Descriptive Statistics

The MI Procedure

Descriptive Statistics

Suppose Y is the n×p matrix of complete data, which may not be fully observed, n₀ is the number of observations fully observed, and n_j is the number of observations with observed values for variable Y_j.

With complete cases, the sample mean vector is

${\overline y}=\frac{1}{n_{0}} \sum {y_{i}}$

and the CSSCP matrix is

$\sum { ( y_{i} - {\overline y} ) ( y_{i} - {\overline y} )' }$

where each summation is over the fully observed observations.

The sample covariance matrix is

$S=\frac{1}{\, n_{0}-1 \,} \sum { ( y_{i} - {\overline y} ) ( y_{i} - {\overline y} )' }$

and is an unbiased estimate of the covariance matrix.

The correlation matrix R containing the Pearson product-moment correlations of the variables is derived by scaling the corresponding covariance matrix:

R = D^-1 S D^-1

where D is a diagonal matrix whose diagonal elements are the square roots of the diagonal elements of S.

With available cases, the corrected sum of squares for variable Y_j is

$\sum { ( y_{ji} - {\overline y_{j}} )^2 }$

where ${\overline y_{j}}=\frac{1}{n_{j}} \sum {y_{ji}}$ is the sample mean and each summation is over observations with observed values for variable Y_j.

The variance is

$s_{jj}^2=\frac{1}{\, n_{j}-1 \,} \sum { ( y_{ji} - {\overline y_{j}} )^2 }$

The correlations for available cases contain pairwise correlations for each pair of variables. Each correlation is computed from all observations that have nonmissing values for the corresponding pair of variables.

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