Collinearity Diagnostics

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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

$d^2_{1} {\ge} d^2_{2} {\ge} ... {\ge} d^2_{p}$

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 / d_j, j = 1, 2, ... , p. The condition number of the X matrix is defined as the largest condition index, d₁ / 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.

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