Page 262 - Applied Statistics with R

262 CHAPTER 13. MODEL DIAGNOSTICS

̂
E[ ] = ,

and variance

̂
⊤
2
Var[ ] = ( ) −1 .
̂
In particular, an individual parameter, say had a normal distribution

̂
2
∼ ( , )

where was the matrix defined as
−1
⊤
= ( ) .
We then used this fact to define

̂
−

√ ∼ − ,

which we used to perform hypothesis testing.
2
So far we have looked at various metrics such as RMSE, RSE and to deter-
mine how well our model fit our data. Each of these in some way considers the
expression

2
∑( − ̂ ) .

=1
So, essentially each of these looks at how close the data points are to the model.
However is that all we care about?

• It could be that the errors are made in a systematic way, which means
that our model is misspecified. We may need additional interaction terms,
or polynomial terms which we will see later.
• It is also possible that at a particular set of predictor values, the errors are
very small, but at a different set of predictor values, the errors are large.
• Perhaps most of the errors are very small, but some are very large. This
would suggest that the errors do not follow a normal distribution.