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Small-sample degrees of freedom with multiple imputation. (English) Zbl 0942.62025

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

MSC:

62F15 Bayesian inference
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