RealGM

Adjusted Plus Minus has been around for a while now, but is only available for this decade (post-2002?), and has been used to determine (or help determine) a player's place among the ranks in the NBA, by comparing how much better their teams fare with them on the court than without them. ORtg and DRtg are used to judge a team's offense and defense, in terms of points scored per 100 possessions, and can be attributed to single players as well. But shouldn't those statistics relate?

A team's +/- per 48 minutes can be calculated from that team's points scored, opponents' points scored, and total minutes played. But we can get to the same number, or at least close enough, from the team's ORtg, DRtg and Pace, which estimates the number of possessions that team has per 48 minutes.



So, subtracting the former from the latter, we get the average margin of victory, adjusted per 48 minutes accounting for OT games lasting longer. In 2003-04, the Minnesota Timberwolves scored 7753 points, while letting their opponents score 7303 points, in 19755 total minutes.



Compared to the 5.487804878 average margin of victory the actually had. Using the T-Wolves' ORtg and DRtg, 105.9 and 99.7 respectively, and their Pace - 89.0 - we get to an average margin of scoring of 5.518 points per 48 minutes, the Team Adjusted Plus Minus. Not too far from the other calculations, and certainly close enough considering the rounding involved in the prior calculations of Pace, ORtg and DRtg.

But that doesn't tell us anything new.

Taking that number, and subtracting how the T-Wolves did without a specific player per 48 minutes, a number that should be achievable using a similar formula, should result in that player's APM. And this is where I get stuck. A team's ORtg or DRtg does not seem to equal the weighed average of that team's players' ORtg or DRtg, but it seems like there should be a correlation between them.

So how can I go from individual ORtg and DRtg? What am I missing? Is there something innately wrong with what I'm trying to do? How badly did I screw up with interpretations? Am I even in the right direction?

Any help would be much appreciated!

There is correlation between APM and individual ORtg & DRtg, but I wouldn't be too confident about it's meaning. First, seems to me like it would end up saying that Steve Kerr is a candidate for offensive GOAT.

Second, it's almost certainly going to be a weaker correlation than statistical +/- (which as I'll point out, is pretty questionable itself). That's a stat specifically developed to give weights to various statistics in proportion to how they correlate to +/- success.

Here's a link to the "statistical +/-" category of blog posts on b-r.com. You'll find somewhere around there the definitions.

http://www.basketball-reference.com/blog/?cat=22

As far as how good statistical +/- is, I'm not that confident. It tends to give really strong weight to steals, which I think just shows the flaw in the method: If you break down the +/- fine enough, of course steals are going to be much more valuable than say, blocks, because steals always end opponent possessions and often result in scores. We know though that is is big men that generally do the best in +/- which goes along with the long held belief that a shot blocker is more valuable than a thief. The problem with the statistical +/- break down in relation to these two stats is two-fold 1) shotblockers impact play even when they don't block shots, so they get underrated, 2) there's no box score stat for "attempted steals", and we know that a guy who gambles too much at steals causes huge problems for his team, so the bad gamblers get overrated.

I'll also say that I think the best way to approximate +/- in history is to track down games a player missed, and compare the difference between the games with him and without him.

With all that said though, quite possibly you have the ability to determine a ORtg/DRtg-based +/- approximation using nothing but a data set and code you write (whereas statistical +/- would be tougher, and games played vs missed would be impossible), and I would be interested to see the results.

Ortg = Points Scored by a Team (Or Lineup) divides by the number of possessions they used, multiplied by 100.

Drtg= Points given up by a Team (Or Lineup) divides by the number of posessions they used, multiplied by 100.

A team's Ortg can be broken up into a weighted sum of the Ortgs of individual lineups.

Lineups as we know are many up of players. APM tries to break up the performance of all of the lineups of a team into the sum of the performances of the individual players. For example, suppose the Lakers play exactly two lineups, Lineup X and Lineup Y.

