One More Nail In the Anti-Per Minute Argument’s Coffin?

One of the core tenets of basketball statistical analysis is the usage of per minute stats. When compared to per game stats, per minute stats are highly valuable in the evaluation of individuals. This is because per minute stats puts players of varying playing time on the same level. Using per game stats, starters will always dwarf bench players due to the extended time they get to accumulate various stats. Meanwhile per-minute stats allows to compare players independent of minutes, allowing for a more even approach in player evaluation.

Recently a debate has come up on the validity and usefulness of per minute stats. I’ve quoted the main parts below, but even abbreviated it’s a long read. If you have the time, I suggest reading it now so the rest of this article will make more sense. For those on a limited time constraint, a quicker summary is here:

Hollinger & Kubatko: “Hey per minute stats are a great way to evaluate players! In fact we’ve done a few studies and it seems that a player’s per minute stats increase slightly when they get more minutes. At the worst we can conclude that they should stay relatively the same.”

FreeDarko: “Per minute stats won’t stay the same if a player gets more minutes, because there is a division between greater and lesser players. A player that only gets 10-25 minutes per game is playing against lesser caliber players. Hence when that player sees an increase in playing time, he’s playing against steeper competition, so his stats should decrease.”

Tom Ziller: “That’s not true. Here is every 10-25 minute player in the last 10 years that saw an increase in minutes. Most of them (70%) saw an increase in per-minute production. To discount any of this data being from young players getting better as they age, I looked at 8+ year vets, and saw that about the same ratio of players increased (69%).

Brian M.: “Tom, the problem with all this data is a causality vs. correlation issue. It’s possible that these players saw more minutes first then improved. But it’s also possible that these players improved first which allowed their coach to play them more minutes.”

Brian’s case is a good one. To use an analogy, imagine I come across a person who calls himself Merlin Appleseed. He claims that just by touching apples he can magically make them taste better. He opens up a box of apples saying that he never touched any of them. He picks out 10, and imbues them with his magic. He asks me to taste each of them. I find all of them to be delicious. He says “here’s the same box I got my apples from. Now I want you to take 10 at random while blindfolded. You can compare them to my magic apples. I bet mine taste better.” I do just as he asks, and indeed my random set of apples are less tasty than his. So does Merlin Appleseed have magical power?

Maybe. Unfortunately this test wouldn’t be able to confirm or deny his magical power. Since Merlin gets to choose his apples, he might be selecting the best ones! To test Merlin’s abilities I would need something to gauge how good his apples are expected to taste. One way to do this would be to find comparable apples that have the same color, size, blemishes, etc. Then I can compare the taste of his apples to my apples. If Merlin’s has the magical powers he claims, then his apples will taste better than my apples.

Similarly with Tom’s study, Brian is saying that by selecting players who have seen an increase in minutes we might be choosing the best apples. This is because players who improve on a per minute basis could be given more playing time by their coaches. Therefore to show whether or not these players have improved, I need to find how good they’re expected to be. Then I can compare their actual performance to their expected performance. If FreeDarko’s theory is true, that role players should decrease their per minute production with more minutes, then they should perform worse than their expected values.

To separate the control group from the test group, I’ll only use players with an even numbered age for the control, and odd numbered ages for the test group. Since this study is intended for role players, which was defined by Ziller, I limited my control group to player seasons where:
* The player age was an even number.
* The player appeared in 41 games or more.
* The season was 1981 or greater.
* The player averaged 10-25 mpg.

Now I can calculate the expected production of the players in my group, by looking at per minute production (PER) over playing time (mpg).

Control Group

Just as expected, the graph tends to go from the bottom left (low production = low minutes) to the top right (high production = high minutes). That is players who receive more minutes are more productive. From the 1840 player-seasons in my data, I’m able to calculate the expected PER based on mpg (PER = .2158*mpg + 8.2941). So if a player averaged 10 mpg, you would expect his PER to be 10.45. This equation is represented by the red line on the graph.

Now that our control group is defined, I need to create the test group. Again this group was defined by Ziller as role players who saw an increase in minutes. I selected player seasons where:
* The player’s age was an odd number.
* The player appeared in 41 games or more.
* The season was 1981 or greater.
* The player averaged 10-25 mpg the year before.
* The player increased his mpg by 5+ from the year before.

Since I have the expected values based on mpg, all that is left is to compare their actual production to the control group. In our test group 185 players did better than their expected PER, while 177 did worse. On average each player gained 0.17 PER. This is a tiny gain, not enough to show that players increase production with more minutes. However it clearly shows that they didn’t decline and at least matched the predicted PER.

Another way to see how our prediction did is to calculate the regression (trendline) of this group, and compare it to the expected equation. The red line in the graph below shows the regression of PER/MPG for our control group.

Test Group

* Control: PER = .2158*mpg + 8.2941
* Test: PER = .2185*mpg + 8.3917

The test group, which has both the higher slope and y-intercept, will slightly outperform the control group. But not by much. The average player who saw 40 mpg, will see a .20 increase in PER, which is negligible. In other words, the test group has neither exceeded nor fallen short of our expectations, but rather has met them.

In the end what does this prove? Specifically this study removes the correlation between the role player group and players that saw extra minutes due to improvement. It debunks the thought that there is some kind of division between per minute stats, where the per minute stats of high minute players are more a representation of actual talent than those who play few minutes per game. But combined with the past works of Hollinger, Kubakto, and Ziller, among others, it makes an overall stronger statement. Players who receive 10 or more minutes per game are likely to keep the same per minute stats no matter what the increase in playing time is. Therefore per minute stats remains far superior to per game stats in terms of comparing and evaluating players.


