5.11. Suppose multiple imputations are created using the method of Problem 5.10 D times, and let y d and Ud be the values

5.11. Suppose multiple imputations are created using the method of Problem 5.10 D times, and let y
ðdÞ  and UðdÞ  be the values of y
 and U for the dth imputed data set. Let
y
 ¼ PðDÞ d¼1 y
ðdÞ  =D, and T be the multiple imputation estimate of variance of
y
. That is, T ¼ U
 þ ð1 þ D1ÞB; where U
 ¼ P D d¼1 UðdÞ  =D; B ¼ P D d¼1 ðy
ðdÞ  
y
Þ 2 :

(a) Show that, conditional on the data, the expected value of B equals the variance of y
. 92 ESTIMATION OF IMPUTATION UNCERTAINTY

(b) Show that the variance of
y
 (conditional on n;r and the population Y values) is D1VarðY
Þþð1  D1ÞVarðy
RÞ, and conclude that
y
 is more efficient than the single-imputation estimate y
:

(c) Tabulate values of the relative efficiency of
y
 to y
R for different values of D, assuming large r and N=r.

(d) Show that the variance of
y
 (conditional on n;r, and the population Y values) is greater than the expectation of T by approximately s2 Rð1  r=nÞ 2 =r.

(e) Assume r and N=r are large, and tabulate true coverages and significance levels of the multiple imputation inference. Compare with the results in Problem 5.10, part (d).