## Collinearity Diagnostics

The **Collinearity Diagnostics** table is
illustrated by Figure 39.22.

**Figure 39.22:** Collinearity Diagnostics Table

**Number**
- is the eigenvalue number.

**Eigenvalue**
- gives the eigenvalues of the
**X**'**X** matrix.
**Condition Index**
- is the square root of the ratio of the largest
eigenvalue to the corresponding eigenvalue.
**Variance Proportion**
- is the proportion of the variance of each estimate
accounted for by each component.

Detailed collinearity diagnostics use the eigenstructure
of **X**'**X**, which can be written as

**X**'**X** = **V** **D**^{2} **V**'
where **V** is an orthogonal matrix whose columns
are the eigenvectors of **X**'**X**, and
**D**^{2} is a diagonal matrix of eigenvalues

After scaling (**X**'**X**) to correlation form,
Belsley, Kuh, and Welsch (1980) construct the condition
indices as the square roots of the ratio of the largest
eigenvalue to each individual eigenvalue,
*d*_{1} / *d*_{j},
*j* = 1, 2, ... , *p*.
The *condition number* of the **X** matrix
is defined as the largest condition index,
*d*_{1} / *d*_{p}.
When this number is large, the
data are said to be *ill conditioned*.
A condition index of 30 to 100 indicates
moderate to strong collinearity.
For each variable, the proportion of the variance
of its estimate accounted for by each component
*d*_{j} can be evaluated.
A collinearity problem occurs when a component
associated with a high condition index contributes
strongly to the variance of two or more variables.
Thus, for a high condition index (>30), the corresponding row
should be examined to see which variables have high values.
Those would indicate near-linear dependence.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.