  • “It’s a pretty simple concept, but one that has largely escaped most NBA front offices: The idea that what a player does on a per-minute basis is far more important than his per-game stats. The latter tend to be influenced more by playing time than by quality of play, yet remain the most common metric of player performance.” — John Hollinger
  • The great thing about this study is that I can perform it again, this time using the “odd” aged players as the control and the “even” aged players as the test group. This time the prediction equation was PER = .2039*mpg + 8.4439. And again our test players slightly outperformed the average. This time 192 did better than their expected PER, while only 161 did worse. On average each player gained 0.23 PER.
  • This article doesn’t mean that every player that has good per minute stats should see more playing time. It’s very clear that basketball stats don’t capture a player’s total ability. A player that does well on a per minute basis may have other flaws, such as poor defense, which prevent him from contributing more. This also isn’t an endorsement for any single per minute ranking system, like PER, WOW, etc. There are flaws in each of these in addition to being unable to account for attributes not captured in box scores.
  • Summary of the events that led to this article.

Back in 2005, I wrote an article outlining some of the pioneers in per minute research.

In the 2002 Pro Basketball Prospectus John Hollinger asked and answered the question ?Do players do better with more minutes?? For every Washington player, Hollinger looked at each game and separated the stats on whether or not he played more than 15 minutes. He found that when players played more than 15 minutes, they performed significantly better than when they played less. To check his work, he used a control group of 10 random players, and each one of those improved significantly as well.

The knock on Hollinger?s study is the small sample size, containing less than 25 guys from only one season. Enter Justin Kubatko, the site administrator of the NBA?s best historical stat page Earlier this week Justin decided to re-examine the theory using a bigger sample size. Taking players from 1978-2004, he identified 465 that played at least a half season and saw a 50% increase in minutes the year after. Three out of four players saw an increase in their numbers as they gained more minutes, although the average increase was small (+1.5 PER).

Two independent studies have shown that NBA players get better when they get more minutes. A conservative interpretation is that per-minute numbers are universal regardless of playing time. So if a player averages 18 points per 40 minutes, he?ll do about that regardless of how many minutes he plays. A more liberal summary would say that underused players will see an improvement in their per-minute numbers if given more court time. A player that only averages 20 minutes a game is likely to be a little bit better if given 35. So the straight dope is per minute stats are a fantastic way to evaluate NBA players.

Recently, this research was questioned by the writers of freedarko.

The problem with this line of reasoning is that it assumes the homogeneity of court time. It assumes that if a player scored 20 points in 20 minutes, he would also score 40 points in 40 minutes. That there will by systematic differences between these two situations is almost too obvious to point out. It’s the difference between sharing the ball with Jordan Farmar while being guarded by Kenny Thomas, and sharing the ball with Kobe Bryant while being guarded by Ron Artest.

Insofar as the problem here is one of rotation, small-scale adjustments in minutes played shouldn’t create major distortions (it isn’t unrealistic to think that if Tim Duncan played 5 extra minutes per game, his per-minute production, as influenced by the level defense he’d face, would basically be the same). But when PER catapults bench players into the starting five (or vice-versa), be on the look-out for inflation. Call this the Silverbird-Shoals Hypothesis, or the THEOREM OF INTERTEMPORAL HETEROGENEITY (TOIH).

Enter Sactown Royalty’s Tom Ziller, to refute Free Darko’s theory.

Shoals and Silverbird are arguing that because low-minutes high-PER guys typically play against fellow bench players, their PER is higher than it would be if they played starter minutes. They aren’t arguing (as some surmised) that PER is useless, just that it is prone to inflation. The argument, from seemingly everyone on the ‘anti per-minute statistics’ side, is that if you increase a player’s minutes, his efficiency will suffer.

There’s a problem with this oft-repeated claim: It’s not true.

Thanks to the data-collection efforts of Ballhype’s own Jason Gurney, I’m going to try to ensure this claim never gets stated as fact ever again. Using seasons from 1997-98 to the present, we identified all players whom played at least 45 games in two consecutive seasons and whom saw their minutes per game increase by at least five minutes from the first season to the second. The players must have played between 10 and 25 minutes per game in the first season, to ensure we were not dealing with either folks who went from none-to-some playing time or superstar candidates who took over an offense and thus got a minutes boost. This is aimed at roleplayers whose role becomes more prominent — exactly the candidate FD’s Theorem of Intertemporal Heterogeneity implies will suffer from increased minutes.

Since I seem to express myself more clearly via Photoshop, here is the result of our mini-study.

No, increased minutes do not seem to lead to decreased efficiency. In fact, the data indicates increased minutes lead to… increased efficiency. More than 70% of the players in the study (there were 251 in total) saw their PER (which is, by definition, a per-minute summary statistic) increase with the increase in minutes. Players whose minutes per game increased by five saw an average change of +1.38 in their PER. The correlation between increased minutes and change in PER in this data set was +0.20.

One step further: Players who had at least five years of experience including their first-season in this study and got the requisite 5-minute increase (106 such players) saw an average change of +1.26 in their PER. It’s not just young kids who happen to improving and getting more minutes all at the same time — vets who get more minutes typically see their per-minute production rise. A full 67% of these players so positive changes in PER with the increased minutes. (And this answers one of Carter’s concerns with existing studies.) Let’s bump this up to players who had at least eight years of experience going into their minutes increase; we had 52 such cases. The average change in PER: +1.31. Of these players, 69% saw their PER increase with more minutes.

Case closed right? Well not if Brian M. has something to say about it.

Imagine we wanted to test the relationship between duration of exercise and reports of fatigue. We have two experimental conditions, one group jogs for 10 minutes and the other for 30 minutes. We predict that the group that jogs 30 minutes will report more fatigue.

