Inference
Imagine we keep the x’s fixed, but resample the errors. The new fit would give us some idea of the variability of the coefficients. The variability depends on the arrangement of x’s (more correlation = more change) and the $\sigma^2$ .
Testing if single variable is required
Often we want to know if a particular variable is necessary in a given model (this is not the same as asking if a variable is related to the response). We can test this by comparing the ratio of a coefficient and its standard error to a t-distribution.
Testing if any variable is required
Can test whether any variable is necessary/useful by looking at F-value (automatically generated by R).
Testing if subset is required
Often want to test if a smaller model is as good. Compare $RSS$
of the models. $F = (RSS_sub - RSS_full) /ds^2$
, where $d$
is number of variable dropped, $s^2$
estimate of $\sigma^2$
from full model. Use anova
in R to do calculations.