Partial leverage

In regression analysis, partial leverage (PL) is a measure of the contribution of the individual independent variables to the leverage of each observation. That is, if h_i is the i^th element of the diagonal of the hat matrix, PL is a measure of how h_i changes as a variable is added to the regression model. It is computed as:

${\displaystyle \left(\mathrm {PL} _{j}\right)_{i}={\frac {\left(X_{j\bullet [j]}\right)_{i}^{2}}{\sum _{k=1}^{n}\left(X_{j\bullet [j]}\right)_{k}^{2}}}}$

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.

In statistics, high-leverage points are those that are outliers with respect to the independent variables. In other words, high-leverage points have no neighbouring points in ${\displaystyle \mathbb {R} ^{p}}$ space, where p is the number of independent variables in a regression model. This makes the fitted model likely to pass close to a high leverage observation. Hence high-leverage points have the potential to cause large changes in the parameter estimates when they are deleted—i.e., to be influential points. 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, which is

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

See also

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

References

• Tom Ryan (1997). Modern Regression Methods. John Wiley.
• Neter, Wasserman, and Kunter (1990). Applied Linear Statistical Models (3rd ed.). Irwin.CS1 maint: multiple names: authors list (link)
• 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.CS1 maint: multiple names: authors list (link)
• 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.

External links

• Partial Leverage Plot, Dataplot manual, Statistical Engineering Division, National Institute of Standards and Technology

 This article incorporates public domain material from the National Institute of Standards and Technology website https://www.nist.gov.