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I tried VIF on the Longley dataset to look for multicollinearity. (I have used a custom function returned in https://beckmw.wordpress.com/2013/02/05/collinearity-and-stepwise-vif-selection/comment-page-1/#comment-1788)

Case 1. Without VIF, model showed Population, GNP, GNP.deflator not statistically significant by looking at the p-value.

lm(formula = Employed ~ ., data = longley)

Multiple R-squared: 0.9955, Adjusted R-squared: 0.9925

Case 2. I tried the VIF using the above function, It has removed GNP, GNP.deflator and Year. Whereas the Year variable was highly significant without VIF, p-value was 0.003037.

(If VIF is more than 10, multicollinearity is strongly suggested.)

require(fmsb)

VIF(lm(Employed~., data=longley)) 

VIF is 221 using fmsb package.

keep.dat <- vif_func(in_frame=longley[,-7],thresh=5,trace=T)

form.in<-paste('Employed ~',paste(keep.dat,collapse='+'))

form.in

fit<-lm(form.in,data=longley) summary(fit)

Multiple R-squared:  0.9696, Adjusted R-squared:  0.962 (using usdm pkg)

Multiple R-squared:  0.9696, Adjusted R-squared:  0.962 (using fmsb pkg)

Questions:

  1. Why the VIF removed the Year while doing VIF, since it was highly significant without applying VIF?

Before VIF:

## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -3.482e+03  8.904e+02  -3.911 0.003560 ** 
## GNP.deflator  1.506e-02  8.492e-02   0.177 0.863141    
## GNP          -3.582e-02  3.349e-02  -1.070 0.312681    
## Unemployed   -2.020e-02  4.884e-03  -4.136 0.002535 ** 
## Armed.Forces -1.033e-02  2.143e-03  -4.822 0.000944 ***
## Population   -5.110e-02  2.261e-01  -0.226 0.826212    
## Year          1.829e+00  4.555e-01   4.016 0.003037 **

After VIF:

 Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
 (Intercept)  -1.323091   4.211566  -0.314  0.75880    
 Unemployed   -0.012292   0.003354  -3.665  0.00324 ** 
 Armed.Forces -0.001893   0.003516  -0.538  0.60019    
 Population    0.605146   0.047617  12.709 2.55e-08 ***
  1. When there ise multicollinearity between two predictors, should we not remove one and retain the other? Here it seems to be removing both the variables.

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First, instead of automatically removing variables using vif or any function, you should use collinearity indexes and proportion of variance explained to get a better understanding of what is going on. In R these are available in the colldiag function in the perturb package.

Second, when you have collinearity, there are a number of possible remedies. E.g.

  • Using a penalized method like ridge regression
  • Getting more data
  • Removing variables
  • Doing partial least squares regression
  • Doing principal components regression
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