  Researchers regularly perform conditional prediction using imputed values of
missing data. However, applications of imputation often lack a firm foundation
in statistical theory. This paper originated when we were unable to find
analysis substantiating claims that imputation of missing data has good
frequentist properties when data are missing at random (MAR). We focused on the
use of observed covariates to impute missing covariates when estimating
conditional means of the form E(y|x, w). Here y is an outcome whose
realizations are always observed, x is a covariate whose realizations are
always observed, and w is a covariate whose realizations are sometimes
unobserved. We examine the probability limit of simple imputation estimates of
E(y|x, w) as sample size goes to infinity. We find that these estimates are not
consistent when covariate data are MAR. To the contrary, the estimates suffer
from a shrinkage problem. They converge to points intermediate between the
conditional mean of interest, E(y|x, w), and the mean E(y|x) that conditions
only on x. We use a type of genotype imputation to illustrate.
