• Multicollinearity in Regression Models is an unacceptably high level of intercorrelation among the independents, such that the effects of the independents cannot be separated. Under multicollinearity, estimates are unbiased but assessments of the relative strength of the explanatory variables and their joint effect are unreliable. (That is, beta weights and R-squares cannot be interpreted reliably even though predicted values are still the best estimate using the given independents). As a rule of thumb, intercorrelation among the independents above .80 signals a possible problem. Likewise, high multicollinearity is signalled when high R-squared and signficant F tests of the model occur in combination with non-significant t-tests of coefficients.
• That is, whereas perfect multicollinearity leads to infinite standard errors and indeterminant coefficients, the more common situation of high multicollinearity leads to large variances and covariances, large confidence intervals, and insignificant significance coefficients. Power is low (the chance of Type II errors is high - thinking you do not have a relationship when in fact one exists - failure to reject the null hypothesis that the coefficients are not different from zero). R-square is high. The coefficients and their standard errors will be sensitive to changes in just a few observations.
• Tolerance is defined as 1 - R-squared, where R-squared is the multiple R of a given independent regressed on all other independent variables. If the tolerance value is less than some cutoff value, usually .20, the independent should be dropped from the analysis due to multicollinearity. This is better than just using simple r > .80 since tolerance looks at the independent variable in relation to all other independents and thus takes interaction effects into account as well as simple correlations. In SPSS 13, select Analyze, Regression, linear; click Statistics; check Collinearity diagnostics.

• Variance inflation factor, VIF. Note, the variance-inflation factor, VIF, may be used in lieu of tolerance as VIF is simply the reciprocal of tolerance. The rule of thumb is that VIF > 4.0 when multicollinearity is a problem. Some authors use the more lenient cut-off of VIF >= 5 when multicollinearity is a problem. In SPSS 13, select Analyze, Regression, linear; click Statistics; check Collinearity diagnostics.

• Condition indices. Discussed more extensively in the section on regression, condition indices over 15 indicate possible multicollinearity problems and over 30 indicate serious multicollinearity problems. In SPSS 13, select Analyze, Regression, linear; click Statistics; check Collinearity diagnostics.

[此贴子已经被作者于2006-5-18 1:05:15编辑过]

Multicollinearity in Structural Equation Models (SEM)

• Standardized regression weights: Since all the latent variables in a SEM model have been assigned a metric of 1, all the standardized regression weights should be within the range of plus or minus 1. When there is a multicollinearity problem, a weight close to 1 indicates the two variables are close to being identical. When these two nearly identical latent variables are then used as causes of a third latent variable, the SEM method will have difficulty computing separate regression weights for the two paths from the nearly-equal variables and the third variable. As a result it may well come up with one standardized regression weight greater than +1 and one weight less than -1 for these two paths.

• Standard errors of the unstandardized regression weights: Likewise, when there are two nearly identical latent variables, and these two are used as causes of a third latent variable, the difficulty in computing separate regression weights may well be reflected in much larger standard errors for these paths than for other paths in the model, reflecting high multicollinearity of the two nearly identical variables.

• Covariances of the parameter estimates: Likewise, the same difficulty in computing separate regression weights may well be reflected in high covariances of the parameter estimates for these paths - estimates much higher than the covariances of parameter estimates for other paths in the model.

• Variance estimates: Another effect of the same multicollinearity syndrome may be negative variance estimates. In the example above of two nearly-identical latent variables causing a third latent variable, the variance estimate of this third variable may be negative.

[此贴子已经被作者于2006-5-18 1:03:39编辑过]