But we must assign people to the two groups randomly in order for the data to have any bearing on the hypothesis. If we systematically assign people who are in better shape to the 30 minute jogging condition, we may find that in fact, if anything, people report less fatigue with longer durations of exercise. But the study is flawed in a fundamental way and so the data don?t tell us much of anything. At most what the results of this poor experiment tell us is that the effect of exercise duration on reported fatigue is not so strong that it overrides the differences in health between the two groups. But that is a really limited conclusion, especially if we don?t even have means to quantify how much the two groups differed in health to begin with.

Knicks 2007 Report Card (A to Z): Nate Robinson

KnickerBlogger: New Yorkers absolutely loved Nate Robinson when he first came to the Knicks. Coming out of the University of Washington, Robinson was a lilliputian guard with colossal physical abilities. Last year Robinson did what you’d expect from an undersized shooting guard. He led all Knick guards in eFG% (51.3%) and 3P% (39.0%) and showed despite his short stature he could get to the line (TS% 55.2%, second among Knick guards). Due to his efficient scoring ability, Robinson was second on the team in points per 40 minutes (19.0 pts/40) only behind Eddy Curry. Not just a one dimensional scorer, among Knick guards Robinson was the best in respect to offensive rebounds (1.6 OREB/40) and turnovers (2.1 TO/40), and second best in respect to steals (1.5 STL/40). Yet despite all that, Robinson is no longer a fan favorite. So what happened?

Simply put, Nate Robinson is his own worst enemy. Along with his diminutive stature and his youthful enthusiasm, Robinson comes with a childlike temperament. There’s a fine line between having a zest for the game and acting like a grade schooler. Robinson not only crosses that line, he lives on it. Less than one month into the season, Nate attempted an in game alley-oop dunk on a fast break, only to be called for traveling on the play. Throwing away points on a losing team for the sake of showboating is among the game’s cardinal sins.

Robinson exacerbated his image problem by perpetually arguing with officials. It’s annoying when a marquee player like Tim Duncan disputes every call, but it’s downright unbearable when a bench guy like Robinson does it. Unfortunately, Nate gave himself plenty of opportunities to argue with officials as his foul rate (4.7 PF/40) was equal to Marbury (2.7 PF/40) and Crawford’s (2.1 PF/40) combined.

Robinson’s immaturity causes his actions to be viewed by the public through tinted glasses. Take for instance Nate’s role in the Denver melee. In the past plenty of Knicks have improved their public image through fisticuffs. Fighting improved Starks, Childs, and L.J.’s popularity among Knick fans. Although Nate was an instigator in the event, it’s hard to believe that a player with a calmer outward demeanor like Eddy Curry would have been seen in the same light. Had Curry been involved, the local airwaves would be talking about his moxie and willingness to defend his teammate. But Robinson was vilified for his role. It’s ironic considering a few years ago, Knicknation was up in arms when no one came to the rescue of Tim Thomas after Jason Collins slammed him to the floor.

To be fair, Nate’s negatives aren’t all in his head. His defense is suspect, and his assist rate is minuscule for a guard. While says the Knicks are 2.4 points per 100 possessions better defensively with Robinson on the floor, opposing PGs are better than average (16.3 oPER) when Nate guards them. To the eye Robinson struggles mightily against the pick & roll, and other than the steals he doesn’t do anything particularly well on defense. I would rate him a mediocre to average defender.

Most people expect Robinson to be a point guard due to his height, but he’s really more of a shooting guard. Even accounting for that, his assist rate is subpar. As I said earlier, the Knick offense allows all the guards to play the point interchangeably. But it seems that Robinson isn’t sharing enough with his teammates. To put things in perspective, his 2.7 AST/40 is about the same as David Lee’s 2.4 AST/40 who rarely touches the ball. Nate does have the ability to make the spectacular play, and can pass the ball on his drives. It just that he desires to take the shot instead of making the pass. Normally you wouldn’t mind that from a guard that shoots as efficiently as Robinson. But then again Robinson suffers from his poor image, one that being a greedy guard certainly fits in with. In a way, for Nate Robinson hell is other people.

KnickerBlogger’s Grade: C, due to bad behavior.

2008 Outlook: With Nate Robinson entering his third season, it’s time to evaluate whether his poor decision making in the past was just youthful exuberance, or if it will continue to be a Rasheed Wallace like permanent petulance. I don’t expect Nate Robinson to turn into John Stockton, because he’s such an excitable person. What I would like to see is for Nate to take his job a little more seriously.

Robinson played 21.4 min/g under Larry Brown, and 21.2 min/g under Isiah Thomas. It seems that two coaches, who had very different views & philosophies, saw Robinson in the same light. If Nate wants to shed his role as spark off the bench, he’ll need to shed his image as a circus act crammed into a basketball uniform. It’ll be interesting to see how Nate plays in the preseason. I can envision Isiah giving Robinson more minutes due to his strong summer showing. If Nate can continue his productive ways, it could mean more playing time when the season starts. That would be a good thing, since the Knicks are paper thin at shooting guard, and they could use Robinson’s production.

Dave Crockett

In many ways KB’s take on Robinson has been by far the most “fair and balanced” (pardon the regrettable and unintended pun) I’ve read. I agree with his take on Robinson in total, but I also wish to offer a complementary perspective that’s less about Robinson’s performance than Robinson as a character in the theater that is professional sports. It’s easy to forget that sports is more than the simple pursuit of competitive dominance since that is precisely what the regular visitors to this blog come to read about and discuss. But, pro sports is also improv theater and all good theater (or “good copy,” to use the parlance of journalists) needs “heroes,” “bad boys,” and “villains.” As the great fat sage, Charles Barkley, is purported to have once said, “They can love you or they can hate you. Both sell tickets.”

