By Andrew Mooney, Boston.com Correspondent

It’s hard to know where the problems with this year’s Celtics begin and end. Defense has been lacking, the new acquisitions are underperforming, and above it all hangs the specter of age, gradually taking its toll on the tired legs of Paul Pierce and Kevin Garnett. Over a third of the way through the season, the Celtics sit below .500, and if they hope to turn things around, they need to identify the sources of their problems now.

A good place to start the evaluation is with their personnel. There is a simple way to measure player value that looks beyond traditional box score statistics: taking the difference between how a team performs with a player on the court and off the court, or plus-minus.

Of itself, this is somewhat of a crude metric, as it doesn’t factor in game situation or who else is on the court. That’s where adjusted plus-minus—the best holistic measure of NBA player performance that advanced stats has to offer—comes in. APM accounts for teammate and opponent strength, as well as the pace at which a team and its opponents play. It’s not meant to be applied to individual games and is most accurate over the course of two or three years, but over the span of a season, APM can give a reasonable estimate of what people want to measure most of all: who helps a team win and lose games, encompassing offense, defense, and even intangibles. (For a more complete discussion of APM, its calculation, and its flaws, click here.)

Jeremias Engelmann compiles and regularizes these numbers for every team in the NBA, and I’ve listed the data for the 2012-13 Celtics below, limiting it to those players who have played 20 percent of the team’s total minutes. For the purposes of interpretation, each number signifies the amount of points per 100 possessions a player adds or takes away relative to a league-average player (APM = 0).

The names at the top are no surprise—one doesn’t need an advanced metric to tell you Kevin Garnett, Rajon Rondo, and Paul Pierce are the Celtics’ three best players. What’s interesting, though, is just how much better Garnett has been than the rest of his team; the gap between him and second-place Rondo is larger than the difference between fourth-place Courtney Lee and tenth-place Leandro Barbosa.

Garnett ranks seventh in the league in APM, above All-NBA types like Kobe Bryant, Russell Westbrook, and Carmelo Anthony. His contributions have come mostly on the defensive end, which becomes apparent every time he’s on the bench; the Celtics have allowed 12 more points per 100 possessions with KG off the court than on it.

The performance of his supporting cast has been…underwhelming. Jeff Green and Brandon Bass, both thought to be potentially key contributors before the season, have been among the least valuable players on the roster.

Things may improve with the return of Avery Bradley, but not sufficiently for the Celtics to make a serious playoff push unless the rest of the team shows quick and drastic improvement. Granted, it was through about this many games last year that people were proclaiming a premature end to the current era of Celtics basketball. We’ll see if this team is capable of a similar late-season turnaround.

By Andrew Mooney, Boston.com Correspondent

A quick recap: in my previous two posts, I’ve done some digging into the recent history of the MLB free agent market, investigating how teams have priced an additional win this year and in years past. Comparing my work with similar research done by David Cameron of FanGraphs, I thought it possible that teams had reached an equilibrium in the way they valued free agents; the rate at which teams paid for an additional WAR found by Cameron for 2008 was essentially the same as what I found for 2012.

To check my findings, I repeated the analysis I did for the 2012 market for 2009-11. Below, I’ve shown both my results (in red) and the figures Cameron arrived at for 2002-08 (in blue).

The proper reaction to the drop-off between the end of Cameron’s analysis and the beginning of mine is the raise of a skeptical eyebrow; even the financial crisis of 2008 wasn’t enough to precipitate that sort of change. If the climate of the MLB free agent market had shifted that dramatically over the course of one year, we wouldn’t need a blog post four years after the fact to notice it.

I thought I had replicated Cameron’s methods fairly well, but our respective results suggest otherwise. The only way I can think of in which Cameron’s work differs from mine is that he says he uses a weighted average of a player’s past three seasons of WAR, instead of just a straight mean, though he doesn’t say exactly what those weights are. With that information, I might have been able to hit the target a little more closely.

Still, my work does corroborate one of Cameron’s findings: from year to year, the market rate paid to free agents is rising, perhaps even faster than it has in the past. From 2009-12, I found an annual average increase of 22 percent in dollars paid per WAR, over twice what Cameron derived for 2002-08. It would appear, then, that the existence of the equilibrium I suggested last week was nothing more than a mirage. Though Cameron and I started from different places methodologically, we both found that, for free agents, the rich have indeed been getting richer.

By Andrew Mooney, Boston.com Correspondent

Last week, I wrote a post attempting to put a price tag on what MLB teams are willing to pay for an additional win in free agency. In 2009, Dave Cameron of FanGraphs conducted the same sort of analysis, with a little bit more rigor than I used. Instead of using just one year, Cameron averaged the WAR from a free agent’s past three seasons, then multiplied by 0.95 to correct for age-related decline to arrive at a team’s “expectation” for the value he would produce the following season.

The dollars per win figures that Cameron arrived at for 2002-2008 can be found in his article linked above. As Cameron notes, each successive year brought with it about a ten percent increase in the market rate teams have paid for an additional win, topping out at $4.5 million in ’08.

But, combining this with the findings of my previous post, it appears the price of a win on the free agent market has remained essentially the same since 2008; I found that an additional WAR cost teams about $4.6 million on average in this year's Hot Stove season.

To be sure, I redid the analysis using Cameron’s method—three years of WAR and a correction for aging—and came to roughly the same result: as of last week, teams paid $453 million for an expected 95.5 WAR in 2013, which comes out to $4.75 million per win. The prevailing market rate has risen just 5.6 percent in the last four years, a marked contrast to the ten percent annual increases in the seven years that preceded them.

