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 space, where 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.
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.
is more than 2 times larger than the mean leverage
h
¯
=
1
n
∑
i
=
1
n
h
i
Relation to Mahalanobis distance
Leverage is closely related to the Mahalanobis distance (proof). Specifically, for some
n
×
p
{\displaystyle n\times p}
matrix
X
{\displaystyle \mathbf {X} }
, the squared Mahalanobis distance of
x
i
{\displaystyle {{\boldsymbol {x}}_{i}}}
(where
x
i
⊤
{\displaystyle {\boldsymbol {x}}_{i}^{\top }}
is
i
t
h
{\displaystyle {i}^{th}}
row of
X
{\displaystyle \mathbf {X} }
) from the vector of mean
μ
^
=
Relation to influence functions
In a regression context, we combine leverage and influence functions to compute the degree to which estimated coefficients would change if we removed a single data point. Denoting the regression residuals as
Partial leverage (PL) is a measure of the contribution of the individual independent variables to the total leverage of each observation. That is, PL is a measure of how
h
i
i
{\displaystyle h_{ii}}
changes as a variable is added to the regression model. It is computed as:
(
P
L
j
)
i
=
(
X
j
∙
[
j
]
)
i
2
∑
k
=
1
n
(
X
j
∙
[
Software implementations
Many programs and statistics packages, such as R, Python, etc., include implementations of Leverage.
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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
captures the actual influence of that observations' deviations from its fitted value on the regression parameters. The formula then divides by
(
1
−
h
i
i
)
{\displaystyle (1-h_{ii})}
to account for the fact that we remove the observation rather than adjusting its value, reflecting the fact that removal changes the distribution of covariates more when applied to high-leverage observations (i.e. with outlier covariate values). Similar formulas arise when applying general formulas for statistical influences functions in the regression context.
In other words, an observation's leverage score determines the degree of noise in the model's misprediction of that observation, with higher leverage leading to less noise. This follows from the fact that
against the remaining independent variables. Note that the partial leverage is the leverage of the
i
t
h
{\displaystyle {i}^{th}}
point in the partial regression plot for the
j
t
h
{\displaystyle {j}^{th}}
variable. Data points with large partial leverage for an independent variable can exert undue influence on the selection of that variable in automatic regression model building procedures.
