# Variance inflation factor (VIF) Variance inflation factor (VIF) is a calculation we can make to check for collinearity among the predictors in a [[Multiple linear regression|multiple linear regression]]. Let $\mathbf{X}$ is the $n \times p$ design matrix for the regression, and let $X_1, ..., X_p$ represent the individual predictors. Then, let $R^2_{X_k|\mathbf{X}}$ represent the [[Coefficient of determination|coefficient of determination]] of a regression where $X_k$ is the outcome and all the other variables in $\mathbf{X}$ are predictors. The idea is, if there is high collinearity with $X_k$, then $R^2_{X_k|\mathbf{X}}$ will be high or close to one. This correlation will cause the variance of $\hat{\beta_k}$ to increase by a factor of $\text{VIF}(X_k)$: $ \text{VIF}(X_k) = \frac{1}{1 - R^2_{X_k|\mathbf{X}}} $ --- # References - [[Applied Linear Regression#4. Interpretation of Main Effects]]