Leverage (statistics)

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 are those observations, if any, made at extreme or outlying values of the independent variables such that the lack of neighboring observations means that the fitted regression model will pass close to that particular observation.^[1]

Modern computer packages for statistical analysis include, as part of their facilities for regression analysis, various quantitative measures for identifying influential observations: among these measures is partial leverage, a measure of how a variable contributes to the leverage of a datum.

Linear regression model

Definition

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

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

the i-th diagonal element of the projection matrix $\mathbf{H} = \mathbf{X} \left( \mathbf{X}^{\mathsf{T}} \mathbf{X} \right)^{-1} \mathbf{X}^{\mathsf{T}}$, where $\mathbf{X}$ is the design matrix. The leverage score is also known as the observation self-sensitivity or self-influence,^[2] as shown by

$h_{ii} = \frac{\partial\hat{y}_i}{\partial y_i},$

where $\hat{y}_i$ and ${y}_i$ are the fitted and measured observation, respectively.

Bounds on leverage

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

Proof

First, note that H is an idempotent matrix: $H^2=X(X^{\top}X)^{-1}X^{\top}X(X^{\top}X)^{-1}X^{\top}=XI(X^{\top}X)^{-1}X^{\top}=H .$ Also, observe that $H$ is symmetric. So equating the ii element of H to that of H ², we have

$h_{ii}=h_{ii}^2+\sum_{j\neq i}h_{ij}^2 \geq 0$

and

$h_{ii} \geq h_{ii}^2 \implies h_{ii} \leq 1 .$

Effect on residual variance

If we are in an ordinary least squares setting with fixed X, regression errors $\epsilon_i,$ and

$Y=X\beta+\epsilon$

$\operatorname{Var}(\epsilon)=\sigma^2I$

then $\operatorname{Var}(e_i)=(1-h_{ii})\sigma^2$ where $e_i=Y_i-\hat{Y}_i$ (the i th regression residual).

In other words, if the model errors $\epsilon$ are homoscedastic, an observation's leverage score determines the degree of noise in the model's misprediction of that observation.

Proof

First, note that $I-H$ is idempotent and symmetric. This gives,

$\operatorname{Var}(e)=\operatorname{Var}((I-H)Y)=(I-H)\operatorname{Var}(Y)(I-H)^{\top}=\sigma^2(I-H)^2=\sigma^2(I-H).$

Thus $\operatorname{Var}(e_i)=(1-h_{ii})\sigma^2 .$

Studentized residuals

The corresponding studentized residual—the residual adjusted for its observation–specific residual variance—is then

$t_i = {e_i\over \widehat{\sigma} \sqrt{1-h_{ii}\ }}$

where $\widehat{\sigma}$ is an appropriate estimate of $\sigma.$

See also

• Projection matrix – whose main diagonal entries are the leverages of the observations
• Mahalanobis distance – a measure of leverage of a datum
• Cook's distance – a measure of changes in regression coefficients when an observation is deleted
• DFFITS
• Outliers – observations with extreme Y values

References

1. Everitt, B. S. (2002). Cambridge Dictionary of Statistics. Cambridge University Press. ISBN 0-521-81099-X.
2. Cardinali, C. (June 2013). "Data Assimilation: Observation influence diagnostic of a data assimilation system" (PDF).