Correlating Sounder Player Ratings and Salary

Last week we took a look at the aggregated Sounders player ratings for the season available from a few different sites. With all of the necessary caveats about the data being subjective and inconsistently available, the results were still decent. Other than some frequent reserves whose ratings were noticeably inflated from playing a large number of games against other reserves, the data fit with what I'd call the wisdom of the crowds regarding the play of the Rave Green over the course of the season. And the fact that some noticeably low-rated players received less and less playing time over the course of the season — and vice versa — suggests that the results match up with what Sigi thinks, as well.

This week I want to correlate those ratings to player salary to see if we can find any relationship, and if so what it might say about the return on investment we're getting on the roster. Since none of our sources have added any ratings for the games since we last updated — including the previous Chivas game, the midweek Saprissa game, or this weekend' game at Houston — our final aggregates are unchanged (and incidentally, it's easy to complain when I'm not doing the work, but come on Prost and Josh. Where's the ratings love?).

Before looking at the data, we should recognize that given the small number of data points we're looking at, we're not expecting great results. And in fact I'm making the population even smaller by kicking out players that accumulated a big chunk of their ratings against reserves (this includes Wahl, Montano, Scott, Seamon, Boss, and Estrada). On the scale of relevance of statistical analysis, this belongs somewhere between "nonsense" and "toy". Our hope would be that there's a positive correlation (that is, that players that make more money are producing better-rated performances) and that there's some reasonable goodness of fit (that is, that the variation in salaries has a significant relation to the variance in performances. Or in practical terms, that the results are closely clustered to the best fit line). Also, my hypothesis going in would be that the relationship would be a low-order polynomial (quadratic or cubic) meaning that salaries would go up at a higher rate as better performances are sought. That is to say, you might pay a certain amount to improve from a player who averages 5.5 to a player who averages 6.5. But you'd need to pay more to move from a 6.5 to a 7.5.

So if we do a little Excel regression and calculate the best fit line, what do we get? Well, I was pleasantly surprised by the results:

For those who don't do a lot of linear regression, this is a two-dimensional scatter plot, in which you assign an axis to each of two variables (in this case the y-axis is salary and the x-axis is our season-average player rating). Once you set one up, you can then have some software attempt to find a 'best fit line', which is a line that most closely relates the two axes according to the data we have. Note that even though I hypothesized that the relationship between ratings and salaries would be polynomial, in fact I just went with a straight line. That's because — while I did in fact get a better-fitting line using a quadratic function — the line got weird toward the low end of the salary, because we have a few low-rated players making more money than you'd expect (thanks, Pete). It taught the algorithm that you actually pay more for really poor players, and while that might be how some other franchises operate, we try to avoid that here. And the fit with a straight line was decent. But it does mean that it will recommend a salary for better players that I think is lower than a more appropriate function would give (see Nkufo, for example).

With that out of the way, how's it look? Well, the correlation is positive, which we'd hoped for. And an R-squared of 0.36 isn't half bad given what we have to work with. So out best fit formula is around "y=153x - 755" meaning essentially that for every rating point a player gives you, you should pay them 153K, and then subtract out 755K. This means that at a certain lower threshold (around 5) it'll recommend that you actually pay someone negative money (or that they pay you), but we can just floor anything below 40K to that, since it's the league minimum. And then grumble that terrible players really should be paying us back.

Players above the line are those that the formula suggests are being overpaid. Players below are those that are being underpaid. Remember that I think the line underestimates the recommended salary for higher-rated players, so don't worry too much about Nkufo. On the other hand, it understates the degree to which Alonso is underpaid.

So here's the Sounders' roster of seniorish players (meaning minus the players I pulled out for having ratings too high for playing against scrubs), including their average rating, current salary (the total, not the cap hit) in thousands, what their 'recommended' salary would be under our formula (floored at 40K), and what the difference is.

Any surprises there? Think Montero would look more underpaid if we weighted towards more recent results? Should Keller get the Nkufo caveat about the line shape, or does 177K seem about right? Disagree that Fernandez is the most overpaid guy on the team (hard to argue when we're paying 300K for a sub, but presumably he's got a more significant role in the club's future plans)? Does the Fucito-vs-Nyassi question still look unresolved?