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Fit Analyses |
When an explanatory variable is nearly a linear combination of other explanatory variables in the model, the affected estimates are unstable and have high standard errors. This problem is called collinearity or multicollinearity. A fit analysis provides several methods for detecting collinearity. Tolerances (TOL) and variance inflation factors (VIF) measure the strength of inter- relationships among the explanatory variables in the model. Tolerance is 1- R^{2} for the R^{2} that results from the regression of the explanatory variable on the other explanatory variables in the model. Variance inflation factors are diagonal elements of (X'X)^{-1} after X'X is scaled to correlation form. The variance inflation measures the inflation in the variance of the parameter estimate due to collinearity between the explanatory variable and other variables. These measures are related by VIF = 1 / TOL. If all variables are orthogonal to each other, both tolerance and variance inflation are 1. If a variable is closely related to other variables, the tolerance goes to 0 and the variance inflation becomes large.
When the X'X matrix is singular,
least-squares solutions for the parameters are not unique.
An estimate is 0 if the variable is a linear
combination of previous explanatory variables.
The degrees of freedom for the
zeroed estimates are reported as 0.
The hypotheses that are not testable
have t tests printed as missing.
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