Suppose we calculate that Lineup X has an Ortg of 125, and Lineup Y has an Ortg of 110. We also know that Lineup X was involved in 100 offensive possessions, lineup Y in 120. I can then calculate the Lakers Ortg for the season as:

octave:9> [125*100 + 110*120]/(100+120)
ans = 116.82

Roughly 117. Now, suppose Lineup X and Lineup Y differ in exactly only one player. Say lineup X has Kobe @ SG, lineup Y has Shannon Brown @ SG. Naively you might say that since Lineup X has an Ortg of 125 and Lineup Y has an Ortg of 110, Kobe is worth roughly +15 points to the Lakers Ortg, at least relative to Shannon Brown.

Regression analysis techniques like APM attempt to make this basic intuition more rigorous...the goal is to look at the performance of many different Lakers lineups and then assign ratings to each player, to see how much they increase/decrease the team's Ortg. Also, rather than looking at only Laker lineups, regression analysis would look at the performance of all lineups in the league, to then assign individual ratings.

So in summary....we are ALWAYS allowed to break up a team's Ortg into a weighted sum of the lineup Ortgs...this is perfectly fine and kosher.
However, APM wants to do more, and break up lineup ratings into weighted sums of player ratings (a step that requires some mathematical and statistical assumptions if we want it to be valid.)
Again, the point of this is rather than saying only "Lineup X is our strongest offensive lineup!", we want to understand who in the lineup is bringing what to the team. Or rather than saying, "Team A is a scoring powerhouse!" we want to understand who is responsible for making the team so potent offensively.

Anyway, to get from APM to Ortg or Drtg....well first of all, APM by itself should be interpreted as "how many net points does Player Z produce for me per 100 possessions"? So this is a sum of both an offensive and defensive term...the standard APM rankings you see online do not separate out Offensive APM from Defensive APM. So you'd need to do a slightly different regression that gives you both offensive and defensive values for each player. If you then do a possession-weighted average of the individual offensive terms, and then add some league-wide average Ortg, you should probably get something similar to the team's actual Ortg.

But of course the nice thing about this is that you've separated out the offensive performance of the entire team into a sum of the individual player contributions...and thus know who is helping and who is hurting you (again, assuming the statistical and mathematical modeling assumptions are correct.)

If some of this doesn't make sense, I invite you to read and work through the tutorial Eli Witus (a guy who works with the Rockets now) put up:
http://www.countthebasket.com/blog/2008 ... lus-minus/

Thanks for the reply.

I actually read most (all?) of those SPM blog posts when they came out, and like you pointed out, it's good, but not without it's flaws, as all stats are. One thing you didn't mention was that non-box-score players, like Battier and Bowen, are bound to be underrated by any statistic that attempts to relate those same stats to point differential.

I was reading through the POY project (great job, by the way, had I actually known about RealGM earlier, I would have liked to be a part of it...), and I think something that happened there is prone to happen in other player comparisons - once no +/- stats are available, it can come down to "my PER is bigger" and utterly subjective arguments. The +/- stats at least try to show a player's contribution to winning, without relying on Pts/Reb/Ast and the like, which I really liked about it, even if it tends to be quite circumstantial.

Ripp - that's basically exactly what I was looking for, but a lot less complicated, and an a lot rougher estimation. A LOT.

Unfortunately for me, I don't have the statistical know-how to do linear regressions or what not, and the only way I can get from stat A to stat B is logic and simple math, but I'll try to work with what I've got, and if the results are too out of whack, maybe I was doing something right. But more likely, I lucked my way to a good result or two.

If you are easily irritated, skip to the dotted line!

As far as the next step for goes, until someone can enlighten me about the correlation between individual O/DRtg and team O/DRtg, I think I have two options - calculate new team O/DRtg by using a weighed average of the individual players' ratings, and see what the result is, or just using the existing team ratings. For the T-Wolves, the calculated ORtg is 106.8 and the actual ORtg is 105.9, and the calculated DRtg is 99.9 instead of 99.7, so I guess I'll go that way.

So, recalculating the TPM - 6.141, another half a point a game in their favor. Not ideal, but since I already threw my credibility out the window, I shall keep marching on.

To separate a specific player from his teammates, the simplest, stupidest way is to do a weighed average of everyone else's O&DRtg. Yay for me, that's what I'm looking for (KISS!). The T-Wolves Minus KG has an ORtg of 105.8744 and a DRtg of 101.4228, and a TPM of 3.961924, assuming the keep the same pace. So Garnett's WPM (Weird Plus Minus) comes out at 2.18. Epic Fail.

