"The Wages of Wins" on r and r-squared
In "The Wages of Wins," the authors regressed wins against salary, found an r-squared of .18, and concluded that, because .18 is low, there is a very weak relationship between payroll and performance, and therefore "teams can’t buy the fan’s love."
I have posted a few times (like here) disagreeing, and arguing that the “r” gives you more useful information than the r-squared. In that regression, the r is about .42, which is high enough to be significant in a baseball sense.
Author Stacey L. Brook disagrees. In a post on the authors’ blog today, he writes,
"Recently, some individuals who claim to have knowledge about statistics have questioned [our] conclusion. Specifically … these individuals have suggested that using the correlation coefficient – otherwise known as r – is a more “real-life” statistic to use in looking at how payroll and wins are related in Major League Baseball. As you can guess, we disagree."
(By the way, I'm not sure Dr. Brook is not necessarily addressing his post to my argument specifically. For all I know, it's someone else entirely. There’s no link in his post and his use of the plural – "some individuals" – suggests it's more than just one person.)
Why do Dr. Brook and his colleagues disagree? They say that using r "exaggerates the relationship" between the two variables. They quote a professor who agrees. They say that r-squared says that "18% of the variance" in performance is explained by salary, and the percentage of variance is the appropriate measurement to consider.
The last claim sounds reasonable, until you realize that "variance" is not being used in the normal English sense of the word. It's a technical, statistical term meaning the square of the standard deviation.
Variances are unintuitive. If the standard deviation of weight is 30 pounds, the variance of weight is 900 square pounds. If the standard deviation of professors' salaries is $10,000, the variance of professors' salaries is 100 million square dollars. And if the standard deviation of team wins is 11.6 wins, the variance of team wins is 136 square wins.
The 18% makes sense only in context of the squares of what you're actually trying to measure. If salary explains 42% of performance, then salary explains 18% of performance squared. But we sabermetricians don't care about what performance squared; we care about performance. And that's why the .42 is more meaningful than the .18.
In the early 20th century, economist Alfred Marshall famously explained how to study economics:
"(1) Use mathematics as shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics."
If we concentrate on the r = 0.42, we can follow Marshall's advice. Translating into English, and using an example that's important in real life, we can say,
"If a team spends one extra standard deviation (in 2006, about $25 million) in salaries, it should have expect to improve its performance by 0.42 standard deviation of wins (about 4.5 wins)."
If you want to burn even more of the mathematics, and you make a few additional assumptions (for instance, that wins and salary are both normally distributed), then I think you can even say,
"If a team becomes the Nth highest-spending team in the league, it will, on average, be 42% as many wins above or below .500 as the Nth winningest team in the league."
That last sentence follows Marshall's prescription; it has no math, it's significant in real life, it's in English, and it's understandable to any GM, whether he knows statistics or not.
And it's based on the r.