In statistics and in particular in regression analysis, leverage is a measure of how far away the independent variable values of an observation are from those of the other observations. High-leverage points, if any, are outliers with respect to the independent variables. That is, high-leverage points have no neighboring points in R p {\displaystyle \mathbb {R} ^{p}} space, where p {\displaystyle {p}} is the number of independent variables in a regression model. This makes the fitted model likely to pass close to a high leverage observation. Hence high-leverage points have the potential to cause large changes in the parameter estimates when they are deleted i.e., to be influential points. Although an influential point will typically have high leverage, a high leverage point is not necessarily an influential point. The leverage is typically defined as the diagonal elements of the hat matrix.
Definition and interpretations Consider the linear regression model y i = x i ⊤ β + ε i {\displaystyle {y}_{i}={\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\beta }}+{\varepsilon }_{i}} , i = 1 , 2 , … , n {\displaystyle i=1,\,2,\ldots ,\,n} . That is, y = X β + ε {\displaystyle {\boldsymbol {y}}=\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}} , where, X {\displaystyle \mathbf {X} } is the n × p {\displaystyle n\times p} design matrix whose rows correspond to the observations and whose columns correspond to the independent or explanatory variables. The leverage score for the i t h {\displaystyle {i}^{th}} independent observation x i {\displaystyle {\boldsymbol {x}}_{i}} is given as:
h i i = [ H ] i i = x i ⊤ ( X ⊤ X ) − 1 x i {\displaystyle h_{ii}=\left[\mathbf {H} \right]_{ii}={\boldsymbol {x}}_{i}^{\top }\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}{\boldsymbol {x}}_{i}} , the i t h {\displaystyle {i}^{th}} diagonal element of the ortho-projection matrix (a.k.a. hat matrix) H = X ( X ⊤ X ) − 1 X ⊤ {\displaystyle \mathbf {H} =\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }} . Thus the i t h {\displaystyle {i}^{th}} leverage score can be viewed as the 'weighted' distance between x i {\displaystyle {\boldsymbol {x}}_{i}} to the mean of x i {\displaystyle {\boldsymbol {x}}_{i}} 's (see its relation with Mahalanobis distance). It can also be interpreted as the degree by which the i t h {\displaystyle {i}^{th}} measured (dependent) value (i.e., y i {\displaystyle y_{i}} ) influences the i t h {\displaystyle {i}^{th}} fitted (predicted) value (i.e., y ^ i {\displaystyle {\widehat {y\,}}_{i}} ): mathematically,
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