Number of models

At this point, it is clear that including more possible explanatory variables into this model, we need to then fit even more models. For example, including just 4 explanatory variables meant that we had to fit 15 models. To see this, consider the total number of possible variable combinations.

\(\newline\) 4 variables: \[{4\choose 1}+ {4 \choose 2} + {4\choose 3} + {4 \choose 4}=15\]

\(\newline\) 5 variables: \[{5\choose 1}+ {5 \choose 2} + {5\choose 3} + {5 \choose 4}+ {5 \choose 5}=31\] Therefore, if we had 5 variables instead of 4 then we would had to have fit more than twice as many models (15 in comparison to 31).

With 20 variables, we are already in excess of 1 million possible combinations of variables.