Robinson, through a combination of his own immaturity as well as the fickle nature of media and fans, has gone from being a precocious but impish bad boy to something of a villain in just two full seasons. Though Robinson has clearly been the catalyst for his own fall from the good graces of many Knicks fans I also think he’s suffered from a demand for a steady of supply of villains that is becoming insatiable. Most of the time in professional sports players move seamlessly between the basic “villain,” “bad boy,” and “hero” roles for any number of reasons through a process that is reasonably organic and not always totally predictable. (I suspect many readers aren’t old enough to remember when Muhammad Ali was a villain to much of the American sporting public. He was hated in no uncertain terms. He had perhaps the most amazing role transformation ever.) But increasingly, the theater of pro sports has come to resemble the theater of pro ‘rasslin’ in its predictability, its cardboard cutouts of who gets assigned to which roles and for how long.

In Robinson’s case, since the Denver fight I see him being typecast as a particularly crappy villain archetype, and I really hope he’s allowed to work his way out of it. I call it the “Jeff George” villain archetype. Sometimes a player opens himself up to fan/media disdain by doing something over-the-top or exposing himself as a jerk and for whatever reason isn’t allowed much of a shot at redemption. Soon, the guy just can’t do anything right. The media fits him with a black top hat and a curly-Q mustache and it becomes obvious to the audience that he’s the guy to hate. (Note: I’m talking about sports-related stuff here NOT criminal or near-criminal behavior.) If you remember former NFL QB Jeff George, he was by most accounts a pompous jerk; universally reviled by fans, media, opposing players, even teammates and coaches. You would think by the way people couldn’t wait to denounce him that the NFL was not littered with similarly unbearable jerks. But of course it was, and is. As much as I truly loath Kansas City Star (and former columnist Jason Whitlock, I must agree with his sentiment that no one can point to anything George ever said or did that was uniquely awful.

Robinson, though not having “achieved” anything approaching the pariah status of George, seems to be quickly approaching the “can’t do anything right” status that is the hallmark of the Jeff George villain archetype. Hell, watch any Knick’s telecast with Mike Breen (even before the fight) and you’ll see what I mean. Regardless of what Robinson actually did on the court Breen would raise questions about his immaturity and decision-making, typically citing his ball-handling, shot selection, and his role in the Denver fight as prima facie evidence. So a poor shooting night or any turnover became proof of Robinson’s immaturity and poor decision-making. Yet somehow a good shooting/low turnover night did not indicate maturity or improved decision-making. The “Nate Robinson cautionary tale” always spins such a night as proof of how much talent Robinson is potentially squandering by his immaturity and poor decision-making.

My outlook for Robinson in 2008 completely mirrors KB’s in most respects. I believe Robinson is quite important to the Knicks playoffs chances. Not only are the Knicks thin at the SG, my entirely intuitive suspicion is that Crawford’s injury last season may be the first in a string of small-but-ongoing leg-related ailments that may keep him shuttling in and out of the lineup. So I believe the Knicks need Robinson to improve; it’s not a luxury. To do so he will have to start with the man in the mirror. Whether he is the new Jeff George or the new Bozo the Clown he simply must learn to focus on things that help the team win and leave the nonsense alone–period. But, I also urge the fans not to give up on this kid. He’s already a useful player and has the chance to get even better.

Brian Cronin – Man, Dave just reminded me of how annoying Mike Breen can be sometimes. The man is a GREAT announcer, but I think he works better on national telecasts, where he is not close to the situation, because man, he certainly seems to have soured upon the Knicks.

Breen reminds me of the stereotypical middle age guy complaining about how the NBA is “all thugs” nowadays. Those guys annoy me so much.

Anyhow, as to Robinson, the guy definitely exhibits some weird behavior, but since the fight, I thought he was actually a lot calmer than before the fight, and he seemed like a real nice asset to the team as an outside shooter. I hated when he tried to control the offense at times (that is not his specialty), but as a guy there to hit the outside shot, I like him there more than most other Knicks, and I think he will be a useful player this season.

Has the United States Made the Adjustment?

Yesterday, the United States brought their record in the FIBA Americas Championships to 3-0 with a 50 point throttling of Canada, 113-63.

Through the first three games, the US is averaging a winning margin of 52 points per game.

While these early opponents aren’t all that impressive, the dominance of the victories IS, and it is a very good sign for the return of United States competitiveness in international play. And really, it seems to be a simple solution to their past problems – the US seems to have actually taken the situation SERIOUSLY for the first time in some years. Read More

Trading David Lee for Kobe Bryant Straight-Up: Shrewd Sabermetrics or Laugh Test Flunkie?

In Basketball on Paper, Dean Oliver devoted an entire chapter to comparing the individual rating systems of several NBA analysts. He argued something that I, and most people who do informed analysis, subscribe to: Any system of statistical analysis cannot only be internally consistent, but must also pass the “laugh test.” A statistical model can be built elegantly and beautifully and pass many confidence intervals within its own logical parameters, but if it’s results are absurd, then there’s obviously a need to return to the proverbial drawing board. Oliver thought of the “laugh test” as a litmus. It’s a very broad, absolutely basic determinant of whether a statistic is logical or not. If your rating system projects the best players with the best numbers, then it’s probably onto something. On the other hand, if your rating system argues that Jerome James is a better center than vintage Shaquille O’Neal, then you better recheck your assumptions.

