### Multivariate Inferences

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 p-dimensional 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 complete-data estimates:

Suppose that is the within-imputation covariance matrix, which is the average of the m 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 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 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 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):