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