Multivariate Inferences

The MIANALYZE Procedure

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 $\hat{Q_i}$ and $\hat{U_i}$ are the point and covariance matrix estimates for a vector-valued parameter Q (such as a multivariate mean) from the ith imputed data set, i=1, 2, ..., m. Then the combined point estimate for Q from the multiple imputation is the average of the m complete-data estimates:

${\overline Q}=\frac{1}m \sum_{i=1}^m \hat{Q_i}$

Suppose that ${\overline U}$ is the within-imputation covariance matrix, which is the average of the m complete-data estimates

${\overline U}=\frac{1}m \sum_{i=1}^m \hat{U_i}$

and suppose that B is the between-imputation covariance matrix

$B=\frac{1}{m-1} \sum_{i=1}^m (\hat{Q_i}-{\overline Q}) (\hat{Q_i}-{\overline Q})'$

Then the covariance matrix associated with ${\overline Q}$ is the total covariance matrix

$T_{0}={\overline U} + (1+\frac{1}m)B$

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

$F_{0}=(Q-{\overline Q})' T_{0}^{-1} (Q-{\overline Q})$

with degrees of freedom p and

v=(m-1)(1+1/r)²

where

$r=(1+\frac{1}m) trace(B {\overline U}^{-1}) / p$

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 F₀ is not easily derived. Especially for small m, the between-imputation covariance matrix B is unstable and does not have full rank for $m \le p$ (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 Q are equal. Under this assumption, a more stable estimate of the total covariance matrix is

$T=(1+r) {\overline U}$

With the total covariance matrix T, the F statistic (Rubin 1987, p. 137)

$F=(Q-{\overline Q})' T^{-1} (Q-{\overline Q}) / p$

has an F distribution with degrees of freedom p and v₁, where

v₁ = (1/2) (p+1) (m-1) (1+[1/r])²

For $t=p(m-1) \leq 4$ ,PROC MIANALYZE uses the degrees of freedom v₁ in the analysis. For t=p(m-1) > 4, PROC MIANALYZE uses v₂, a better approximation of the degrees of freedom given by Li, Raghunathan, and Rubin (1991).

v₂ = 4 + (t-4) ( 1+ [1/r] (1-[2/t]) )²

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