Partial leverage

In statistics, high-leverage points are those that are outliers with respect to the independent variables. Leverage 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

$H = X(X'X)^{-1}X'. \,$

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:

$\left(\mathrm{PL}_j\right)_i = \frac{\left(X_{j\bullet[j]}\right)_i^2}{\sum_{k=1}^n\left(X_{j\bullet[j]}\right)_k^2}$

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.

See also

• Partial residual plot
• Partial regression plot
• Variance inflation factor for a multi-linear fit
• Scatterplot matrix

External links

• Partial Leverage Plot

References

• Tom Ryan (1997). Modern Regression Methods. John Wiley. 
• Neter, Wasserman, and Kunter (1990). Applied Linear Statistical Models (3rd ed.). Irwin. 
• Draper and Smith (1998). Applied Regression Analysis (3rd ed.). John Wiley. 
• Cook and Weisberg (1982). Residuals and Influence in Regression. Chapman and Hall. 
• Belsley, Kuh, and Welsch (1980). Regression Diagnostics. John Wiley. 
• Paul Velleman; Roy Welsch (November 1981). "Efficient Computing of Regression Diagnostiocs". The American Statistician (American Statistical Association) 35 (4): 234–242. doi:10.2307/2683296. JSTOR 2683296. 

 This article incorporates public domain material from websites or documents of the National Institute of Standards and Technology.