Here, I’m simply converting his initial work into an interactive visualization.

The size of each tile is based on the player’s Value over Replacement Player, a total-value-to-the-team measure.  The color of each tile is based on each player’s Advanced Statistical Plus/Minus, a per-minute rate.  Really bad players (below -2.5 ASPM, replacement level) will not be on the chart at all–Charlotte has a number of these, and barely any over replacement level.

Notes:

• Manu was dominant when he was playing–when he returns, the Spurs will likely be among the league’s elite once again.
• Who is the big 2 in New York again?
• Ryan Anderson: good player
• Carrying otherwise poor teams: Millsap, Lowry, Jennings
• Philly, the Spurs, and Indy are very balanced

Click on the tiles to see who the player is and what his VORP and ASPM numbers are.

For this review, I will consider a player to not have played (i.e. “Without You”) if the player played 20% or less of their average MPG. This allows us to look into the coach’s rotation decisions a bit more, while still maintaining the spirit of the WOWY analysis. For Dallas, this primarily effects DeShawn Stevenson and Rodrigue Beaubois. For Miami, Mike Miller and James Jones are effected quite a bit.

As has been discussed in previous WOWY posts, I have stabilized the regression with 30 games of team-average play to mitigate noise in the regression.

I present 2 tables for each team: equally-weighted, and Bayesian-weighted. The equally-weighted regression counts all games played this season equally. It’s probably the best for evaluating the value of each individual player. The Bayesian-weighted regression weights more recent games more heavily to better predict future performance. I have subjectively added an additional weight of 1.5x for playoff games (because of differing rotation choices in playoffs) and 2x for previous games against each other. Of course, those previous games against each other were so long ago they are still relatively negligible in the regression, compared to the playoff performance.

Dallas Mavericks, Equally-Weighted

Dallas Mavericks, Bayesian Weighting

Miami Heat, Equally-Weighted

Miami Heat, Bayesian Weighting

Ummm… the Heat look scarey.  Really scarey.  Since Udonis Haslem came back, they’ve been utterly dominant. That said, the sample size with Haslem back is only 4 games, and the 15.5 number is probably overstated. Probably.  However, they were pretty good without him, and swept Chicago when he came back.

For the actual series prediction, I used the standard deviations generated by these WOWY numbers.

As much as I hate to do it, I’ve got to pick Miami to win this.  I believe the regression may be overstating Miami’s chances, some, but not THAT much.  Just about any way I tweak the numbers, Miami in 6 is still the most likely outcome.  So that’s my prediction: Miami in 6.

Here’s the information used to generate these numbers.  Difficulty is adjusting for opponent, location, and rest.

Dallas Mavericks Game Log

Miami Heat Game Log

• On Bayesian Predictive Efficiency Ratings
• NBA Final Regular Season Bayesian Efficiency Ratings

Here are the Bayesian Ratings, updated through the end of the conference finals:

And here is my most intricate Team-Level Google Motion Chart to date:

 
(Source Data on Google Spreadsheets)
 

What is Plus/Minus?

Plus/Minus, at its core, simply compares how the team fares with the player on the court with how the team fares with the same player off the court.  If the team does better with the team on the court than off, the player is probably pretty good.  If the team does better with the player on the bench, the player is probably not as good.

The concept of Plus/Minus originated, I believe, with hockey.  It was first used there in the 1950s, according to Wikipedia.  Since the 1960s it has been in common use.

Plus/Minus was adapted to basketball some time thereafter; I have not been able to track down when it was first used.

For the 2007-2008 season, the NBA began tracking Plus/Minus in box scores.  Results may be found on NBA.com (along with player pairs, trios, etc.) for every year since 05-06.  BasketballValue.com has complete results (termed “1 year unadjusted overall rating”) for every year since 07-08.

