DFFITS

Definition

The DFFITS consider the influence of the $i$-th observation on $\widehat{y_i}$

Single Observation

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ $DFFIT{S}_i=\frac{|{\mathbf{x}}_i^{⊺}(\hat{\mathbf{\beta }}-{\hat{\mathbf{\beta }}}_{(i)})|}{s_{(i)}\sqrt{h_{ii}}}=\frac{1}{s_{(i)}}\frac{|e_i|\sqrt{h_{ii}}}{1-h_{ii}}=\frac{|r_i^{\ast }|\sqrt{h_{ii}}}{\sqrt{1-h_{ii}}}$ where $s_{(i)}$ is the externally studentized residual

Set of Observations

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ $DFFIT{S}_K=\frac{1}{s_{(K)}}\frac{{\mathbf{x}}_i^{⊺}({\mathbf{X}}^{⊺}\mathbf{X}{)}^{-1}{\mathbf{X}}_K^{⊺}(I-{\mathbf{H}}_K{)}^{-1}{\mathbf{e}}_K}{\sqrt{h_{ii}}}$ where $s_{(K)}$ is the externally studentized residual and $\mathbf{H}$ is the matrix of leverages