Multiple imputation

Multiple imputation repeatedly draws missing values from fitted predictive distributions, analyzes each completed dataset, and combines the resulting estimates and uncertainty.

A recorded variable included in imputation models to help predict missing values or explain response patterns, even if it is absent from the final substantive model. Useful auxiliaries can improve precision and make missing at random more plausible; they cannot establish that assumption from the observed data.

For $M>1$ completed-data estimates $Q_m$ with variances $U_m$, pool using $\overline{Q}=M^{-1}{\sum }_m{Q}_m$, $\overline{U}=M^{-1}{\sum }_m{U}_m$, $B=(M-1{)}^{-1}{\sum }_m(Q_m-\overline{Q}{)}^2$, and $T=\overline{U}+(1+M^{-1})B$. The total variance includes within-imputation uncertainty, between-imputation uncertainty and finite-$M$ simulation uncertainty.

Use the pooled estimate and total variance from Rubin's rules. A basic large-complete-sample scalar interval is $\overline{Q}\pm t_{\nu ,1-\alpha /2}\sqrt{T}$, with $\nu =(M-1)[1+\overline{U}/((1+M^{-1})B){]}^2$ for $B>0$. Small complete-data samples need an appropriate degrees-of-freedom adjustment; the normal limit applies as the available degrees of freedom become large.

Specify a conditional distribution for each incomplete variable given the others, cycle through parameter draws and missing-value draws, and retain multiple completed datasets after suitable convergence checks. Preserve observed entries. Adequate models, appropriate support and the intended analysis structure matter; arbitrary conditional models need not be compatible with one joint distribution. Standard applications assume missing at random given included predictors. Use Rubin's rules to pool estimates and uncertainty.