Issues with Raw Plus/Minus

It has long been recognized that raw Plus/Minus is not particularly indicative of each player’s actual performance or value.  The largest issue is that the stat is totally context-dependent: if I was on the court with Lebron James, Dwyane Wade, and Chris Bosh, I’d probably have a pretty good Plus/Minus despite the fact I shouldn’t be playing at all.  And even if I was a really good player, I still wouldn’t have good raw Plus/Minus numbers if I were playing with Jamario Moon, Anthony Parker, Ramon Sessions, and J.J. Hickson.  In addition, the opponents matter as well.  Some players just play in garbage time against the other team’s scrubs.  Their Plus/Minus numbers may be pretty good, but only because they were playing against Brian Scalabrine!

This big issue, the dependence of Plus/Minus on the other 9 players on the court, gave rise to a new statistic aimed at adjusting for this problem: Adjusted Plus/Minus.

Adjusted Plus/Minus: the Basics

Each period of time when a group of 10 players are on the floor may be expressed as:

ObservedEfficiencyDifferential = P1 + P2 + P3 + P4 + P5
                                 - P6 - P7 - P8 - P9 - P10 + HCA

where P1 to P5 are on one team, and P6 to P10 are on the other.

Next, compile that equation for EVERY matchup of lineups through the whole season.  You now have somewhere around 60,000 independent equations and somewhere around 400 unknowns (the number of players that played that year).  If you solve this system of equations for P1, P2 … Pn , weighting each equation by the number of possessions played in that stint, you have normal, raw APM.

However, some players have a very small sample size, so they may muddy the final results by basically taking up all of the residual in the few minutes they played.  Many APM approaches, such as that used by BasketballValue.com (the site to go to for APM results), lump the rarely-used players into an “other players” bucket and just average them out.  This should improve the estimates on the players that are actually rated.

Let me digress here to point the reader to the definitive series on Adjusted Plus/Minus, mostly over at 82games.com:

• Numbers Game (Washington Post, Winston & Sagarin, 4/2004)
• Measuring how NBA players help their teams win (Dan Rosenbaum, 4/2004)
• 2005/2006 Adjusted Plus Minus Results (David Lewin, 2006)
• 2004/2005 Adjusted Plus Minus Results (David Lewin, 2006)
• Adjusted Plus-Minus: An Idea Whose Time Has Come (Steve Ilardi, 10/2007)
• Adjusted Plus-Minus: 2007-2008 Midseason results (Ilardi & Barzilai, 2008)
• Adjusted Plus-Minus Ratings: New and Improved for 2007-2008 (Ilardi & Barzilai, 2008)
• Calculating Adjusted Plus/Minus (Eli Witus, 2008)
• Offensive and Defensive Adjusted Plus/Minus (Eli Witus, 2008)

(Those last 2 provide nice “beginners” guides on how to do the actual construction of APM.)

Adjusted Plus/Minus: Collinearity & Sample Size

What are the issues with normal Adjusted Plus/Minus?

Well, the biggest one is that the sample size within one season is not enough for stability.  The “New and Improved” article above discusses some of the issues.  Basically, the big issue is one of collinearity.

Collinearity occurs in this data because of the way coaches use rotations.  P1 and P2 may always go into and come out of the game together–so which one causes the results?  Their numbers would be identical, except for the few times they don’t come in or go out together–and those few cases would dominate each of the players’ results.  So, say they always played together except 1 possession all year.  If the team scored (efficiency = 200) then the player out there at that time would be rated +200 above the other player.  They together may have to sum to 0 (known from all of the other stints together), but P1 is rated +100 and P2 -100 because of the 1 time they did not play together.  Now obviously, this never happens to this degree, but it does happen somewhat.

A second case is when P1 and P2 are only substituted for each other.  Suppose P1 and P2 both play center.  Suppose P1 is Dwight Howard and P2 is Marcin Gortat.  They only sub for each other, for just about the whole season.  When this is the case, we only really can detect how they relate to each other, not how the 2 of them relate to their teammates.  For the season, the team may be +8.  There is no way to know whether the center position is +10 and the rest is -2, or the center position is -2 and the rest of the team is +10.  What numbers are returned for the team are subject to the vagaries of the few minutes when the situation is different.

