La multicolinealidad debe considerarse un problema si la R cuadrada de Xi contra las demas Xs es mayor a la R cuadrada de la gregresion.

Regresion original (p. 50)

reg <- lm(mtcars$mpg ~  mtcars$am + mtcars$wt + mtcars$hp, data=mtcars)
summary(reg)
## 
## Call:
## lm(formula = mtcars$mpg ~ mtcars$am + mtcars$wt + mtcars$hp, 
##     data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4221 -1.7924 -0.3788  1.2249  5.5317 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 34.002875   2.642659  12.867 2.82e-13 ***
## mtcars$am    2.083710   1.376420   1.514 0.141268    
## mtcars$wt   -2.878575   0.904971  -3.181 0.003574 ** 
## mtcars$hp   -0.037479   0.009605  -3.902 0.000546 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.538 on 28 degrees of freedom
## Multiple R-squared:  0.8399, Adjusted R-squared:  0.8227 
## F-statistic: 48.96 on 3 and 28 DF,  p-value: 2.908e-11

Regresion de “am” contra todas las demas.

regx1 <- lm(mtcars$am ~  mtcars$wt + mtcars$hp, data=mtcars)
summary(regx1) 
## 
## Call:
## lm(formula = mtcars$am ~ mtcars$wt + mtcars$hp, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.6309 -0.2562 -0.1099  0.3039  0.5301 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.547430   0.211046   7.332 4.46e-08 ***
## mtcars$wt   -0.479556   0.083523  -5.742 3.24e-06 ***
## mtcars$hp    0.002738   0.001192   2.297    0.029 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3423 on 29 degrees of freedom
## Multiple R-squared:  0.5597, Adjusted R-squared:  0.5293 
## F-statistic: 18.43 on 2 and 29 DF,  p-value: 6.833e-06

Regresion de “wt” contra todas las demas.

regx2 <- lm(mtcars$wt ~  mtcars$am + mtcars$hp, data=mtcars)
summary(regx2) 
## 
## Call:
## lm(formula = mtcars$wt ~ mtcars$am + mtcars$hp, data = mtcars)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.83338 -0.24390 -0.05175  0.15592  1.24801 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.576956   0.255062  10.103 5.22e-11 ***
## mtcars$am   -1.109360   0.193215  -5.742 3.24e-06 ***
## mtcars$hp    0.007437   0.001406   5.289 1.14e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5207 on 29 degrees of freedom
## Multiple R-squared:  0.7351, Adjusted R-squared:  0.7168 
## F-statistic: 40.24 on 2 and 29 DF,  p-value: 4.315e-09

Regresion de “hp” contra todas las demas.

regx3 <- lm(mtcars$hp ~  mtcars$am + mtcars$wt, data=mtcars)
summary(regx3) 
## 
## Call:
## lm(formula = mtcars$hp ~ mtcars$am + mtcars$wt, data = mtcars)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -60.05 -31.56 -15.55  22.43 131.64 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -88.55      48.37  -1.831   0.0775 .  
## mtcars$am      56.23      24.48   2.297   0.0290 *  
## mtcars$wt      66.02      12.48   5.289 1.14e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 49.06 on 29 degrees of freedom
## Multiple R-squared:  0.5211, Adjusted R-squared:  0.4881 
## F-statistic: 15.78 on 2 and 29 DF,  p-value: 2.31e-05

Conclusion

Multicolinealidad no es un problema en la regresion original.