What Leads to Winning? Model Building Part 2

This post follows directly from my last; you should go read that now.  In terms of improving our NFL winning model, since I already mentioned it, let’s start with opponent’s penalty rate.  If I add that to the model I started with (the same one that Brian reports), I move up to an R-squared of .777.  That’s higher than the .76 I started with, but is it higher enough?  That question is, in essence, what statistics was invented to answer, and something usually ignored by many people.  If one number is higher than another, then by golly it’s better, right?  It isn’t always true.  In the case of R squared, it is impossible to get a lower number when you add a variable; it can only stay the same or go up.  And if it doesn’t go up much, then we say that the new variable doesn’t significantly explain more than our previous model.  Maybe it explains an extra half of a percent of the variance, but that isn’t really ‘enough’ in the statistical sense.  We test this by running an ANOVA between the two models (at least that’s one way to do it, and how I do it).  In the case of opponent’s penalty rate, the added explanatory power is indeed significant; this is supported by looking at the significance value for the opponent penalty rate beta value, which is very small.  Also, it isn’t necessary but it’s nice to see that it’s roughly the opposite of the team’s penalty rate, -.456 versus .47, meaning that a penalty on the other team helps as much as a penalty to my team hurts.  Symmetry isn’t necessary, but it seems plausible here.