Why did I bring up D-Howard and Marcin Gortat?  Because the exact situation outlined above actually occurred.  In the 09/10 season, Howard and Gortat basically only subbed for each other.  Every lineup for the Magic that played more than 13 minutes total the whole year featured exactly 1 of Gortat or Howard manning the center position.  A similar situation occured in 08/09, though not quite as drastic.  What happened?  Here’s a quick table:

2010-2011 offers the clearest answer as to their actual rating, because Gortat was traded mid season, breaking up the tandem. Finally, we get a read that isn’t totally obscured by collinearity.  Note how the difference between Gortat and Howard stayed pretty consistent in ’08 and ’09, but they varied inversely with the rest of the team tremendously.

Interlude: Reliability vs. Validity

Before I go further, a quick review of reliability and validity is in order.  Columbia University has an excellent discussion of this subject online; if you don’t want to read it, I’ll attempt to summarize here.

“Reliability refers to a condition where a measurement process yields consistent scores (given an unchanged measured phenomenon) over repeat measurements.”  This is a measure that quantifies how much random fluctuations can interfere with getting consistent results.  As we have seen above, the collinearity of APM samples greatly decreases APM’s reliability.

“Validity refers to the extent we are measuring what we hope to measure (and what we think we are measuring).”  APM is completely valid, because it is directly measuring the result.  However, it may not be as valid for players who switch teams, because their value will almost certainly change based on fit/role in the system.  Box-score based stats are not as valid, because they measure a proxy rather than the subject directly.

This image, from the Columbia article, sums it up:

Reliability and Validity

Adjusted Plus/Minus: Stabilization Techniques

So, we have collinearity/reliability problems with APM.  What are the possible solutions?  Several were alluded to or discussed in the articles listed above.  I’ll take them 1 by 1.

1. Long-Term APM

The obvious one is just to use more years/data.  Many players change teams between years, and that greatly helps the collinearity problem.  Several long-term APM’s have been calculated.  Steve Ilardi did in his “New and Improved” article; he posted a full, equally-weighted six-year APM on the APBRmetrics forum.  More recently, Jeremias Engelmann ran 4 year 07-10 APM’s and posted on APBRmetrics (results here).  Aaron Barzilai on BasketballValue.com gives 2 year APM’s.

That helps.  However, the long-term APM may still have a few collinearity issues, and more importantly, it only tells us one thing: how good the given player averaged over the possessions they played in that span.  So yeah, it’s nice to know that KG was the best of the last 6 years, or that Lebron was the best of the last 4.  But perhaps I want more specificity?  Also, there can be issues with players that only played a portion of the time span–if a player plays only for the last year in the time span, and the other players have gone down hill from their average over the time span, the player that only played 1 year will be artificially & inaccurately inflated.  We’ve traded some validity for quite a bit more reliability.

There are at least 3 more approaches.

2. Weighted Long-Term APM

The next simplest approach is to take several years of data and weight the year desired more heavily than the other years.  This approach will give a more reliable estimate for the year in question, and doesn’t lose as much validity as the equally-weighted approach.  The collinearity is again reduced by the multi-year data.  Of course, we’re still dealing with, perhaps, 40% of our sample coming from years besides the one we wanted to measure.  This is the approach Steve Ilardi took in Adjusted Plus-Minus Ratings: New and Improved for 2007-2008. He posted updated numbers for this approach in a thread on APBRmetrics, and there was a long discussion on the method as well.

There are a few avenues for improvement with the “weighted years” approach that have not been explored, at least publicly.  The first would be to add an aging curve as a pre-processing adjustment.  If we’re trying to measure Kevin Garnett now, we should adjust his prior performance downward before running the regression.  Another area, one that’s quite basic, would be to run cross validation to determine the weights: take the prior year data and PART of the current year data, and explore what weights to use for best prediction of the rest of the current year data.  I’d like to see more research done on this sort of stabilization; I think it holds a lot of promise.

