How does multiple imputation produce standard errors that reflect missing-data uncertainty?

## The three phases Multiple imputation replaces one guess per gap with a distribution of guesses, and then keeps track of how much the answer depends on which guess you drew. **1. Imputation.** Build a model for each incomplete variable given the others, and generate `m` completed datasets by drawing from it. The draw must be stochastic: a predicted mean plus a randomly drawn residual, or a sample from the predictive distribution, and ideally with the imputation model's own parameter uncertainty included. If you fill with the conditional mean instead, all `m` datasets come out nearly identical. **2. Analysis.** Run the identical analysis on each completed dataset, exactly as if it were the real data, producing `m` estimates `Q_i` of the quantity of interest and `m` variance estimates `U_i`. **3. Pooling.** Combine with Rubin's rules. ## Rubin's rules The pooled point estimate is the simple average: `Qbar = (1/m) * sum Q_i` The within-imputation variance is the average of the individual variances — the uncertainty you would have had with no missing data: `Ubar = (1/m) * sum U_i` The between-imputation variance is the sample variance of the estimates themselves — how much the answer moved depending on which plausible completion you drew: `B = sum (Q_i - Qbar)^2 / (m - 1)` And the total variance is `T = Ubar + (1 + 1/m) * B` with the standard error being `sqrt(T)`. The `(1 + 1/m)` inflation accounts for estimating `Qbar` from finitely many imputations rather than infinitely many; it is why small `m` is penalised rather than silently rewarded. Inference then uses a t-reference distribution with degrees of freedom derived from the ratio of the between and total terms, so heavy missingness both widens the interval and thickens its tails. ## Fraction of missing information The quantity `((1 + 1/m) * B) / T` estimates the fraction of missing information (FMI) — the share of the total uncertainty attributable to the gaps rather than to ordinary sampling noise. It is not the same as the fraction of cells missing: a well-predicted variable with 30% of its values absent can have a small FMI, while a poorly-predicted one with 10% missing can have a large one. FMI is the number to quote when someone asks how much the missing data cost you. ## Choosing m Five to ten completions was the classic advice and is adequate when little information is missing. Modern practice scales `m` with FMI — a common rule of thumb sets the number of imputations near a hundred times the fraction of missing information, so an FMI of 0.3 suggests around 30. More imputations cost only computation and make the pooled standard error more stable, so err upward. ## Building the imputation model The imputation model must be at least as rich as the analysis model. Concretely: - **Include the outcome.** Imputing predictors without the outcome draws values as if the two were unrelated, which attenuates the very association you are trying to estimate. - **Include auxiliary variables** that predict either the missing value or the fact that it is missing, even if the analysis will not use them. They make the MAR assumption more plausible and reduce FMI. - **Preserve structure.** Interactions, non-linear terms and skew present in the analysis model should be representable by the imputation model, or the completions will be systematically too tidy. - **Respect the variable type.** A binary field should get plausible binary values, not fractional ones. This requirement is called congeniality: an imputation model less rich than the analysis model biases results toward the simpler structure it assumed. ## Checking imputations Compare the distribution of imputed values against the observed values, per variable and per completion. They need not match — under MAR they legitimately differ, since the missing rows differ on the observed variables — but wildly implausible values, impossible categories or negative quantities point at a broken imputation model. Also confirm the `m` estimates genuinely differ; if `B` is near zero when a lot of data is missing, the draws are probably not stochastic. ## What it does not do Standard multiple imputation assumes missing at random: that the missing values behave like the observed ones once you condition on the variables in the imputation model. It is not a cure for MNAR. It also cannot invent information that was never collected — it is machinery for propagating honest uncertainty, not for manufacturing certainty. ## The common failure The most frequent mistake is to average the `m` completed datasets into one filled dataset and then analyse that. This collapses to a single, unusually smooth imputation, destroys the between-imputation variance and reproduces exactly the falsely narrow standard error that multiple imputation exists to prevent. You pool the *estimates*, never the datasets.