The MI Procedure

Descriptive Statistics

Suppose $\mb {Y} = ( \mb {y}_{1}, \mb {y}_{2}, \ldots , \mb {y}_{n} )^{}$ is the $(n{\times }p)$ matrix of complete data, which might not be fully observed, $n_{0}$ 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

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

and the CSSCP matrix is

$\sum { ( \mb {y}_{i} - \overline{\mb {y}} ) ( \mb {y}_{i} - \overline{\mb {y}} )^{} }$

where each summation is over the fully observed observations.

The sample covariance matrix is

$\mb {S} = \frac{1}{\, n_{0}-1 \, } \sum { ( \mb {y}_{i} - \overline{\mb {y}} ) ( \mb {y}_{i} - \overline{\mb {y}} )^{} }$

and is an unbiased estimate of the covariance matrix.

The correlation matrix $\mb {R}$ , which contains the Pearson product-moment correlations of the variables, is derived by scaling the corresponding covariance matrix:

$\mb {R} = \mb {D}^{-1} \mb {S} \, \mb {D}^{-1}$

where $\mb {D}$ is a diagonal matrix whose diagonal elements are the square roots of the diagonal elements of $\mb {S}$ .

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.