Okay, a few more approaches to go.

3. Statistically-Stabilized APM

In his seminal article on this subject, Dan Rosenbaum questioned the use of un-stabilized Adjusted Plus/Minus because of its very noisy nature.  His approach was to create a box score metric and use that to stabilize the regression.  The box score metric was generated by regressing box score data onto un-stabilized APM results; this sort of box score metric has become known as Statistical Plus/Minus (SPM) and has bred a whole series of versions.  That’s another story.  Anyway, Rosenbaum combined that SPM with his un-stabilized APM after each were run; the result was weighted toward either SPM or APM depending on which had the lower standard error.  I’m not entirely sure how this worked mathematically and how appropriate the approach was, but I’m pretty sure he knew what he was doing.  Thus far, I have not seen anyone else use a similar approach.

Now, we’ll really get math intensive:

4. Regularized Adjusted Plus/Minus

The last approach I have seen in public is the Regularized Adjusted Plus/Minus (RAPM) first proposed by Joe Sill and presented at the MIT Sloan Sports Analytics Conference. This approach takes advantage of a mathematical method known as Tikhonov Regularization or Ridge Regression.  The method essentially adds a penalty factor to the regression for results being far away from the mean.  This penalty factor, called lambda, is chosen based on cross validation, usually K-fold cross validation.  The data is broken into a number of segments (folds) and 1 at a time is removed and various choices for lambda explored.  The penalty factor is chosen such that maximum out-of-sample accuracy is attained.  This should remove most collinearity/noise, but lose only a small amount of the validity of the measure–except, one major issue is that because all players are regressed toward league mean, players with few minutes are considered average, and to rate out really badly, a player must have had a bunch of minutes to verify that the player was indeed that bad.

Jeremias Engelmann’s site stats-for-the-nba.appspot.com contains several versions of RAPM for different purposes.  He’s got 1 year RAPM for the last 5 or 6 years, a 3.x year rating that has best out-of-sample accuracy for this year, some longer term ratings, ratings for the Euroleague, and ratings for several stats (such as rebounding).  The framework is all RAPM.  There are a few issues, as mentioned above: the regression of rarely-used players to league mean is a big one, the lack of aging adjustment for multi-year ratings another, and not weighting multi-year ratings towards recency for best predictive power a third.  APBRmetrics threads are here and here.

Other Potential Variations on APM

I think those 4 stabilization variations are the ones that I have seen in the public domain.  There’s probably significantly more accurate/intricate models in NBA teams’ hands.  Winston, Rosenbaum, Lewin, Ilardi, Barzilai, Witus, and Sill are all either working for teams now or have worked for teams in the past.

There are a number of potential avenues for research that have not been explored.

• One area is team lineup APM.  BasketballValue.com runs lineup APM’s, but due to small sample sizes and the tremendous number of lineups/variables used, using one of the stabilizing techniques are absolutely required to get good results.  Many lineups play just a few minutes a year, so lumping a bunch of the less-used lineups together may be necessary.  Wayne Winston also works with lineup APM’s quite a lot on his blog; I will note Winston often does not seem to respect enough the small sample size/error issues with his research.
• Another major area would be looking at player pairs, trios, quartets, etc. to investigate synergistic effects between players.  Again, sample size issues are likely a risk here.
• Bayesian Adjusted Plus/Minus, which is a generalization of the RAPM approach, has been talked about but has not been implemented in the public domain.  Whereas RAPM regresses toward league average, Bayesian APM could regress toward any orthogonal prior–toward a value based on any sort of independent information.  For instance, we could regress toward a value suggested by a player’s playing time, adjusted for how good his team is.  Or, we could regress toward a Statistical Plus/Minus rating.  Or–well, the possibilities are endless.  The key to Bayesian Adjusted Plus/Minus is proper validation of the methods used to avoid over-fitting the data.
• Another area for research is the handling of positions–do some players play better in some positions than others?  The collinearity when researching this gets even worse, but it is an area that needs investigation.
• As mentioned several times above, using aging adjustments to pre-process data looks like a promising approach for using long-term data sets to yield more of a point estimate (i.e. What’s KG like right now?)
• A seemingly untouched area involves projections based on adjusted plus/minus numbers.  Once the difficulties with APM calculations have been ironed out, projections and good aging curves are a key area to study.  I discussed this subject once on the APBRmetrics forum, but never got very far.

