Multicollinearity and misleading statistical results.

Multicollinearity represents a high degree of linear intercorrelation between explanatory variables (EVs) in a multiple regression model. Because of its presence, the results of regression analysis go wrong. The diagnostic tools of multicollinearity include variance inflation factor (VIF), condition index (CI) and condition number (CN), and variance decomposition proportion (VDP). Multicollinearity can be presented by the coefficient of determination (Rh2) for a multiple regression model with one EV (Xh) as the model's response variable and the others (Xi[i≠h]) as its EVs. The variances (σh2) of the regression coefficients constituting the final regression model are proportional to VIF (1-Rh2/1). Hence, an increase in Rh2 (strong multicollinearity) inflates σh2. The inflated σh2 produce unreliable probability values and confidence intervals of the regression coefficients. The square root of the ratio of the maximum eigenvalue to each eigenvalue from the correlation matrix of standardized EVs is termed as CI. CN is the maximum of CI. Multicollinearity is present when VIF is higher than 5 to 10 or condition indices are higher than 10 to 30. However, they cannot indicate EVs with multicollinearity. VDPs obtained from the eigenvectors can identify the variables with multicollinearity by showing the extent of the inflation of σh2 according to each CI. When two or more VDPs, which correspond to a common CI higher than 10 to 30, are higher than 0.8 to 0.9, the EVs associated with the VDPs are multicollinear. Excluding multicollinear EVs makes statistically stable multiple regression models.