Multivariate inference based on Wald tests can be done with m 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 pdimensional parameter (such as a multivariate mean) from the imputed data set, i = 1, 2, …, m. Then the combined point estimate for from the multiple imputation is the average of the m completedata estimates:

Suppose that is the withinimputation covariance matrix, which is the average of the m completedata estimates:

And suppose that is the betweenimputation covariance matrix:

Then the covariance matrix associated with is the total covariance matrix

The natural multivariate extension of the t statistic used in the univariate case is the F statistic

with degrees of freedom p 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 m, the betweenimputation 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 betweenimputation and withinimputation 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 F statistic (Rubin, 1987, p. 137)

has an F distribution with degrees of freedom p 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):