I’m sure there are some other areas of potential research that I have overlooked, but those are what came to mind.

This wraps up my review of Adjusted Plus Minus and Stabilization Techniques.  I would like to make this a working document, so feel free to contact me to add something, point out an area that needs to be written better, or any other comment.  I hope this proves a useful reference to everyone in the advanced basketball stats community!

2011 NBA: the 750 best offensive performances, Wordle form

Notes:

• Some players are in there many times: Dwyane Wade has 17, though none are very easy to see.  Chris Paul and Lebron James had 16 each.  Kevin Love has 14, Kevin Durant 13 (wait–this is missing last night… should be 14), Carmelo Anthony has 13, and then Monta, Blake, Deron, D-Rose, Dirk, and Kevin Martin follow with 12 appearances each.
• The names are colored with the team’s primary color.
• The names are nominally alphabetical from left to right.
• Lebron James’ 51 was the best offensive game of the year.  Ty Lawson’s 3-Pt barrage was second.  After that comes: KD’s efficient 43, Kyle Lowry’s 36 on 18 shots, Paul Millsap’s heroic game vs. Miami, C.J. Miles crushing Minnesota (would be higher except for strength of opponent being factored in!), K-Love, Luol Deng, and Amare’s best.
• The size of the type is proportional to offensive VORP minus 7.0.  Lebron’s 51 was worth about 17 points over replacement on offense.
• Where has Kobe been this year?

I picked these up off of the team’s logo and/or website colors. Nearly all of the teams also use white as well.

Here are the brightest stars in the NBA, by dominant superstar performances this year. I’m giving 3 points for a top 50 performance, 2 points for a top 100 performance, and 1 point for a top 250 performance.

The NBA's Brightest Stars: Wordle Visualization

There have been 25,877 individual performances so far in this NBA season, so just making this chart is significant. Chris Paul has shown the brightest, with 4 Top 50 performances, 4 Top 100 performances (including the top 50–so none between 50 and 100) and 11 total top 250 performances.

There are 4 players who have had 3 Top 50 performances–you know them. These are players who can take over the game completely: Howard, James, Durant, and Ellis. (Yes, Ellis also has several of the worst performances this year.)

There are 10 players that were a flash-in-the-pan. Players that for one, and for only one night, played a game for the ages–and then it was over. You’ll remember them, too: Ty Lawson‘s 10-10 3′s streak. Bargnani going for 41-7-6 on the Knicks. Matt Barnes pouring in 24-7-6 in just 24 minutes without missing any shots. The other 7: Amare Stoudemire, Marcus Thornton, Rudy Gay, Toney Douglas, Carlos Boozer, Andre Miller, and Andris Biedrins. Yes, Amare only had 1 top 250 performance this year.

On the other hand, there were some workmanlike superstars, players with a bundle of top-250 performances this year: Chris Paul had his 11, Kevin Love turned in a massive 10, Howard had 9, Lebron and LaMarcus Aldridge had 8, and Monta Ellis and Zach Randolph had 7 each. No one else had more than 5 top-250 performances.

One final note: Tony Allen had 2 top 50 performances, and 3 top 100 performances. His NBA-wide 13th-best performance of the year line: 9-12 from the floor and 9-12 from the line for 27 pts, with 4 Rbs, 1 Ast, 5 steals, and 3 blocks at OKC to carry Memphis to a win on February 8th.