# 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]

## Definition

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

the *i*-th diagonal element of the projection matrix
, where
is the design matrix (whose rows correspond to the observations and whose columns correspond to the independent or explanatory variables).

## Interpretation

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

which states that the leverage of the *i*-th observation equals the partial derivative of the fitted *i*-th dependent value
with respect to the measured *i*-th dependent value
. 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 follows directly from the computation of the fitted values as .

## Bounds on leverage

### Proof

First, note that *H* is an idempotent matrix:
Also, observe that
is symmetric (i.e.:
). So equating the *ii* element of *H* to that of *H* ^{2}, we have

and

## Effect on residual variance

If we are in an ordinary least squares setting with fixed X and homoscedastic regression errors

then the *i-*th regression residual

has variance

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.

### Proof

First, note that is idempotent and symmetric, and . This gives

Thus

### Studentized residuals

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

where is an appropriate estimate of

## Related concepts

### Partial leverage

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.

### Mahalanobis distance

Leverage is closely related to the Mahalanobis distance[3] (see proof: [4]).

Specifically, for some matrix the squared Mahalanobis distance of some row vector from the vector of mean , of length , and with the estimated covariance matrix is:

This is related to the leverage of the hat matrix of after appending a column vector of 1's to it. The relationship between the two is:

## Software implementations

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

Language/Program | Function | Notes |
---|---|---|

R | `hat(x, intercept = TRUE)` or `hatvalues(model, ...)` | See |

## See also

- 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

## References

- Everitt, B. S. (2002).
*Cambridge Dictionary of Statistics*. Cambridge University Press. ISBN 0-521-81099-X. - Cardinali, C. (June 2013). "Data Assimilation: Observation influence diagnostic of a data assimilation system" (PDF).
- Weiner, Irving B.; Schinka, John A.; Velicer, Wayne F. (23 October 2012).
*Handbook of Psychology, Research Methods in Psychology*. John Wiley & Sons. ISBN 978-1-118-28203-8. - Prove the relation between Mahalanobis distance and Leverage?