Modeling and Multivariate Methods - SAS

Chapter 17 Correlations and Multivariate Techniques 457
Computations and Statistical Details
Inverse Correlation Matrix
The inverse correlation matrix provides useful multivariate information. The diagonal elements of the
inverse correlation matrix, sometimes called the variance inflation factors (VIF), are a function of how
closely the variable is a linear function of the other variables. Specifically, if the correlation matrix is denoted
R and the inverse correlation matrix is denoted R -1 , the diagonal element is denoted r ii and is computed as
where R i
2 is the coefficient of variation from the model regressing the i th explanatory variable on the other
explanatory variables. Thus, a large r ii indicates that the ith variable is highly correlated with any number of
the other variables.
Distance Measures
The Outlier Distance plot shows the Mahalanobis distance of each point from the multivariate mean
(centroid). The Mahalanobis distance takes into account the correlation structure of the data as well as the
individual scales. For each value, the distance is denoted d i and is computed as
where:
r ii 1
= VIF i
= ---------------
2
1 – R i
d i
= ( Y i – Y)'S – 1 ( Y i – Y)
Y i is the data for the i th row
Y is the row of means
S is the estimated covariance matrix for the data
The reference line (Mason and Young, 2002) drawn on the Mahalanobis Distance plot is computed as
AxF × nvars where A is related to the number of observations and number of variables, nvars is the number
of variables, and the computation for F in formula editor notation is:
F Quantile(0.95, nvars, n–nvars–1, centered at 0).
If a variable is an exact linear combination of other variables, then the correlation matrix is singular and the
row and the column for that variable are zeroed out. The generalized inverse that results is still valid for
forming the distances.
The T 2 distance is just the square of the Mahalanobis distance, so T 2 i = d 2 i . The upper control limit on the
T 2 is
( n – 1) 2
UCL = ------------------ β
n
a;
p 2 -- ;-------------------
n – p – 1
2
where
n = number of observations