Multiple Imputation

Multiple imputation (MI) generates $m$ (commonly 5 to 40) complete datasets by imputing each missing value multiple times, drawing from an explicit imputation model (such as a Bayesian posterior predictive distribution or, in the common Multiple Imputation by Chained Equations approach, a sequence of conditional regression models for each variable with missing data), so the imputed values vary appropriately across the $m$ datasets to reflect genuine uncertainty about what the missing values actually were. The intended analysis is performed separately and identically on each of the $m$ completed datasets, producing $m$ point estimates ${\hat{\theta }}_1,\dots ,{\hat{\theta }}_m$ and their variances, which are then combined using Rubin's rules, the overall point estimate is the average $\bar{\theta }=\frac{1}{m}\sum {\hat{\theta }}_j$, and the total variance combines the within-imputation variance (the average of the $m$ individual sampling variances) with the between-imputation variance (how much the point estimates themselves varied across the $m$ datasets, reflecting genuine imputation uncertainty), $T=\bar{W}+\left(1+\frac{1}{m}\right)B$; this two-part variance combination is precisely what allows multiple imputation to produce valid, appropriately wider standard errors than any single-imputation method, correctly reflecting that missing data genuinely add uncertainty beyond ordinary sampling variability, under the assumption that the data are missing at random (or missing completely at random) with respect to the variables included in the imputation model.

• Multiple imputation's central statistical advantage over any single-value imputation method (mean, deterministic regression, single hot-deck) is that it produces appropriately larger, more honest standard errors, single imputation methods systematically understate uncertainty by treating an imputed guess as if it were genuinely observed data, a problem multiple imputation is specifically designed to fix through its between-imputation variance component.
• Multiple Imputation by Chained Equations (MICE), also called fully conditional specification, is the most widely used practical algorithm for generating multiply imputed datasets when several variables have missing values simultaneously, it imputes each incomplete variable in turn using a regression model conditional on all other variables (observed or currently imputed), cycling through all incomplete variables repeatedly until the imputations stabilize.
• Multiple imputation's validity, like nearly all standard imputation methods, depends on the missing at random (or missing completely at random) assumption, if the true missingness mechanism is missing not at random, meaning the probability of a value being missing depends on the unobserved value itself even after conditioning on everything in the imputation model, multiple imputation as standardly implemented can still produce biased results.