While no single computation can perfectly encompass the entire contribution of a basketball player, John Hollinger developed a system to sum up a player’s boxscore contribution and express them in one number. Player Efficiency Rating (PER) is a sophisticated equation that goes so far as to adjust for the yearly value of possession and the pace a team plays. In Hollinger’s analogy, PER serves as a way of considering players from different positions, allowing an “apples to oranges” comparison. But while PER is a handy little number, what it doesn’t do is convert statistical efficiency into actual wins. That’s where Dave Berri’s Wages of Win (WoW) steps in. WoW takes the same boxscore statistics that PER uses and converts it to a formula that measures how many wins a player produces. This metric can evaluate a player’s total contribution over the course of a season and break it down per minute. Like PER, WoW serves as a way to summarize a player’s contribution in one number.

Now, let’s ask PER who were the most productive basketball players on the planet this past season. PER picks these as its starting five:

1. Dwyane Wade SG 29.2
2. Dirk Nowitzki PF 27.9
3. Yao Ming C 26.7
4. Tim Duncan C 26.4
5. Kobe Bryant SG 26.3

Nothing to laugh at here. In fact, it’s a pretty amazing team. Wade is the best player, slightly ahead of Dirk, who is just a bit ahead of Ming, Duncan, and Bryant, who are in a dead heat for third best. If you were starting a basketball team and were given first pick at any player in the NBA you couldn’t go wrong by picking any of these five players. They’re the best of the best. Granted, PER isn’t intended to be the final word on basketball performance, but it is a good starting point for figuring out relative worth. Would you trade your 15 PER performer for a 29 PER man? Almost certainly. Of course you’d take into account team composition, need, age, defense, contract terms, but all else being equal, you’d be doing your team a service by having the greater PER over the lesser. And if the PER was almost twice greater, like say Dwyane Wade over Jamal Crawford, well, then there’s really no thinking involved. Of course you’d rather have Wade. It’s a no-brainer. In fact, by this measure, you’d rather have Wade than any single player on the Knicks current roster.

Now, WoW gets to pick its own top five. Note that in order to compare WoW to PER we’re using Wins Produced per 48 Minutes (WP/48), since these are both rate stats:

1. David Lee PF .403
2. Jason Kidd PG .403
3. Marcus Camby C .371
4. Shawn Marion F .370
5. Carlos Boozer PF .351

Look at that again. David Lee led the NBA in wins produced rate. Um…really. So according to this sophisticated, statistical model, the most productive professional basketball player on the planet is David Lee. The best. On. The. Planet. Let me say that being a die-hard Knicks fan, I will be the first to argue that Lee is an All-Star caliber forward. He’s cool, he’s great. He’s an out-of-the-box rebounding, ambidextrous-finishing, no-look passing, efficiency machine. He’s awesome! It’s just that, you know, he really doesn’t create much offense. He’s more of a great glue guy than a centerpiece. And that’s why he’s not exactly a superstar.

Now, I really love the guy. Don’t get me wrong. I wouldn’t trade our man for the world. Oh, wait. Yes. Yes, I would. I’d trade David Lee in a heartbeat. For Tim Duncan. Or Yao Ming. Or Dwyane Wade. Or Kobe Bryant. Or Dirk Nowitzki. Or Lebron James. Or Amare Stoudemire. Or…OK, you get the point. I’d trade him for at least a dozen players who aren’t just All-Stars, they’re legitimate championship-level franchise cornerstones. Yet, right there in plain black and white, Wages of Win’s assumptions fail Oliver’s “laugh test.” WoW argues that Lee is the best player in the entire league, and that’s ridiculous.

WoW makes a very big deal about bucking conventional wisdom. And sure enough, statistical analysts are the ones who’re supposed to be bucking said conventional wisdom. At the Wages of Wins Journal, Berri argues that “perceptions of performance in basketball do not match the player’s actual impact on wins” because “less than 15% of wins in the NBA are explained by payroll.” However payroll isn’t a good measuring stick of perception due to the complexities of a closed system like NBA free agency. There are a host of factors on why a player may be overpaid from the talent available to the desperation of the team involved. In other words conventional wisdom thinks Rashard Lewis is overpaid at $126M, too.

So although conventional wisdom has a tendency to be wrong in some areas, figuring out sport superstars is not one of its weaknesses. There usually is a consensus on the league’s best players from both statistical analysis and conventional wisdom. The cream of the crop in the NFL are Peyton Manning, LaDanian Tomlinson, and Larry Johnson whether you go by the numbers or eyes. In MLB it would be Albert Pujols, Ryan Howard, Manny Ramirez, David Ortiz, Alex Rodriguez, and Johan Santana. At the top of the ladder of player evaluation, conventional wisdom is pretty much dead on.

According to WoW, David Lee (.403) is a far more productive player than Kobe Bryant (.242). Since teams with more productive players win more games than other teams, then Lee is better for your basketball team than Bryant. But why stop there? The Knicks could trade Renaldo Balkman (.272) straight up for Dwyane Wade (.255) and lose productivity. That’s right. WoW is arguing that if a Lee for Kobe, and a Balkman for Wade trade went through, then the Knicks would be a worse team for it. They’re arguing that Bryant and Wade, at the cost of our two young, talented forwards will hurt the Knicks’ productivity. You’ve got to be kidding me.

As the Knicks GM, would I pull the trigger on a Lee for Bryant deal? Is there even a debate? Who wouldn’t? Oh, right, WoW wouldn’t. WoW doesn’t even think it’s close. We can all disagree on which player is the very best (or the most productive), but WoW’s results are “laughable.” Dave Berri has criticized PER in the past, but before people can begin to take WoW as seriously as a tool for evaluating player performance as PER, it’s obviously going to have to address what caused this terrible absurdity in its rating process.

Can Curry and Randolph coexist?

