Multivariate inference based on Wald tests can be done with
imputed data sets. The approach is a generalization of the approach taken in the univariate case (Rubin 1987, p. 137; Schafer 1997, p. 113). Suppose that
and
are the point and covariance matrix estimates for a
-dimensional parameter
(such as a multivariate mean) from the
imputed data set,
= 1, 2, ...,
. Then the combined point estimate for
from the multiple imputation is the average of the
complete-data estimates:
Suppose that
is the within-imputation covariance matrix, which is the average of the
complete-data estimates:
And suppose that
is the between-imputation covariance matrix:
Then the covariance matrix associated with
is the total covariance matrix
The natural multivariate extension of the
statistic used in the univariate case is the
statistic
with degrees of freedom
and
where
is an average relative increase in variance due to nonresponse (Rubin 1987, p. 137; Schafer 1997, p. 114).
However, the reference distribution of the statistic
is not easily derived. Especially for small
, the between-imputation covariance matrix
is unstable and does not have full rank for
(Schafer 1997, p. 113).
One solution is to make an additional assumption that the population between-imputation and within-imputation covariance matrices are proportional to each other (Schafer 1997, p. 113). This assumption implies that the fractions of missing information for all components of
are equal. Under this assumption, a more stable estimate of the total covariance matrix is
With the total covariance matrix
, the
statistic (Rubin 1987, p. 137)
has an
distribution with degrees of freedom
and
, where
For
, PROC MIANALYZE uses the degrees of freedom
in the analysis. For
, PROC MIANALYZE uses
, a better approximation of the degrees of freedom given by Li, Raghunathan, and Rubin (1991):