Multivariate Inferences |
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):