I must admit that my initial gut reaction to the Randolph trade was not exactly great. And I still don’t really like it. The obvious parallel here is the disastrous Francis trade, in which the Knicks acquired a talented but flawed player with a huge contract who duplicated almost exactly the skill set of a player already on the roster. Unlike the Francis trade, there is no question the Knicks won big on the talent end of this trade. But is there any hope that Curry and Randolph might coexist any better than Marbury and Francis did? On closer inspection, it’s not as poor a match as your gut reaction might have you think. Not that I’m doing jumping jacks over here, but let me explain.

The immediate concern is that Randolph’s prodigious scoring duplicates what Curry brings to the table. However, the story is not quite that simple. Curry is exclusively a low post player; last season he attempted 79% of his FGAs close to the basket and shot those at a stellar .667 eFG%. On the 21% of his FGAs that were further out, he shot an embarrassing .243. However, Randolph is more of a perimeter player. Last season he attempted a full 59% of his FGAs on jumpers and dropped them in at a .417 clip, which is actually pretty good efficiency on a jump shot for a big guy. (By way of comparison, in Frye’s rookie season he attempted 64% of his FGAs on jumpers and shot an identical .417 clip. The similarity here is actually pretty eerie.) A relatively paltry 41% of Randolph’s FGAs came in the paint, and his eFG% on those inside attempts was .551– good, but not Eddy Curry good.

So there is a relatively natural division of labor here: Curry is exclusively the workhorse in the paint, whereas Randolph has an effective face-up game to complement his effective post game. It is plausible that Randolph could become the more perimeter oriented complement to Curry that Frye was supposed to be while still doing considerable damage in the paint as well. In fact, admittedly having not seen much of Portland over the past few seasons, checking out his youtube clips reveals a player who is surprisingly quick and nimble with an effective face up game and a sneaky knack for scoring. He is not quite the methodical bruiser I had in mind, in spite of his hefty physique. For instance, did you know Zach Randolph could do this? It seems that the offensive talents of Randolph and Curry do indeed have a fighter’s chance of coexisting. If it works out it would be an awfully tough duo to contain.

While we’re comparing the two, Randolph is also a much better passer than Curry. He had twice as many assists per 100 possessions (7.9) and more than 6 fewer turnovers per 100 possessions (11.6) than Curry last season. In fact, contrary to appearances, Randolph’s turnover rate is entirely benign. His turnovers per 40 minutes were only so high last season because of his monstrous usage rate. Compare Randolph’s turnovers per 100 possessions with other high usage big men last season and you find that it’s actually par for the course. Only one guy sticks out like a sore thumb on this list. Can you guess who it is?

player usage rate TO / 100poss
Nowitzki 26.8 8.3
Garnett 25.2 9.9
Bosh 23.8 10.5
Brand 22.3 10.9
Boozer 24.9 11.2
Randolph 30.2 11.6
Gasol 23.3 11.6
J. O’Neal 25.8 11.9
Duncan 25.5 11.9
Shaq 26.3 12.1
Yao 29.9 13.2
Stoudemire 22.5 14.2
Curry 23.1 17.7

All this means that Randolph is the more versatile, and ultimately superior, offensive option even though he does not dominate the low post like Curry does. This may explain why Randolph’s usage rate has been consistently higher than Curry’s over their respective careers. Defenses have a harder time denying Randolph possession because of his more diversified game, which could be important for the Knicks given that guards not named Jamal Crawford have sometimes had difficulty feeding Curry the ball. Randolph does not need a guard to feed him in the low post in order to be dangerous, which is key in late game situations.

What about defense? By reputation, Randolph is a slouch. It doesn’t help his case that last season he blocked as many shots per 40 minutes as Nate Robinson. (Yes, you read that right.) But here are his defensive +/- numbers since 02/03:

season defensive +/-
02/03 +5.8
03/04 +2.0
04/05 -2.8
05/06 +1.5
06/07 +1.7

As always, +/- is an imperfect tool that is difficult to interpret. But nonetheless, over the past 4 seasons a relatively consistent pattern emerges for Randolph. His defensive +/- suggests that on average his teams have been better defensively when he’s off the court, but only slightly so– by less than one basket per 48 minutes 100 possessions. However, all of those teams since 03/04 have been in the bottom third in defensive efficiency, which qualifies the interpretation of the +/- numbers. What they suggest is that Randolph isn’t so bad on defense that he makes an already poor defensive team much worse. That isn’t quite the same as concluding that Randolph is even a passable defender. On the other hand, it maybe suggests that Randolph won’t make the Knicks worse on D than they already are. But is he bad enough that he could drag down a defense that is otherwise average or above average? I don’t think the existing data allows a firm conclusion on that question one way or the other. It’s clear that he is not a stalwart on D but it’s not clear if his weaknesses are relatively benign, entirely prohibitive, or somewhere inbetween.

At least the guy is a terror on the boards. He was among the league leaders with a 17.6 rebound rate, which figures to bolster New York’s existing strength in rebounding. The Knicks are already an elite offensive rebounding squad (2nd in the NBA last season), and Randolph should help improve the defensive rebounding (11th). A front court of Randolph (17.6), Lee (20.7), and Balkman (16.4) could be genuinely dominant on the glass on both ends of the court. And of course this is the one area in which Randolph clearly and uncontroversially complements Curry.

So setting aside for now the inconvenient truths that Randolph comes with a huge contract and a history of jail time and punching opponents and teammates alike… he may not be as poor a fit on the court for the Knicks as you thought on first glance. Now, if we could just trade Eddy Curry for Tyrus Thomas and Joakim Noah, then we’d really be cooking.

Crawford Out for the Season

You already know about that Crawford, but did you know about this one?

