Multicollinearity and misleading statistical results

Abstract

Multicollinearity represents a high degree of linear intercorrelation between explanatory variables in a multiple regression model and leads to incorrect results of regression analyses. Diagnostic tools of multicollinearity include the variance inflation factor (VIF), condition index and condition number, and variance decomposition proportion (VDP). The multicollinearity can be expressed by the coefficient of determination (Rh2) of a multiple regression model with one explanatory variable (Xh) as the model's response variable and the others (Xi [i ≠ h]) as its explanatory variables. The variance (σh2) of the regression coefficients constituting the final regression model are proportional to the VIF. Hence, an increase in Rh2 (strong multicollinearity) increases σh2. The larger σh2 produces 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 explanatory variables is referred to as the condition index. The condition number is the maximum condition index. Multicollinearity is present when the VIF is higher than 5 to 10 or the condition indices are higher than 10 to 30. However, they cannot indicate multicollinear explanatory variables. VDPs obtained from the eigenvectors can identify the multicollinear variables by showing the extent of the inflation of σh2 according to each condition index. When two or more VDPs, which correspond to a common condition index higher than 10 to 30, are higher than 0.8 to 0.9, their associated explanatory variables are multicollinear. Excluding multicollinear explanatory variables leads to statistically stable multiple regression models.

Keywords: Biomedical research; Biostatistics; Multivariable analysis; Regression; Statistical bias; Statistical data analysis.

Fig. 1.

The effects of the coefficient of determination (Rj 2 ) from a regression model $[X_{ij} = {\gamma }_0 + \sum _{l=1}^k {\gamma }_{l }{X}_{il} + {ϵ}_i (i = 1, 2, \dots , n; l = 1, 2, \dots , k; l \ne j\left)\right]$ on the variance of a regression coefficient of interest [Var(β_j)]. The presence of multicollinearity (an increase in R_j²) inflates Var(β_j). X_ij: jth explanatory variable of a regression model [Y_i = β₀ + β₁X_i1 + β₂X_i2 + , … , + β_kX_ik + ε_i (i = 1,2, … , n)].