Statistical Modelling: Variance Inflation Factor (VIF) in Regression Analysis. Detecting and Addressing Multicollinearity
Gain insights into the Variance Inflation Factor (VIF) and its role in identifying multicollinearity in regression models. Learn how to interpret VIF values, diagnose correlated independent variables, and enhance model stability.
Variance Inflation Factor (VIF) is a statistical measure that quantifies the extent of multicollinearity in a regression analysis.
Multicollinearity occurs when independent variables in a regression model are highly correlated with each other, which can lead to unreliable and unstable estimates of the regression coefficients.
The VIF of a particular independent variable measures how much the variance of the estimated regression coefficient is inflated due to multicollinearity. A high VIF indicates a high degree of multicollinearity, suggesting that the corresponding independent variable is highly correlated with other predictors in the model.
The VIF is calculated as follows:
VIF = 1 / (1 — R²)
where R² represents the coefficient of determination of the regression model when the independent variable in question is regressed on all the other independent variables in the model.
To understand the mathematical formulation of VIF, let’s consider a multiple linear regression model with p independent variables:
Y = β₀ + β₁X₁ + β₂X₂ + … + βₚXₚ + ɛ
where Y is the dependent variable, X₁, X₂, …, Xₚ are the independent variables, β₀, β₁, β₂, …, βₚ are the regression coefficients, and ɛ is the error term.
To calculate the VIF for the ith independent variable, we regress Xᵢ on all the other independent variables in the model:
Xᵢ = α₀ + α₁X₁ + α₂X₂ + … + αᵢ₋₁Xᵢ₋₁ + αᵢ₊₁Xᵢ₊₁ + … + αₚXₚ + ɛᵢ
where Xᵢ is the ith independent variable, X₁, X₂, …, Xᵢ₋₁, Xᵢ₊₁, …, Xₚ are the other independent variables, α₀, α₁, α₂, …, αᵢ₋₁, αᵢ₊₁, …, αₚ are the regression coefficients, and ɛᵢ is the error term.
We then calculate the coefficient of determination (R²) of this regression model, which represents the…