The Bias and Efficiency of Incomplete-Data Estimators in Small Univariate Normal Samples

Abstract

Widely used methods for analyzing missing data can be biased in small samples. To understand these biases, we evaluate in detail the situation where a small univariate normal sample, with values missing at random, is analyzed using either observed-data maximum likelihood (ML) or multiple imputation (MI). We evaluate two types of MI: the usual Bayesian approach, which we call posterior draw (PD) imputation, and a little used alternative, which we call ML imputation, in which values are imputed conditionally on an ML estimate. We find that observed-data ML is more efficient and has lower mean squared error than either type of MI. Between the two types of MI, ML imputation is more efficient than PD imputation, and ML imputation also has less potential for bias in small samples. The bias and efficiency of PD imputation can be improved by a change of prior.