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 Fit Analyses

## 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 D2 V' where V is an orthogonal matrix whose columns are the eigenvectors of X'X, and D2 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, d1 / dj, j = 1, 2, ... , p. The condition number of the X matrix is defined as the largest condition index, d1 / dp. 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 dj 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.

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