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. 261-275

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)

Data

Standardizing the Predictors

Correlation Matrix of Predictors and Inverse

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)

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

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

Police Applicants Height Data - I

Police Applicants Height Data - II

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 30

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)] to obtain

Reduced Regression Fit

Transforming Back to Transformed X-scale

Comparison of Coefficients and SEs Original Model Principal Components

Predicted Values