How many imputations do you need? A two-stage calculation using a quadratic rule

When using multiple imputation, users often want to know how many imputations they need. An old answer is that 2 to 10 imputations usually suffice, but this recommendation only addresses the efficiency of point estimates. You may need more imputations if, in addition to efficient point estimates, you also want standard error (SE) estimates that would not change (much) if you imputed the data again. For replicable SE estimates, the required number of imputations increases quadratically with the fraction of missing information (not linearly, as previous studies have suggested). I recommend a two-stage procedure in which you conduct a pilot analysis using a small-to-moderate number of imputations, then use the results to calculate the number of imputations that are needed for a final analysis whose SE estimates will have the desired level of replicability. I implement the two-stage procedure using a new Stata command called how_many_imputations (available from SSC) and a new SAS macro called %mi_combine (available from the website missingdata.org).