Page 284 - Applied Statistics with R

284 CHAPTER 13. MODEL DIAGNOSTICS

coef(model_2)[2]

## x
## -0.9507397

We will say that this point has high leverage, is not an outlier due to its small
residual, and has a very small influence.
Lastly, the added point in the third plot has a large effect on the slope, which
is now:

coef(model_3)[2]

## x
## -0.5892241

This added point is influential. It both has high leverage, and is an outlier due
to its large residual.

We’ve now mentioned three new concepts: leverage, outliers, and influential
points, each of which we will discuss in detail.

13.3.1 Leverage

A data point with high leverage, is a data point that could have a large influence
when fitting the model.
Recall that,

̂
⊤
⊤
= ( ) −1 .
Thus,

̂
⊤
⊤
̂ = = ( ) −1
Now we define,

⊤
= ( ) −1 ⊤
which we will refer to as the hat matrix. The hat matrix is used to project
onto the subspace spanned by the columns of . It is also simply known as a
projection matrix.