Hoops Lab: The Professor Talks Hoop

How Does Adjusted Plus Minus Work?

Last week I wrote about adjusted plus minus (APM), why I find it so useful in player evaluation, and gave you a very brief overview of how it works. This week I'll go a bit further into how it works and de-mystify some of the nuts and bolts so more of you can get a feel for it. I'll also tell you about some big weaknesses in the stat, and what has been done to address them.

One of my favorite articles about APM has a cool picture of a bull's-eye dart board that helps illustrate the difference between box score and +/- stats, for good or for ill. In the article the author uses the terms "valid" and "reliable" to make his point. Using the bull's-eye picture, valid means that your stat is aimed in the right direction (centered around the bull's-eye) but may not necessarily be very consistent. Reliable, on the other hand, means that your stat is very consistently repeatable (e.g. all of your darts hit the same spot) but the spot that they hit might not be the center of the bull's-eye. The ideal stat would be both consistent AND reliable (shots centered around the bulls eye, clustered close together).

Box score stats like points or rebounds are reliable, but since they don't accurately measure the whole game they aren't necessarily valid. On the flip side, +/- stats DO measure the whole game impact, but because of how they are calculated they aren't necessarily reliable for a single season. This is the big weakness of +/- stats. There aren't enough different player combinations that play enough minutes in any given season to fully solve the complex math in an APM calculation without making some assumptions - assumptions that add error to the calculation. That's why if you look at any 1-year APM calculation, like for example this 2008 APM calculation, you'll see a column called "SE" next to the one for "+/-". SE stands for standard error, and for Dwight Howard in that 2008 example his standard error of 10.7 is almost as big as his +/- of 12.7. In other words, all the single-season APM could tell us was that Howard measured out somewhere between best in the NBA and average - not really that useful. A single season just isn't enough data. So, the NBA statheads have come up with a few ways to make the APM measure better.

The first approach is to do multi-year studies. With multi-year studies the calculation has a lot more information to work with and the standard errors are a LOT lower, and thus the rankings they give are more accurate and realistic (or both valid and reliable). In this 6-year APM study, for example, you see standard errors down below 1.0 (instead of the 10.7 error you saw in the Howard example above). One of the problems with the multi-year approach, though, is that you can't examine any single season or the changes from one season to the next using that method. So, statisticians have attempted different methods to get more accuracy within a single season.

One way to get lower standard errors in a single year is to use values from previous years to stabilize your calculation. Here's an a href="http://82games.com/ilardi2.htm">alternative APM study for 2008, where the statistician used data from previous years to help lower his error for a single season. Here, Howard measures out as +4.5 with an error of +1.9 - well within the range of the previously mentioned 2008 APM calculation, but with a much lower standard error than the previous single-season standalone method.

Another way to lower standard errors for a single season is by "regularizing" the calculation, which a statistician named Englemann does on his advanced stats website. Regularized APM (RAPM) assumes that all low-minute players are roughly as good (or bad) as each other, which keeps the small amount of minutes that a star might play with a relative scrub from unduly influencing his value. This method has proven to be a better predictor than raw APM, even from a single season.

Thus, if I want to know who the most impactful player of the last four years has been, I go to a multi-year 2007-2011 APM study. This would tell me that LeBron James has had the biggest impact with good reliability. However, if I want to know who had the biggest impact of last season, I look to a 2011 single-season RAPM study that was stabilized using previous years - not quite as reliable as the multi-year study, but still pretty strong, and it tells me that Dirk Nowitzki was the most impactful player of 2011 with LeBron back in fourth. And if I want to know who has had the biggest impact of 2012 thus far, I go in realizing that the condensed season hasn't allowed for fully dependable APM results yet, but that again the RAPM study with previous years stabilization is probably my best bet, with a little bit of raw on/off-court +/- thrown in for some added flavor.

These days we have a lot of tools we can use to evaluate our players, and adjusted plus minus is one of the coolest and most in-depth ways that we have to do it once you get over the initial learning curve. I'm telling you, it sounds harder than it is, and once you capture the basic concept of the +/- approach it becomes a tool you'll always have in the toolkit.

Around the League:

•Old teams sprinting to finish line: The Spurs (8-game winning streak) and Celtics (5-game winning streak) entered their game on Wednesday night as two of the hottest teams in the league, and on Wednesday they played to a virtual draw with the Spurs eking out a 1-point victory when (reigning NBA Player of the Week AND Month) Paul Pierce's free throw-line jumper rimmed out. Between them, the Spurs and Celtics have beaten both the Heat and the Thunder in recent weeks, and the Celtics get to take their crack at the Bulls on Thursday. The point is, the season is ending in the exact opposite fashion as it begun, with the veteran teams now hitting their strides and playing at high levels while some of the younger teams have slid a bit. Just a trend to watch, as you consider your roster moves in this fantasy playoffs season.

