Leverage (statistics)

In the linear regression model, the leverage score for the i-th observation is defined as:

${\displaystyle h_{ii}=\left[\mathbf {H} \right]_{ii},}$

the i-th diagonal element of the projection matrix ${\displaystyle \mathbf {H} =\mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}}$, where ${\displaystyle \mathbf {X} }$ is the design matrix (whose rows correspond to the observations and whose columns correspond to the independent or explanatory variables).

The leverage score is also known as the observation self-sensitivity or self-influence,[2] because of the equation

${\displaystyle h_{ii}={\frac {\partial {\widehat {y\,}}_{i}}{\partial y_{i}}},}$

which states that the leverage of the i-th observation equals the partial derivative of the fitted i-th dependent value ${\displaystyle {\widehat {y\,}}_{i}}$with respect to the measured i-th dependent value ${\displaystyle y_{i}}$. This partial derivative describes the degree by which the i-th measured value influences the i-th fitted value. Note that this leverage depends on the values of the explanatory (x-) variables of all observations but not on any of the values of the dependent (y-) variables.

The equation ${\displaystyle h_{ii}={\frac {\partial {\widehat {y\,}}_{i}}{\partial y_{i}}}}$follows directly from the computation of the fitted values via the hat matrix as ${\displaystyle {\mathbf {\widehat {y}} }={\mathbf {H} }{\mathbf {y} }}$; that is, leverage is a diagonal element of the design matrix:

${\displaystyle h_{ii}=\mathbf {H} (i,i).}$

${\displaystyle 0\leq h_{ii}\leq 1.}$

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 leverage ${\displaystyle h_{ii}\equiv x_{i}'(X'X)^{-1}x_{i}}$ and the regression residual ${\displaystyle {\hat {e}}_{i}\equiv y_{i}-x_{i}'\beta }$ , one can compare the estimated coefficient ${\displaystyle {\hat {\beta }}}$ to the leave-one-out estimated coefficient ${\displaystyle {\hat {\beta }}^{(-i)}}$ using the formula [3][4]

${\displaystyle {\hat {\beta }}-{\hat {\beta }}^{(-i)}={\frac {(X'X)^{-1}x_{i}'{\hat {e}}_{i}}{1-h_{ii}}}}$

Young (2019) uses a version of this formula after residualizing controls.[5]

To gain intuition for this formula, note that the k-by-1 vector ${\displaystyle {\frac {\partial {\hat {\beta }}}{\partial y_{i}}}=(X'X)^{-1}x_{i}}$ captures the potential for an observation to affect the regression parameters, and therefore ${\displaystyle (X'X)^{-1}x_{i}{\hat {e}}_{i}}$ captures the actual influence of that observations' deviations from its fitted value on the regression parameters. The formula then divides by ${\displaystyle (1-h_{ii})}$ 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.[6][7]

If we are in an ordinary least squares setting with fixed X and homoscedastic regression errors ${\displaystyle \varepsilon _{i},}$

${\displaystyle Y=X\beta +\varepsilon$ ;\ \ \operatorname {Var} (\varepsilon )=\sigma ^{2}I}

then the i-th regression residual

${\displaystyle e_{i}=Y_{i}-{\widehat {Y}}_{i}}$

has variance

${\displaystyle \operatorname {Var} (e_{i})=(1-h_{ii})\sigma ^{2}}$

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.

Many programs and statistics packages, such as R, Python, etc., include implementations of Leverage.

• Projection matrix – whose main diagonal entries are the leverages of the observations
• Mahalanobis distance – a (scaled) measure of leverage of a datum
• Cook's distance – a measure of changes in regression coefficients when an observation is deleted
• DFFITS
• Outlier – observations with extreme Y values
• Degrees of freedom (statistics), the sum of leverage scores