David Stern suspended referee Joey Crawford indefinitely today following Crawford’s ejection of Tim Duncan in Sunday’s Spurs/Mavs matchup on national TV. According to the story, Stern said:

“Especially in light of similar prior acts by this official, a significant suspension is warranted,” Stern said in a statement. “Although Joey is consistently rated as one of our top referees, he must be held accountable for his actions on the floor, and we will have further discussions with him following the season to be sure he understands his responsibilities.”

The article goes on to suggest that Crawford thinks he may have officiated his final NBA game.

I must admit I am pleasantly surprised by the commissioner’s decision to suspend Crawford. The league is usually tight-lipped when it comes to disciplining its zebras, and rightly so. But some public disciplining of Joey Crawford has been due for some time.

The actions of officials are routinely blown out of proportion and discussed without proper appreciation for how difficult their jobs are. However, Joey Crawford is in a category all by himself. I cannot think of a single official that so publicly and consistently crosses the line to the point of being unprofessional. Crawford consistently refuses to follow the “sticks and stones” mantra that the league expects players to follow. Crawford, is to my mind rarely satisfied with simply diffusing a situation. He insists on having the last word. Everyone else must walk away, must not laugh, lest they show him up. But Crawford seems to have no problems showing others up. I have never felt this way about a single other official in any sport. (And I’m a baseball fan first and foremost where the screaming matches with umpires are legendary.)

We entrust officials with control over our games so that a disinterested third party can manage the rules and manage conflict. The league has done an admirable job of making players understand that they must respect this arrangement. But, those to whom such trust is given must also respect the arrangement–not lord it over others. Joey Crawford, on too many occasions for my taste, lost sight of that.

Is Marbury a Loser?

Stephon Marbury has the skills, stats, and salary of a star. Nonetheless, he is perceived by many to be a loser. This greater perception of Marbury-As-Loser is likely formed in part by a constellation of subsidiary perceptions, such as the perception that Marbury is selfish (especially if you are a point guard purist), the perception that he has a poor attitude (especially if you consider wearing a towel on one’s head to be an indicator of poor attitude), and the perception that he is a poor teammate (especially if you’re into the tabloids). Probably the biggest factor in his losing rep, though, is just the fact that in his 10 seasons in the NBA, Marbury’s teams have finished in the lottery 6 times and have never won more than 45 games. In the 4 seasons where Marbury’s teams qualified for the playoffs, they failed to advance past the first round.

All else being equal, that history of futility at the team level could be construed as pretty damning. In fact, it comes off worse than just that. Jason Kidd and Steve Nash both managed to immediately elevate teams that faltered with Marbury just a season before, making it seem as if the team success was there for the taking all along, just waiting for a competent point guard to unleash it. Likewise, the fact that Marbury’s spotty team success has been distributed over four tours of duty in four different cities makes it seem as if the losing is a trend more readily attributable to the player than to his various teams.

However, closer inspection of Marbury’s career reveals numerous counterpoints to the above lines of reasoning. A thorough, season-by-season laundry list of objections one could raise to the traditional Marbury-As-Loser argument has recently been compiled by Dax-Devlon Ross. The Reader’s Digest version is that more often than not, Marbury’s teams have been either awful in terms of raw talent, or ravaged by injury, or both.

Points similar to those Ross enumerates have been raised in various Internet discussions on the Knicks ever since Marbury was traded to New York, but historically the skeptics have remained unconvinced. The bottom line, they insist, is that Marbury has failed to get it done. What is more, they claim, is that even those stats that do seem to reflect well on Marbury are misleading. Marbury’s 20 and 8 are nothing but numbers, empty stats that serve to promote the ego rather than team success. Sure, Marbury can ring up the scoreboard, but in the end his numbers do not translate into a tangible, on-court impact that really helps his teams win. Or so it is claimed.

What is nice is that these sorts of arguments needn’t be as indirect and unresolved as they sometimes seem fated to be. We don’t need to be satisfied with circumstantial evidence or received wisdom in this case.

How do we directly measure a player’s impact on his team’s success? The most straightforward measure is the player’s raw plus/minus statistics, which is a measure of the team’s point differential while a player is on the court vs. the team’s point differential while the player is off the court. But these plus/minus stats are not optimal for isolating the true impact of a given player, since they are subject to several influences beyond the player’s control. Consider, for instance, that you could hold a player’s impact on the court constant but change his plus/minus numbers drastically by changing the quality of his substitute, or by changing the quality of opposition he normally faces, or by changing the quality of teammates he normally plays with. For instance, TJ Ford would probably have a better plus/minus if his backup were Moochie Norris rather than Jose Calderon.

A more sophisticated measure of a player’s impact on his team’s success is adjusted plus/minus. The idea behind adjusted plus/minus is that we use statistical methods to remove the variation in a player’s plus/minus data that results from the other 9 players who happened to be on the court during his various court appearances. In essence, this removes the confounds alluded to above and gives us a pure measure of how much better (or worse) a player makes his team. (More on adjusted plus/minus methodology here and here.)

The major drawback to the adjusted plus/minus numbers is their scarcity. Unadjusted plus/minus numbers are only readily available starting from the 2002-03 season, which is presumably the season when the online game logs necessary for calculating plus/minus stats became available. And then there is the matter of calculating adjusted plus/minus from those rawer stats; thus far, no NBA stats site out there has made it a matter of course to fully integrate adjusted plus/minus numbers into its databases.

Fortunately, David Lewin recently crunched the adjusted plus/minus numbers for the 2004-05 and 2005-06 seasons in a series of articles for The results of the analysis would seem to turn the popular conception of Marbury on its head.

