1. Multicollinearity in Regression Principal Components Analysis
Standing Heights and Physical Stature Attributes Among Female Police Officer Applicants
S.Q. Lafi and J.B. Kaneene (1992). “An Explanation of the Use of Principal Components Analysis to Detect and Correct for Multicollinearity,” Preventive Veterinary Medicine, Vol. 13, pp

2. Data Description
Subjects: 33 Females applying for police officer positions
Dependent Variable: Y ≡ Standing Height (cm)
Independent Variables:
X1 ≡ Sitting Height (cm)
X2 ≡ Upper Arm Length (cm)
X3 ≡ Forearm Length (cm)
X4 ≡ Hand Length (cm)
X5 ≡ Upper Leg Length (cm)
X6 ≡ Lower Leg Length (cm)
X7 ≡ Foot Length (inches)
X8 ≡ BRACH (100X3/X2)
X9 ≡ TIBIO (100X6/X5)

3. Data

4. Standardizing the Predictors

5. Correlations Matrix of Predictors and Inverse

6. Variance Inflation Factors (VIFs)
VIF measures the extent that a regression coefficient’s variance is inflated due to correlations among the set of predictors
VIFj = 1/(1-Rj2) where Rj2 is the coefficient of multiple determination when Xj is regressed on the remaining predictors.
Values > 10 are often considered to be problematic
VIFs can be obtained as the diagonal elements of R-1
Not surprisingly, X2, X3, X5, X6, X8, and X9 are problems (see definitions of X8 and X9)

7. Regression of Y on [1|X*]
Note the surprising negative coefficients for X3*, X5*, and X9*

8. Principal Components Analysis
While the columns of X* are highly correlated, the columns of W are uncorrelated
The ls represent the variance corresponding to each principal component

9. Police Applicants Height Data - I

10. Police Applicants Height Data - II

11. Regression of Y on [1|W]
Note that W8 and W9 have very small eigenvalues and very small
t-statistics
Condition indices are 63.5 and 85.2,
Both well above 10

12. Reduced Model
Removing last 2 principal components due to small, insignificant t-statistics and high condition indices
Let V(g) be the p×g matrix of the eigenvectors for the g retained principal components (p=9, g=7)
Let W(g) = X*V(g)
Then regress Y on [1|W(g)]

13. Reduced Regression Fit

14. Transforming Back to X-scale

15. Comparison of Coefficients and SEs
Original Model
Principal Components