What does this mean exactly? Has the free agent market reached an equilibrium of sorts, in which teams feel they have efficiently priced free agent talent, and annual contract sizes rise only with inflation? This seems hard to believe in a world where Anibal Sanchez scores a five-year, $80 million deal and the move is regarded as sensible, but I suppose it wasn’t so long ago that the same was said for John Lackey.

But blockbuster deals like those received by Albert Pujols and Prince Fielder are outliers, at the extreme end of the distribution. The slowdown in contract inflation might say less about the biggest deals given out each year and more about those signed by mid-level free agents, who make up the majority of each free agent class. It might be that teams have regained a little sanity in the way they evaluate the Melky Cabreras and Russell Martins of the world.

Of course, there may be other factors at work here that don’t show up in simple dollars and cents; perhaps players have started bargaining for more years instead of more dollars recently, or some other confounding factor has caused wages to stop rising so steeply.

The answer to this question might be clearer after investigating the free agent markets from 2009, 2010, and 2011, but at least one source suggests that last year’s free agent pool was paid similarly to this year’s. In my next post, I’ll dig a little deeper into the past few years to see if the trend has remained steady.

By Andrew Mooney, Boston.com Correspondent

With the signings of Josh Hamilton and Zack Greinke last week, the Hot Stove reached its boiling point. Another year, another awe-inspiring spectacle of money being thrown around by front offices desperate for a new toy.

The cash hemorrhages we see every winter for top free agents leads me to wonder if teams ever hope to accrue back the value of a contract like Hamilton’s or Greinke’s. But for every massive free agent contract, there are many more sensible, well-negotiated moves that are not subject to the superstar premium. This fact leads to the broader question: how do MLB teams price the free agents they sign?

For an easy, comparable measure of value, I’ll use WAR. This allows me to assess position players and pitchers together, so I can come to a total rate that teams pay free agents for their per-unit value.

I compiled a list of all the free agents to sign with an MLB team as of Tuesday night, then recorded their WAR from the 2012 season, which I used as a projection for how we might expect them to perform in 2013. Obviously, this is a quick and dirty method for predicting performance—it doesn’t incorporate age or injury history, for one thing—but recent form is essentially what front offices are paying for. I am not suggesting that every player is going to exactly replicate his play from a year ago, but rather that this is the primary criterion by which a team and a player settle on a contract.

Next, I logged the per-year dollar average from each player’s new contract—essentially, what his team will be paying him in 2013. With these pieces of information, I was now able to estimate the going market rate per WAR.

Thus far, teams have allocated about $453 million in 2013 for an expected 98.7 WAR, which comes out to $4.6 million per WAR. The average per-WAR cost for last year’s free agent class was $4.5 million, so the market has remained essentially the same. In fact, assuming two percent inflation from one year to the next, the figures are exactly the same, with the extra $0.1 million simply a product of the general rise in the price level.

For their part, the Red Sox have not exactly been buying low this offseason. The six players the team signed this offseason project to produce 13.7 WAR, at a cost of $116.75 million in 2013. That amounts to $8.5 million per WAR, or nearly double the rate of the rest of the league. This number is a bit inflated, given that the Red Sox are banking on more productive campaigns in 2013 from Mike Napoli and Stephen Drew, but the front office certainly wasn’t playing hardball during its negotiations.

With most of the big free agent names now off the market, this number is not likely to change much in the coming weeks. The Sox will have to depend on the notion that they rightly overpaid for premium performance, something that hasn’t quite held up for the past few free agent classes to come to Fenway.

By Andrew Mooney, Boston.com Correspondent

For the people moaning about how Manti Te’o’s Heisman snub proves the award is exclusively an offensive one, you’re right—not in that Te’o should have won it, but that it goes to the guys who put points on the board. In the 78-year history of the Heisman trophy, Charles Woodson is the only primarily defensive player to win it, and even he got a boost from his time on offense and special teams, scoring four touchdowns in his Heisman-winning campaign.

What interests me more than the debate over the justice of this fact is the specific type of offensive player who wins the award. We all know that running backs and quarterbacks have monopolized the voting, but who has ruled the game when? Inspired by this tweet from Smart Football’s Chris Brown, I decided to visualize the positional trends in Heisman history to see which were the eras of the running back and which were dominated by quarterbacks.

For the first 50 years the Heisman was awarded, halfbacks and running backs ruled the college landscape, collecting the trophy 35 times (70 percent). With an elite back, a team’s running game was still the focus of its offense, as legendary names like Archie Griffin, Earl Campbell, Billy Sims, and Herschel Walker stood at the forefront of the game.

Since that time, however, quarterbacks have begun to assert themselves as more worthy of the nation’s “most outstanding player” designation, mirroring the shift in the sport itself. Assisted by the proliferation of pro-style and spread offenses through the college game, 17 out of the 28 winners since 1985 have been quarterbacks (60 percent), including 11 out of the past 13.

The running game certainly still has a place in college football—just ask Oregon—but it’s hard for voters to ignore the gaudy stats being put up by quarterbacks in spread offenses, who can chew up the yards of a running back on the ground in addition to their skill through the air. Troy Smith, Tim Tebow, Cam Newton, Robert Griffin III, and now Johnny Manziel have benefitted from this type of attack, all in the last seven years. For the time being, the quarterback is ascendant in Heisman voting, though it is anyone’s guess as to what kind of shift the next revolution in the sport will cause.