In the 2004-05 season, Marbury?s first full season in New York, the Knicks tallied a disappointing 33-49 record. Adding insult to injury, Marbury was roundly criticized after a torrid stretch of play in December inspired him to proclaim himself “the best point guard in the NBA.” The comment offended on two fronts. First, it was perceived as a politically incorrect kind of comment to make. (Things have apparently changed since the days of Muhammad Ali.) Second, it was regarded as a laughable claim at best. Sure, Marbury averaged 21.7 ppg and 8.1 apg, and even finished the season with a career second-best 21.9 PER? but those were empty numbers belonging to a loser. His team was no good, he didn?t make his teammates better on offense, and he didn?t play defense.

Marbury?s proclamation may still be regarded as non-PC, but in light of the adjusted plus/minus data, entertaining its truth value doesn?t seem quite so absurd. According to Lewin?s numbers, Marbury was 4th in the league in the 04-05 season in adjusted plus/minus at +12.4 points per 100 possessions, behind only Paul Pierce, Tim Duncan, and Elton Brand.

The simple conclusion to draw from this is that the Knicks’ struggles during the 04-05 season came in spite of, rather than because of, Marbury’s play. In fact, Marbury’s positive influence on the court was strong enough to rank among the league’s best. Therefore? in line with Ross?s arguments? if one were inclined to assign blame for the Knicks disappointing record that season, one would more properly distribute that blame among Marbury?s lackluster teammates, and possibly his coach.

The 2005-06 season is regarded as a down season for Marbury and his Knicks, as Larry Brown rode into town and… well, you know the rest. Marbury averaged career lows in points and assists per 40 minutes (17.9 and 7.0) and his 16.5 PER was his worst since his first two years in the league. His team stumbled to a bitterly disappointing 23-59 season. Marbury?s stock in adjusted plus/minus slipped as well, but he still posted a quite strong +7.57 points per 100 possessions. As with the prior season, Marbury was one of the few bright lights on an otherwise struggling team. It would be difficult to pin the losing aura of the 05-06 season on Marbury?s lapel since he was one of the few forces driving positive, winning play for the team.

Taking the weighted average of adjusted plus/minus over the 04-05 and 05-06 seasons, Lewin found Marbury to rank 8th in the league overall at +10.47 points per 100 possessions. By way of comparison, Nash (+8.47) and Kidd (+8.27) ranked 16th and 17th, respectively.

At this point, one can imagine the familiar response arising: those are just empty numbers; they don?t capture what the players are really doing on the court. But of course, adjusted plus/minus stats are designed exactly for the purpose of measuring a player?s net impact on his team?s success, rather than measuring milestones like points scored and assists recorded that may or may not contribute to a winning effort. It is logically possible for a player to average 20 ppg and 8 apg but still not help his team win. However, it is not possible for a player to rank exceptionally well on adjusted plus/minus but still not help his team win.

If one insists that Marbury is a sieve on defense, it must be the case that he is just that much better on offense. If one insists that Marbury is selfish and does not make his teammates better, it must be the case that Marbury?s style of play is nonetheless extremely effective at the team level, making the style critiques a moot point. If one insists that Marbury has still not been putting his teams over the top, one would seem to be criticizing Marbury for not being the best player in the league. In the end, it is not clear exactly how Marbury has helped his teams play better in recent seasons, but that he has helped them play better seems indisputable.

So, is Marbury a loser? Popular opinion may say yes, but at least in recent seasons, it turns out that popular opinion is wrong.

Of course, as with any other statistic, the adjusted plus/minus data do not come without caveats. A basic caveat is that the data may be rather noisy. Adjusting for the influence of surrounding players means you must measure the influence of surrounding players, and in some instances this may result in small sample sizes and hence the final numbers may come with rather large degrees of statistical variance.

One effect this has is to throw rank orderings into doubt. If two players are close in terms of their mean adjusted plus/minus, but both data points come with considerable statistical variation, then we may not have a lot of faith that the player who ranks higher is really higher in any statistically meaningful sense. Still, in the end, the actual averages that we get are our best guesses as to who should rank higher than whom.

It is also worth noting that in Lewin?s 05-06 analysis, the Detroit Pistons had strange adjusted plus/minus numbers, which Lewin argues was an anomaly due to their extremely rigid substitution patterns. Lewin does not think that this kind of substitution effect significantly skews the data for other teams. Still, it is worth noting that these sorts of problems may occur with the data.

These sorts of caveats do not seem to matter much to the general conclusion made here regarding Marbury, however. For instance, although Marbury was the point guard with the best adjusted plus/minus in 04-05, we probably cannot say with great confidence that his adjusted plus/minus (+12.4 points per 100 possessions) is really statistically distinguishable from Jason Kidd?s adjusted plus/minus from that same season (+11.15 points per 100 possessions). Therefore, one should not take away from this the ironclad conclusion that Marbury was in fact the best point guard in the 04-05 season. But if one is just asking the much more general question of whether Marbury really helps his teams win or not, then the sheer magnitude of his adjusted plus/minus numbers would make it very difficult for someone to argue that, in spite of the data, he is still a “loser.” Even factoring in statistical uncertainty, Marbury comes out looking roses on this analysis.

One final point of interest is that Marbury?s overall strong play may be a recent development. Dan Rosenbaum conducted an adjusted plus/minus analysis on the 02-03 and 03-04 seasons for Although Rosenbaum?s methodology differs somewhat from Lewin?s, in Rosenbaum?s analysis Marbury comes out looking like an average, break-even player during the 02-03 and 03-04 seasons. It is unlikely that such a drastic difference can be chalked up to the relatively minor methodological differences employed in the two studies. Therefore, it may be the case that one or more of the traditional critiques of Marbury (e.g. gives everything back on defense) were in fact legitimate at one point, despite being false in more recent years. It is hard to make any strong conclusions regarding this, though, given the absence of consistent methodologies across seasons and the paucity of data before the 02-03 season.