \(U_i\) is in squared units of \(Q_i\), whereas its square root is in the units of \(Q_i\). For instance, if \(Q_i\) is a mean difference in mmHg, \(U_i\) is in \(\text{mmHg}^2\).

The Pooling Notation

The symbols below label distinct summaries of the imputation-specific results. Their definitions are straightforward, but keeping their roles and units separate is essential.

These are definitions and the standard scalar total-variance expression, not a claim that every inferential detail has been settled by calculating \(T\). The derivation and small-sample adjustments to inference require additional results. Here, the key is to identify what each quantity summarizes: \(\bar U\) averages the estimated variances within completed datasets, while \(B\) summarizes how much the estimates vary across imputations. Both are measured in squared units of \(Q\); \(T\) is also in squared units, and \(\sqrt{T}\) is in the units of \(Q\).

Why the Between-Imputation Variance Has \(M-1\)

The denominator in \(B\) is the sample-variance denominator. Its role can be justified by treating the imputation-specific estimates, under ideal independent draws from the relevant imputation distribution, as draws with a common variance. The sample mean is estimated from those same draws, which costs one degree of freedom in the deviations around the mean.

This equality gives the deviations one linear constraint, so they have \(M-1\) degrees of freedom. It does not make the deviations statistically independent. For example, with \(M=2\), the two deviations are exact negatives of each other. Under independent draws with finite variance, dividing the sum of squared deviations by \(M-1\) makes the sample variance an unbiased estimate of the variance of a draw. This explains the conventional form of \(B\); it does not mean the imputations themselves are always independent in practice. Chained-equation draws may retain dependence if chains have not mixed well, so convergence and adequate Monte Carlo sampling matter.

The \(U_i\) values have a different role. Each is obtained from the complete-data analysis applied to one completed dataset, using that analysis’s variance estimator. These values represent estimated sampling variance conditional on the completed data; they are not the spread of the \(Q_i\) across imputations. The distinction prevents treating uncertainty within a completed-data analysis and variation between imputations as if they were the same source.

Check the Estimand and Scale Before Calculating

The formulas do not repair incompatible analysis results. Before pooling, verify that each completed-data analysis estimates the same target, uses the same parameterization, and returns the same kind of variance estimate. A useful check is dimensional: every \(Q_i\) must have the same units, and every \(U_i\) must have the corresponding squared units.

Assumptions, Diagnostics, and Alternatives

The notation is simple; the validity of the resulting inference depends on the analyses and imputations behind it. At minimum, the missing-data and imputation models must be defensible for the intended target, and the completed-data analysis must estimate that target appropriately. Rubin’s usual variance interpretation also relies on proper representation of imputation uncertainty and sufficiently well-behaved Monte Carlo draws. The notation alone cannot guarantee these conditions.

Practical diagnostics include checking that chains have mixed and stabilized, inspecting whether imputed values are plausible, and verifying that each completed dataset produces the same model parameter and variance definition. Implausibly erratic \(Q_i\) values can reflect an unstable analysis, sparse data, or poor chain behavior. A very small \(B\) is not proof that missing-data uncertainty is negligible: it may also occur when the imputation model or sampling process fails to reflect relevant uncertainty.

Multiple-imputation pooling differs from analyzing just one completed dataset. A single-imputation analysis supplies one \(Q\) and one \(U\), but it has no across-imputation \(B\) to represent differences among plausible completed datasets. Complete-case analysis is another alternative, but it uses a different subset of observations and may target a different quantity or require different assumptions. Neither alternative can be made equivalent to pooling merely by relabeling its standard error.

The scalar notation is suitable when the target is one number, such as a mean difference or regression coefficient. For a vector of parameters, the estimates are vectors and the variance quantities are covariance matrices; joint inference must account for their covariances. Applying scalar formulas independently does not, by itself, provide a valid joint test.

Common Mistakes

• Confusing \(U_i\) and \(B\). \(U_i\) is an estimated variance within completed dataset \(i\); \(B\) is the sample variance of the \(Q_i\) values across imputations.
• Averaging standard errors as if they were variances. The definition of \(\bar U\) averages \(U_i\), not \(\sqrt{U_i}\). The squared units make the distinction visible.
• Pooling estimates of different targets. A common unit is not enough; the estimand and model parameter must also match.
• Calling the deviations independent because they have \(M-1\) degrees of freedom. The deviations have one linear constraint, not statistical independence.
• Assuming \(T\) alone completes inference. Standard errors, degrees of freedom, intervals, and tests require their corresponding inferential rules.