A different approach - A team of KGs! Their TPM would be 17.8. and so KG's WPM will come out as 13.83. Better, but a totally lame way to get there.

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So, seeing as I can't do isht with numbers, maybe a statistical savant will pick up the gauntlet, a soft silk gauntlet, but still, and run some regressions, and get back to us with a better way. I mean, there should be a way, right?

Final thing, on a scale of 1 through 10, how moronic did I come out looking?

rk755 wrote:Thanks for the reply.

I actually read most (all?) of those SPM blog posts when they came out, and like you pointed out, it's good, but not without it's flaws, as all stats are. One thing you didn't mention was that non-box-score players, like Battier and Bowen, are bound to be underrated by any statistic that attempts to relate those same stats to point differential.

Well, APM and SPM are kind of related. APM rewards nonboxscore guys. All SPM is is a way to figure out how to weight boxscore stuff, based on APM values. Basically SPM is similar to PER, but using the APM value to calculate weights (in fact, you just run a standard APM regression, then take the values you get and regress boxscore stats against the APM values, giving you weighting for rebounds, assists, blocks, etc.)

Unfortunately for me, I don't have the statistical know-how to do linear regressions or what not, and the only way I can get from stat A to stat B is logic and simple math, but I'll try to work with what I've got, and if the results are too out of whack, maybe I was doing something right. But more likely, I lucked my way to a good result or two.

It isn't too hard to learn. You can learn how to do linear regression in Excel..there are some tutorials you can google. Or you could learn this free software package called R, and do your regressions there...won't take you more than a few hours, I think.

I think it makes sense for you to do the regressions to understand what is going on...a lot of these formulas I didn't truly, truly understand until I implemented them myself and played around with them.

So, basically, I totally screwed up before, by weighing the individual ORtg by minutes played, instead of possessions used. Stupid, I know. So let's try that, and cross our fingers for better results!

The T-Wolves, yeah, I know, had 7323.49 possessions in 2003-04 (According to Oliver's formula). Assuming they didn't change their pace depending on the players on the court, that would mean that a player's total possessions will equal his USG%*MP/(Team MP/5).

Some Excel wizardry later, and the KG-less T-Wolves will have an ORtg of 104.75, and a DRtg of 101.42. Considering pace, that's a scoring margin of 2.96 per 48 minutes, as opposed to the T-Wolves normal, calculated, 5.896. So that's + 2.93. As to the calculations, here they are:

First, all the players, and their Minutes Played, USG%, ORtg, DRtg and Possessions (Since I don't have Oliver's book, I'll use the rounded USG% for each player, and multiply it by his MP divided by the team's MP divided by 5). The total team possessions came out as 7323.48893 using Oliver's formula, while the actual sum of all player possessions was 7293.88, a discrepancy that is related to the rounding of the numbers at each step. I re-calculated pace using the total team possessions I came up with, just to make sure there are less rounding errors, and came up with 88.9717714 (as opposed to 89.0, big deal, right?).



Next, I calculated points produced, by multiplying the player possessions with their ORtg, just to verify everything(points produced != points scored). The next column is Poss*ORtg, with the last entry, for team, being the weighed average of ORtg, that came out as 106.5, higher than the team's actual 105.7, which would suggest that, at least that season, the T-Wolves were worse than the sum of their parts.



And now to calculate the team D&ORtg without each player. There are probably better ways to do this, but none that came to mind for me, so I just did a weighed average for everyone except the player in question each time, weighed by possessions for ORtg and minutes for DRtg, and here are the results:



Now, using the same method as before to calculate the team's scoring difference, without each player, i.e. (Team ORtg-Team Drtg)*Pace/100, we get the following results for each player -



Now, the first column is the players' names ( ), followed by the TPM - Team's +/- when removing that player, while the last entry is the team's regular, calculated +/-, then a column of the player's DRtg, multiplied by his MP, to recalculate the team's DRtg, the last entry in that column, and finally, for each player, the team's regular +/- minus his own TPM, or as I like to call it - RPM: Rating Plus Minus (if it actually sticks, that will change to RK's RPM ).

