Leverage Point and Outlier Detection :: SAS/STAT(R) 12.1 User's Guide

Leverage Point and Outlier Detection

The regular variable LEVERAGE is defined as

${\mbox{LEVERAGE }} = \left\{ \begin{array}{ll} 0 & {\mbox{ if }} \mr {RD}(\mb {x}_ i) \leq C(p) \\ 1 & {\mbox{ otherwise }} \end{array} \right.$

where $C(p) = {\sqrt {\chi ^2_{p; 1-\alpha }}}$ is the cutoff value. can be set with the leverage CUTOFF= option, and $\alpha$ can be set with the leverage CUTOFFALPHA= option.

If projected robust distances are computed for a data set that has a low-dimensional structure, the default cutoff value is $C(q) = {\sqrt {\chi ^2_{q; 1-\alpha }}}$ where q is the dimensionality of the low-dimensional space. The LEVERAGE is then defined as

${\mbox{LEVERAGE }} = \left\{ \begin{array}{ll} 0 & {\mbox{ if POD}}(\mb {x}_ i)=0 {\mbox{ and PRD}}(\mb {x}_ i) \leq C(q) \\ 1 & {\mbox{ if POD}}(\mb {x}_ i)=0 {\mbox{ and PRD}}(\mb {x}_ i)>C(q) {\mbox{ (called in-plane leverage)}}\\ 1 & {\mbox{ if POD}}(\mb {x}_ i)>0 {\mbox{ (called off-plane leverage)}}\\ \end{array} \right.$

where POD is the projected off-plane distance and PRD denotes the projected robust distance. You can specify a cutoff value with the CUTOFF or the CUTOFFALPHA suboptions of the LEVERAGE option in the MODEL statement.

Residuals $r_ i, i=1,\ldots ,n$ , based on robust regression estimates are used to detect vertical outliers. The variable OUTLIER is defined as

${\mbox{OUTLIER }} = \left\{ \begin{array}{ll} 0 & {\mbox{ if }} |r_ i| \leq k{\hat\sigma } \\ 1 & {\mbox{ otherwise }} \end{array} \right.$

where ${\hat\sigma }$ is the estimated scale in the model and the multiplier of the cutoff value is specified by the CUTOFF= option in the MODEL statement. By default, k = 3.

An ODS table called Diagnostics contains the LEVERAGE and OUTLIER variables.

The ROBUSTREG Procedure

Leverage Point and Outlier Detection