Miscellanea. Small-sample degrees of freedom with multiple imputation

An appealing feature of multiple imputation is the simplicity of the rules for combining the multiple complete-data inferences into a final inference, the repeated-imputation inference (Rubin, 1987). This inference is based on a t distribution and is derived from a Bayesian paradigm under the assumption that the complete-data degrees of freedom, νcom, are infinite, but the number of imputations, m, is finite. When νcom is small and there is only a modest proportion of missing data, the calculated repeated-imputation degrees of freedom, νm, for the t reference distribution can be much larger than νcom, which is clearly inappropriate. Following the Bayesian paradigm, we derive an adjusted degrees of freedom, ν̃m, with the following three properties: for fixed m and estimated fraction of missing information, ν̃m monotonically increases in νcom; ν̃m is always less than or equal to νcom; and ν̃m equals νm when νcom is infinite. A small simulation study demonstrates the superior frequentist performance when using ν̃m rather than νm.