Miscellanea. Small-sample degrees of freedom with multiple imputation

Abstract

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, ν com , are infinite, but the number of imputations, m , is finite. When ν com is small and there is only a modest proportion of missing data, the calculated repeated-imputation degrees of freedom, ν m , for the t reference distribution can be much larger than ν com , which is clearly inappropriate. Following the Bayesian paradigm, we derive an adjusted degrees of freedom, ν̃ m , with the following three properties: for fixed m and estimated fraction of missing information, ν̃ m monotonically increases in ν com ; ν̃ m is always less than or equal to ν com ; and ν̃ m equals ν m when ν com is infinite. A small simulation study demonstrates the superior frequentist performance when using ν̃ m rather than ν m .