What we see, first of all, is that the results aren't as far apart as they are for APM and SPM, with just ~3.5 separating Garnett from the team's "worst" player, Sprewell, but that's mainly because the "On Court" +/- is basically the team's +/-, and, especially for the low minutes players, the "Off Court" +/- isn't really affected by them.. Other than that, KG's the biggest contributor, and only he, Cassell, Fred Hoiberg, Ervin Johnson and Oliver Miller were positive contributors, with the rest holding them back some.

It still might be interesting to compare the results for different players, across the same season or through the times, so in the next post I'll show some of the results.

EDIT: Since doing the T-Wolves' RPMs, I also separated those numbers to Offensive and Defensive RPM (should have from the start...), so here they are:

Continuing, here are the final results, only RPM, ORPM and DRPM, for all of the teams in 2004, by order of finish in the regular season, starting with the top 5 in the East (players within each team ranked by WS).

Indiana Pacers -



New Jersey Nets -



Detroit Pistons -



Miami Heat -



Milwaukee Bucks -



Next up, Hornets, Knicks, Celtics, Cavs and Raptors!

And on it goes...

New Orleans Hornets -



New York Knicks -



Boston Celtics -



Cleveland Cavaliers -



Toronto Raptors -



Philadelphia 76ers -



Atlanta Hawks -



Washington Wizards -



Chicago Bulls -



Orlando Magic -



Next up - the Western Conference!

Last for now, the Western Conference of 2004;

Minnesota Timberwolves (already posted, but what the heck) -



Los Angeles Lakers -



San Antonio Spurs -



Sacramento Kings -



Dallas Mavericks -



Memphis Grizzlies -



Houston Rockets -



Denver Nuggets -



Utah Jazz -



Portland Trail Blazers -



Seattle Supersonics -



Golden State Warriors -



Phoenix Suns -



Los Angeles Clippers -



Phew... That's it for 2004, so let's give out some awards! The "I'm so much better than y'all, it's not even funny anymore" award goes to Kevin Garnett! with a huge 2.9 RPM! The "Just give me the go-dang ball and get out of my way!" award goes to Peja Stojakovic, and his 2.27 ORPM! The "Should I just guard everyone and be don with this?" award goes to Ben Wallace, who edges KG by the slightest of margins! The "So, basketball you say, huh?" award goes to Mr Shimmy, Tippy-Toe Three, Antoine Walker! The "What am I supposed to do with this orange thingy?" award goes to, once again, Antoine Walker! And finally, the "Can you guard this guy for a second while I flirt with this cheerleader" award goes to Tony Parker!

That's it for now, conclusions and more stats will come up. Eventually.

Isn't the player ORTG and DRTG based on boxscore stats weighted by team performance? Like taking the defense, then using the bpg and rpg to figure out which player had the biggest role in that d.

I'm not sure I can trust equations with that.

One version of APM I'd like to see is something like this

- Take the on-court ORTG of a player (82games has them since 05)
- Compare it to league average ORTG

For example 05 Nash has 121.7 on court ORTG. League average for teams is 106.1. So you give the Suns when Nash is on a +15.5 ORTG differential. This makes it a better version of the team ORTG/DRTG stats cause it eliminates the 15mpg they don't play

My problem with APM is it has too many variables. The more variables, the less clear conclusions you can take out of it. The Nash calculation I just did is a lot more simple. It shows how the Suns offense was compared to average when he was on the court and that's it. It eliminates the "what they did without him" part, which is the most variable infused section of APM. What they did without him usually comes down to how well the rest of the team is built to play without their best guy. The Hornets survive a lot less without Paul than the Suns without Nash. But this tells me far more about the supporting guys than Nash and Paul, if that makes sense. So I'd just get rid of the minus and concentrate on what they do, not what their teammates do without them. I understand this makes the results a lot less definitive than +/-, but I'm fine with more vague stats if we can eliminate the variables. The minus stat has way too many variables for me to us it.

