Payroll vs. wins -- still significant in 2006

The “Wages of Wins” blog approvingly quotes a news story from the Southern Ledger arguing a weak relationship between MLB payroll and performance.

First, the news story. Its prime exhibit is a chart of “dollars spent per win,” which is exactly what it sounds like: payroll divided by wins. It shows the Marlins leading, at $220,000 per win (as of this week), and the Yankees trailing, at $2.4 million per win. Ranked by dollars per win, the chart seems a nearly-random mix of good teams and poor teams, which writer Bill Freehling uses to imply that salaries don’t matter all that much.

However, dollars per win is a poor measure of payroll efficiency, because the relationship between pay and performance isn't linear. A team could, in theory, have 25 players making the minimum $327,000, which would make its payroll about $8.2 million. That team could probably still win a few games. Even if it only won 20 games out of 140, that would still be only $410,000 per win, which would appear to lead the league.

For the Yankees to match that kind of efficiency with their $195 million payroll, they’d have to win 475 games out of 140. That would be difficult, even if Derek Jeter's defense improved substantially.

So, clearly, the relationship isn’t linear. Some have suggested “marginal dollars per marginal win” as an appropriate measure. Even that one isn’t perfect, because of diminishing returns on increasingly high salaries. But at least it’s better. In any case, the chart isn’t very informative.

Freehling is on more solid ground when he points out that the teams in “the top-third [of payroll] spent about 117 percent more than the bottom-third but had won just 14 percent more games.”

But even that statistic doesn’t mean much out of context. Is 14% a lot? A little? What about 117%? What kind of relationship do these numbers imply, and do they really mean that salary doesn’t matter much?

14% more wins, in a 162-game season, is about 11 games in the standings. That’s a fair bit of difference between the high-spending teams and the low-spending teams. Not as much as one would think, perhaps, but it’s still 1.5 standard deviations from equal – a low-payroll team has a long way to go to catch up to the richer clubs.

Second, the blog entry itself. David Berri uses the article to confirm “The Wages of Wins’” claim that payroll is not an economically significant factor in predicting wins, because the r-squared of the relationship is only 17%. (As I argued here, the important number is the r, not the r-squared. An r-squared of 17% is an r of 41%, which implies that, in a certain sense, payroll actually explains 41% of wins.)

Berri notes that so far in 2006, the r-squared is 24%, which is statistically significant but doesn't carry much "oomph."

But again, I argue that would make the r around 49%, which is pretty significant.

How signficant? Here’s another way of looking at it.

Start by observing that payroll doesn’t buy wins directly – it can only buy talent. Do a “what if.” Suppose that GMs could evaluate talent perfectly, and always spent exactly the correct amount for the player’s ability. Suppose we also were able to find a function to translate increased payroll into increased ability, whatever that function might be.

In that case, the correlation between salary and wins would be exactly the same as the correlation between ability and wins. So what would the correlation be between ability and wins?

It wouldn't be 100%, because teams don’t always win exactly in accordance with their ability. Just as a fair coin might have more or less than 50% heads, just by luck, a .500 team might go 90-72 or 77-85 just by chance alone. So r would be less than 1.

How much less? Tangotiger tells us here (see comments) that the variance of team ability is .06^2. Based on that, I ran a simulation, and came up with a wins/talent correlation around .83.

(This number happens to be SD(ability)/SD(wins) … which might not be a coincidence. A result from Tangotiger tells us that var(ability)/var(wins) is the correlation between two independent sets of wins with the same ability distribution … this might be a variation of that result.)

So even if payroll completely determined talent, the best we could get would be an r of 0.83.

Stated in point form:

--If payroll had no relationship to talent, we’d get r=0.
--If payroll had 100% relationship to talent, we’d get r=0.83.
--For 2006, we actually get r=0.49.

Or, put into baseball terms:

-- Suppose there were a perfect relationship between payroll and ability. You’d find that a one SD increase in payroll led to a .83 SD increase in wins.

--In real life, a one SD increase in payroll leads to a .49 SD increase in wins.

All things considered, the relationship between payroll and wins seems pretty oomphy to me.