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:
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Suppose that is the within-imputation covariance matrix, which is the average of the
complete-data estimates:
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And suppose that is the between-imputation covariance matrix:
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Then the covariance matrix associated with is the total covariance matrix
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The natural multivariate extension of the statistic used in the univariate case is the
statistic
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with degrees of freedom and
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where
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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
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With the total covariance matrix , the
statistic (Rubin 1987, p. 137)
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has an distribution with degrees of freedom
and
, where
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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):
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