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Biometrika 1999 86(4):948-955; doi:10.1093/biomet/86.4.948
© 1999 by Biometrika Trust
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Miscellanea. Small-sample degrees of freedom with multiple imputation

J BarnardA1 and DB RubinA2

Department of Statistics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA A1 E-mail: barnard@stat.harvard.edu A2 rubin@stat.harvard.edu

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, {nu}com, are infinite, but the number of imputations, m, is finite. When {nu}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}com, which is clearly inappropriate. Following the Bayesian paradigm, we derive an adjusted degrees of freedom, m, with the following three properties: for fixed m and estimated fraction of missing information, m monotonically increases in {nu}com; m is always less than or equal to {nu}com; and m equals {nu}m when {nu}com is infinite. A small simulation study demonstrates the superior frequentist performance when using m rather than {nu}m.

Key Words: Bayesian inference; fraction of missing information; missing at random; missing data mechanism; repeated imputation


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