•Garbage time for bad teams: Just as it was predictable that good teams would start streaking late, it's unsurprising the bad teams are entering garbage time. The NCAA season is over, GMs of bad teams are starting to look towards the draft, and players on bad teams (especially rookies and free agents) are realizing that this is their last opportunity to make a good impression this year. I expect lots of young players, journeymen, and free agents to suddenly start filling the traditional Anthony Randolph or Ramon Sessions role as a free agent pick-up that could swing the league for you. One potential candidate, Gerald Green, is mentioned in the New Additions section below.

•Willing to drop huge names: All of my head-to-head leagues are in the playoffs right now, and the leagues that allow daily line-up changes are very tense because every game counts. As such, in one league I've seen names like Derrick Rose, Stephen Curry, Amare Stoudemire, Eric Gordon, Kevin Martin and Danilo Gallinari on the free agent wire. With teams getting eliminated every week, fantasy owners can no longer afford to stash injured impact players on their bench for potential future gain - the future is now.

•Work the FA wire HARD down stretch: One big moral to the story told by these first few bullet points is that a good owner has to work the free agency wire HARD down the stretch. This is the time of year when you need to be locked in to your favorite sports/fantasy news sites continuously to know which injured vet is about to make his return, or which new injury has opened up another opportunity, or which backup is starting to explode. This is where a lot of leagues will be won.

•Rose's back - almost back? Speaking of this, Derrick Rose (back) has returned to full contact practice and could return to game action as soon as Thursday night. So if you're in a league like one of mine where some fool got desperate enough to drop a top-10 player, you don't need me to tell you to go get him. And if you've been holding onto him through his long absence, your patience is about to be rewarded.

•Bynum's behavior: I mentioned in last week's column that Andrew Bynum's attitude was becoming worrisome - he had already been kicked out of a game for two technicals, and been benched by his coach in another game for poor decisions and effort (including an ill-advised 3-pointer). Now, he's been fined by Lakers GM Mich Kupchak for "numerous infractions". This is in addition to the ankle injury that kept him out of Tuesday's game, but allowed him to return (in a huge way) with a big game on Thursday. All of this volatility during the fantasy postseason is the last thing that any fantasy owner wants, but at this point you're locked into the mercurial center so all you can do is stay tuned to Rotowire for updates on whether or not Bynum will play on a given day.

•Gordon's return:Eric Gordon returned to game action on Wednesday night after having missed the last 51 contests with a knee injury. He played 34 minutes, was rusty from the floor but did draw nine free throw attempts and seemed to settle into the action in the second half. If you stashed Gordon all season or can get him off of your free agent wire, it appears to be a great time to get him back into your lineups.

•Irving's shoulder: Irving will be out for the next seven to 10 days with a shoulder injury that he aggravated on Tuesday. As mentioned above, if you need the games played it is actually justifiable to cut Irving for this reason. But if you do plan to stash him, listen out for the results of his Wednesday MRI to make sure his prognosis hasn't changed for the worse.

•Ridnour's ankle:Luke Ridnour is in a similar situation, with an ankle injury that's estimated to keep him out for at least a week. He has been productive for me in the leagues where I have him, but I'm likely to drop Ridnour and attempt to pick up Jose Barea as soon as I finish this article.

New Additions

•Eric Gordon (83% owned in Yahoo leagues): Gordon is owned in most leagues, but now that he is back on the court he should be owned in all of them. He got cut in a lot of leagues with his long-term injury (I was able to pick him up two days ago in one of my leagues), so if you're in a league where he's still available you should snatch him up.

•Marcus Camby (55% owned): Camby has taken over as the starting center in Houston, and responded with 36 rebounds, 10 blocks, and 7 steals in his last three games as a starter. If he's available in your league, he's pretty much must-add status for the late season push.

•Ben Gordon (50% owned): Gordon has been maddeningly inconsistent (or maybe consistently under-achieving) for the entire season, and just when he showed recent signs of life he immediately went out with another minor injury. But he's back now, still starting with Rodney Stuckey out, and has averaged more than 24 points in those four starts.

•J.J. Barea (24% owned): With Luke Ridnour expected to miss at least a week with an ankle injury, Barea makes a great pick-up if he's healthy enough to take advantage of it. Barea had missed the last six games with a thigh bruise, but he returned on Wednesday night with 15 points, eight assists, six boards and three treys in 37 minutes of action. In his last three games pre-injury he notched one triple-double and had three straight double-digit assist games. Pick him up.

•Gerald Green (14% owned): Garbage-time alert! The Nets are going nowhere, Green has a high-flying circus-style game that sells tickets. Look for Green to get lots of minutes down the stretch, which should allow him to continue the 15-point/6-rebound averages he has posted over the last week.

•Avery Bradley (6% owned): Bradley is a young second year player that recently got the opportunity to start with Ray Allen injured, and averaged almost 15 points and more than a steal in his five starts. Allen is back now, but Bradley still got 34 minutes off the bench and led the Celtics with 19 points on Wednesday night. He appears to be a young guy peaking at the right time, and could be worth an add.

Keeping up with the Professor

If you're interested in my takes throughout the week, you can follow me on Twitter @ProfessorDrz. Also, don't forget that you can catch me on the radio on Rotowire Fantasy Sports Today with Chris Liss and Jeff Erickson on XM 87, Sirius 210.