I just don't think you can calculate basketball like baseball. Players will mean more or less to teams depending on their supporting cast. For example a star play-maker on a team with all off ball, non creators should have a great offensive +/-. When a star's playmaking is duplicated by another perimeter guy, meaning the offense can run smoothly without him, his +/- will be worse. This has nothing to do with the player. Again I think without the minus portion everything becomes clearer. Now you just look at how the offense is with him on, then ask yourself how much talent he's playing with, and make the judgment yourself

The best way would be if you could quantify the supporting cast's talent level - then take the star player's on court ORTG and subtract the support. But how do you quantify a supporting cast? Almost impossible to do statistically. Has to be own judgment. Like saying "05 Nash had Joe Johnson, Marion, and Amare while 09 Paul had West + shooters, therefore Nash had more to work with". You can't quantify the difference between the two teams, you have to figure it out non statistically. Perhaps the boxscore created individual ORTG/DRTG stats could get you within the ballpark

^-- Are you criticizing APM in general, or the calculations rk755 did? If the former, I can address that.

More APM in general

Dr Mufasa wrote:One version of APM I'd like to see is something like this

- Take the on-court ORTG of a player (82games has them since 05)
- Compare it to league average ORTG

For example 05 Nash has 121.7 on court ORTG. League average for teams is 106.1. So you give the Suns when Nash is on a +15.5 ORTG differential. This makes it a better version of the team ORTG/DRTG stats cause it eliminates the 15mpg they don't play

APM pretty much does this. Or rather, a variant that looks at offensive APM and defensive APM. The way you interpret each number is, "how much does player X add to the per possession scoring of a theoretical average basketball team"?
Remember, the idea behind APM is to rank everyone in the league at a time. So implicitly, you are comparing to a league average. It is like having 400 people in a room, figuring out the average guy's height, and then discussion Steve Nash's height relative to the average guy in the room.

My problem with APM is it has too many variables. The more variables, the less clear conclusions you can take out of it.

You are right that in general, the more variables you have, the more data you need to figure stuff out. But really...400 or so variables is a very small number relative to the data you have.

The Nash calculation I just did is a lot more simple. It shows how the Suns offense was compared to average when he was on the court and that's it. It eliminates the "what they did without him" part, which is the most variable infused section of APM. What they did without him usually comes down to how well the rest of the team is built to play without their best guy. The Hornets survive a lot less without Paul than the Suns without Nash. But this tells me far more about the supporting guys than Nash and Paul, if that makes sense. So I'd just get rid of the minus and concentrate on what they do, not what their teammates do without them. I understand this makes the results a lot less definitive than +/-, but I'm fine with more vague stats if we can eliminate the variables. The minus stat has way too many variables for me to us it.

That Nash calculation is sort of a crude way of doing APM...you are effectively summing the Ortg of lineups Nash was in and comparing them to the league average Ortg. Problem is, you are leaving a lot of information unused. How did the non-Nash lineups do relative to the league average? That tells you a lot about Nash.

Think about it in linear algebra terms...if you have lots of equations involving Nash and some not involving him:
N1 = A + B + Nash
N2 = C + B + Nash
N3 = A + C + Nash
N4 = A+ B + C

If you use ALL of the above equations, you can figure out what the values A, B, C and Nash are. If you throw away the that 4th equation (that doesn't describe Nash), you cannot figure out any of the values. Basically you need to use as much information as possible when solving the problem, otherwise even a simple system of equations like the above might be impossible to solve.

And the thing is too, regression isn't just simple linear algebra...you have noise added to each equation. So throwing away data can really make it hard to determine a player's worth quickly.

EDIT: And by quickly, I mean, "without using enormous amounts of data."

Dr Mufasa wrote:I just don't think you can calculate basketball like baseball. Players will mean more or less to teams depending on their supporting cast. For example a star play-maker on a team with all off ball, non creators should have a great offensive +/-. When a star's playmaking is duplicated by another perimeter guy, meaning the offense can run smoothly without him, his +/- will be worse. This has nothing to do with the player. Again I think without the minus portion everything becomes clearer. Now you just look at how the offense is with him on, then ask yourself how much talent he's playing with, and make the judgment yourself

I don't get it...if the team functions just as well when the star playmaker is gone, then either his sub is equally amazing, or in fact the star is overrated. Star players make their presence felt. Like, look at the Hornets with Paul and Collison:
http://basketballvalue.com/teamplayers. ... C&team=NOH

Collison put up great statlines, and I think some people here on RealGM used that to say that Paul is overrated or something. Thing is, APM (rightly, imo) thinks that the gap between Paul and Collison is enormous.

