Partial leverage

Partial leverage

In statistics, high-leverage points are those that are outliers with respect to the independent variables. Influential points are those that cause large changes in the parameter estimates when they are deleted. 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 of the hat matrix "H", where :H = X(X'X)^{-1}X' Partial leverage is used to measure the contribution of the individual independent variables to the leverage of each observation. That is, if "h"i is the "i"th row of the diagonal of the hat matrix, the partial leverage is a measure of how "h"i changes as a variable is added to the regression model.

The partial leverage is computed as::left(mathrm{PL}_j ight)_i = frac{left(X_{jullet [j] } ight)_i^2}{sum_{k=1}^nleft(X_{jullet [j] } ight)_k^2}where:"j" = index of independent variable:"i" = index of observation:"X"j. [j] = residuals from regressing "X"j against the remaining independent variables

Note that the partial leverage is the leverage of the "i"th point in the partial regression plot for the "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.

ee also

* Partial residual plot
* Partial regression plot
* Variance inflation factor for a multi-linear fit
* Scatter plot matrix

External links

* [http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/partleve.htm Partial Leverage Plot]

References

* cite book
title = Modern Regression Methods
author = Tom Ryan
publisher = John Wiley
year = 1997
* cite book
title = Applied Linear Statistical Models
edition = 3rd ed.
author = Neter, Wasserman, and Kunter
year = 1990
publisher = Irwin
* cite book
title = Applied Regression Analysis
edition = 3rd ed.
author = Draper and Smith
publisher = John Wiley
year = 1998
* cite book
title = Residuals and Influence in Regression
author = Cook and Weisberg
publisher = Chapman and Hall
year = 1982
* cite book
title = Regression Diagnostics
author = Belsley, Kuh, and Welsch
publisher = John Wiley
year = 1980
* cite journal
title = Efficient Computing of Regression Diagnostiocs
author = Paul Velleman
coauthor = Roy Welsch
journal = The American Statistician
month = November
year = 1981
volume = 35
number = 4
pages = 234-242


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