Page 285 - Applied Statistics with R

13.3. UNUSUAL OBSERVATIONS 285

The hat matrix, is a matrix that takes the original values, and adds a hat!

̂ =

The diagonal elements of this matrix are called the leverages

= ℎ ,

where ℎ is the leverage for the th observation.

Large values of ℎ indicate extreme values in , which may influence regression.

Note that leverages only depend on .
Here, the number of s is also the trace (and rank) of the hat matrix.

∑ ℎ =

=1
What is a value of ℎ that would be considered large? There is no exact answer

to this question. A common heuristic would be to compare each leverage to
two times the average leverage. A leverage larger than this is considered an
observation to be aware of. That is, if

ℎ > 2ℎ ̄

we say that observation has large leverage. Here,

∑ ℎ
̄
ℎ = =1 = .

For simple linear regression, the leverage for each point is given by

1 ( − ̄ ) 2
ℎ = + .

This expression should be familiar. (Think back to inference for SLR.) It sug-
gests that the large leverages occur when values are far from their mean.

Recall that the regression goes through the point ( ̄ , ̄).
There are multiple ways to find leverages in R.