BTW, I think you are talking more about +/-, not adjusted +/-...+/- is a different thing and can be a bit misleading by itself.

Ripp wrote:

My problem with APM is it has too many variables. The more variables, the less clear conclusions you can take out of it.

You are right that in general, the more variables you have, the more data you need to figure stuff out. But really...400 or so variables is a very small number relative to the data you have.

Don't think we're using the variables term in the same way. Maybe call them "factors". The way I see it we are only trying to isolate one 'factor', that's the player's value. The more factors going creating a stat that aren't the player's doing, the less it tells us. The number 3 can only be created by multiplying 3 x 1. The number 60 can be created by multiplying 60 x 1, 30 x 2, 20 x 3, 15 x 4, 12 x 5, 6 x 10, or you can break it down further into say, 4 x 5 x 3 and hundreds of other combinations. If you just give me the number 60 and told me a multiplication series created it, there's no way of knowing how you got there

That Nash calculation is sort of a crude way of doing APM...you are effectively summing the Ortg of lineups Nash was in and comparing them to the league average Ortg. Problem is, you are leaving a lot of information unused. How did the non-Nash lineups do relative to the league average? That tells you a lot about Nash.

Not as much as it tells you about the supporting players.

Instead of comparing Nash to Paul, let's just compare Paul to a twin Paul. Paul A is the 09 version, where the team had 113 ORTG with him on and 97 with him off. Paul B plays with Joe Johnson, Josh Smith, and Al Horford. With him they have an amazing 120 ORTG, without him they have 114.8 (Johnson's on court ORTG last year) since those Hawks guys can clearly run a great offense by themselves. Still though, that's only +5.2, while the other version was +16. The difference isn't Paul, it's that Johnson, Smith, and Horford can run a way better offense by themselves than the Paul less Hornets in 09

Think about it in linear algebra terms...if you have lots of equations involving Nash and some not involving him:
N1 = A + B + Nash
N2 = C + B + Nash
N3 = A + C + Nash
N4 = A+ B + C

If you use ALL of the above equations, you can figure out what the values A, B, C and Nash are. If you throw away the that 4th equation (that doesn't describe Nash), you cannot figure out any of the values. Basically you need to use as much information as possible when solving the problem, otherwise even a simple system of equations like the above might be impossible to solve.

The problem is there's not enough equations to do it this way. Hypothetically if you could simulate Nash on every team and combination of players, and then do the same for every other star, you could get an accurate gauge since you would be eliminating the variables for every star. As it stands every star plays with different teams and supporting casts which hands them all a bunch of different variables. It's like Nash is playing with A, B, and C and Paul is playing with K, L, and M, then extend that to every star and give them different letters, expanding the alphabet to hundreds of them. You could figure out Nash vs Paul if they both played with A, B, and C. But with Paul playing with K, L, and M there's almost no way to accurately cross compare them just by lining up equations.

The only way to make it work is have the same the variables for all players, at which point they wouldn't be variables, they'd be constants. And at this point it's just not possible to do that statistically.

Have you read the original article describing it? I strongly suggest you read through it:

http://www.82games.com/comm30.htm

Then read the wikipedia page on linear regression:

http://en.wikipedia.org/wiki/Linear_regression

Then download and run it yourself:
http://www.countthebasket.com/blog/2008 ... lus-minus/

Many of the things which you say cannot be done or aren't statistically possible in fact are, assuming you believe the model described on the 82games.com page is approximately correct.

Otherwise we will be talking in circles a bit. Like, if you believe the hypothesis/modeling assumption that the performance of a lineup is roughly the linear sum of the values of the players, then linear regression (or some variant of this) is the right thing to do. If you don't believe this, then you hunt for another technique.

OK, so I'm trying to improve on what I've got so far, by estimating the team's ORtg and DRtg when a specific is actually on the court, rather than just taking the team's "normal" values for the OnCourt part of the equation.

Until now, I calculated how the team did without each player on the court, and subtracted that from how they normally did, regardless of whether or not that player was playing.

