# A Regression Puzzle

Here’s a regression puzzle courtesy of Advanced NFL Stats from a few years ago and pointed to recently by Holden Karnofsky from his interesting new blog, ColdTakes. The nominal issue is how to figure our whether Aaron Rodgers is underpaid or overpaid given data on salaries and expected points added per game. **Assume that these are the right stats and correctly calculated.** The real issue is which is the best graph to answer this question:

Brian 1:…just look at this super scatterplot I made of all veteran/free-agent QBs. The chart plots Expected Points Added (EPA) per Game versus adjusted salary cap hit. Both measures are averaged over the veteran periods of each player’s contracts. I added an Ordinary Least Squares (OLS) best-fit regression line to illustrate my point (r=0.46, p=0.002).Rodgers’ production, measured by his career average Expected Points Added (EPA) per game is far higher than the trend line says would be worth his $21M/yr cost. The vertical distance between his new contract numbers, $21M/yr and about 11 EPA/G illustrates the surplus performance the Packers will likely get from Rodgers.

According to this analysis, Rodgers would be worth something like $25M or more per season. If we extend his 11 EPA/G number horizontally to the right, it would intercept the trend line at $25M. He’s literally

off the chart.

Brian 2:Brian, you ignorant slut. Aaron Rodgers can’t possibly be worth that much money….I’ve made my own scatterplot and regression. Using theexact same methodologyandexact same data, I’ve plotted average adjusted cap hit versus EPA/G. The only difference from your chart above is that I swapped the vertical and horizontal axes. Even the correlation and significance are exactly the same.As you can see, you idiot, Rodgers’ new contract is about twice as expensive as it should be. The value of an 11 EPA/yr QB should be about $10M.

Ok, so which is the best graph for answering this question? Show your work. Bonus points: What is the other graph useful for? Holden points to Phil Birnbaum’s helpful analysis in the comments.

More puzzles at cold-takes.