Back to KG, for example. Using the rounded USG% and individual ORtg and DRtg (since I don't have the formulas), and calculating Pace myself, I came up with the following numbers for Garnett - 1772.718059 Possessions & 1985.444227 Points Produced. Calculating those two numbers for each player on that team, we can calculate the T-Wolves' ORtg without Garnett playing, by multiplying each player's Possessions by his ORtg, other than Garnett, and dividing by the total number of Possessions used by those players. This gives KG a "Team ORtg without" of 104.7532208 (the regular team ORtg I calculated is actually 106.5073705).

For "Team DRtg without" I simply averaged the DRtg for the rest of the players, weighed by their minutes played. For KG that came out as 101.422779.

Those two number, were used to calculate how the team fared without KG, by subtracting the DRtg from the ORtg, multiplying by Pace and dividing by 100. That gave us an estimation of KG's OffCourt +/- (sort of).

To get from that to the overall +/- for each player, I just subtracted that number from the team's regular +/-, without accounting for the fact that the latter number included both the time the player was on the court, and off of it.

I'm trying to correct that, by coming up with "Team ORtg with" and "Team DRtg with" numbers.

For the first one, I assumed that whoever a specific player played with, they performed overall like that player's "Team ORtg without", and had them use every possession that the team had while that specific player was on the court. Again, for KG, he uses 29.6% of the possessions available, and plays for 81.7767653% of the possible time. 81.77...% of the team's total possessions (7323.488934) is 5988.912363, and subtracting KG's possessions from that, we get that the other 4 players on the court with KG had a total of 4216.194304 possessions. With an ORtg of 104.7532208, that means they produced 4416.599327 points. Add Garnett's PProd to that, and we get 6402.043554 PProd with Garnett on the flour. Dividing that by the amount of possessions used, and multiply by 100, and the result is that KG's "Team ORtg with" is 106.8982674, which translates to 1.908486004 points per 48 minutes more than the "Team ORtg without", and 0.347787882 more than what was calculated before.

Now, for the second part, I'm kinda stuck. My question is, if KG has a DRtg of 92, does that mean that, when he's on the court, the team's DRtg is also 92? If that indeed is the case, it OBLITERATES everything done so far, seeing how that gives KG a DRPM (Defensive Ratings Plus/Minus) of ~8.4, about 3 times the RPM he had before this change!

Another thing I would like some help with, is calculating individual ORtg, DRtg, Points Produced, Possessions and USG%, to get more accurate results, so if anyone can direct me to the formulas, it would be much appreciated.

Ripp wrote:Have you read the original article describing it? I strongly suggest you read through it:

http://www.82games.com/comm30.htm

Then read the wikipedia page on linear regression:

http://en.wikipedia.org/wiki/Linear_regression

Then download and run it yourself:
http://www.countthebasket.com/blog/2008 ... lus-minus/

Many of the things which you say cannot be done or aren't statistically possible in fact are, assuming you believe the model described on the 82games.com page is approximately correct.

Otherwise we will be talking in circles a bit. Like, if you believe the hypothesis/modeling assumption that the performance of a lineup is roughly the linear sum of the values of the players, then linear regression (or some variant of this) is the right thing to do. If you don't believe this, then you hunt for another technique.

Not to intrude on your question to another poster, but what I'm about to say may help a bit:

APM does take into consideration the relative strengths and weaknesses of teammates and opponents by compiling all lineup combinations over the course of a season. To a large degree it is successful. What it can't fully control are the limitations of a given team's roster. Taking an example to the extreme:

Assume a team has one above average PG playing 24 minutes a night and exactly zero other players who are remotely capable of running the point. When the PG is on the floor efficiency is 120 for all other SG to C lineup combinations, so the regression would look something like:

120 = B constant + B pg x pg value + B player 2 x player 2 value +... B player 5 x player 5 value.

Sub in a small forward who can't run the point for the PG and efficiency plumets to 80. The regression would be:

80 = B constant + B sf x sf value + all else equal from above.

Assume you run this with several combinations of other players trying to assume the PG role and you continue to get similar 80 pts per 100 possessions against an entirely average team. The PG may be a good pg, but in the scheme of things, his ability is overstated due to the roster scarcity of the skill set he provides to the team.

Now assume the team's PF is traded for another PG in the offseason to pick up the remaining 24 minutes at this position. Given the loss of the PF, the team's pts/poss drop to 105 with the original PG on the floor. The team's pts/poss with the new PG are also 105. Suddenly, the value of the original PG drops and his teammate's value increases even if they are intrinsically the same players as the year before.

APM is a nice top down approach that can assign team performance to individual players. The central weakness of APM is that it has trouble assigning value to duplicative and non-duplicative output that is a function of roster limitation vs. duplicative and non-duplicative output that is a function of a player being really, really bad or good. It can be too roster specific, but it is very good at picking up Battieresque things you can't see in a box score.

SPM has an extra layer of problems. It is a bottom up approach that attempts to count things that lead to pts/poss on a league average. A guy efficiently getting 20 ppg on a good offensive team will have a nice SPM value. The problem arises if there is excess offensive firepower on the team and they don't miss a beat without him. Or maybe one player is the default rebounder to start fast breaks on 50-50 defensive rebounds that fall between teammates. Those defensive rebounds make the designated rebounder look good, but they don't add any incremental value to the team. Going back to the 20 ppg scorer. Assume for a second he got a lot of his baskets off open looks when the team's all-star center was double teamed. Next year he is traded to a team where he can't get open looks and his scoring drops to 15 ppg. So the team context problem still pops up, but in a slightly different way. There is also a problem with the weights used in the linear regression model. Everything is based upon league average, but the value of a rebound, pt, steal, etc could vary materially across teams due to the same types of roster limitations described above, so in this respect, it ignores team context.

Chicago76 wrote:Not to intrude on your question to another poster, but what I'm about to say may help a bit:

APM does take into consideration the relative strengths and weaknesses of teammates and opponents by compiling all lineup combinations over the course of a season. To a large degree it is successful. What it can't fully control are the limitations of a given team's roster. Taking an example to the extreme:

Assume a team has one above average PG playing 24 minutes a night and exactly zero other players who are remotely capable of running the point. When the PG is on the floor efficiency is 120 for all other SG to C lineup combinations, so the regression would look something like:

120 = B constant + B pg x pg value + B player 2 x player 2 value +... B player 5 x player 5 value.

Sub in a small forward who can't run the point for the PG and efficiency plumets to 80. The regression would be:

80 = B constant + B sf x sf value + all else equal from above.

Assume you run this with several combinations of other players trying to assume the PG role and you continue to get similar 80 pts per 100 possessions against an entirely average team. The PG may be a good pg, but in the scheme of things, his ability is overstated due to the roster scarcity of the skill set he provides to the team.

Yep, you are right...there will be large gap between the values of the PG and his backup (the SF).

But this doesn't necessarily mean that the PG will be overrated. If the PG is playing 24+ minutes a game, that means 50%+ of your dataset tells you that the PG is roughly average. The remaining minutes played by his bad backups don't boost his value...they'll just hurt the scores of whoever his backup is. So this sucks for the SF, but I'm not sure it helps the PG. So yes, players being played out of position are probably likely to be hurt by APM.

If you are still unsatisfied with this answer, there are several ways to deal with this issue. You could incorporate more years worth of data, when presumably the SF was playing at his natural position. You could also introduce some sort of statistical regularization...there are number of tricks you can use.

There is another issue that is related, however. Consider the following three situations:

[1] Two players only appear on the floor at the same time and leave at the same time.
[2] Two players share 95% of their minutes together.
[3] Players A and B are substitutes for each other, and when A is on the floor B is off, and vice versa....and we never observe A and B playing with different lineups of players, they always play with the same players.

All three are examples of something called collinearity. Situations 1 and 3 are pretty much impossible to deal with using regression alone...as you can see with Situation 1, there are an infinite number of ways the two players can be valued which are also consistent with the data.

Fortunately, Situation 2 can be addressed simply by collecting more data. But I've not really seen 1 and 3 myself, at least with the data I've been working with.

Now assume the team's PF is traded for another PG in the offseason to pick up the remaining 24 minutes at this position. Given the loss of the PF, the team's pts/poss drop to 105 with the original PG on the floor. The team's pts/poss with the new PG are also 105. Suddenly, the value of the original PG drops and his teammate's value increases even if they are intrinsically the same players as the year before.

Not necessarily...I don't think the starting PGs APM will change. It is possible his APM value stays the same, the new PG has the same value, his teammates stay the same except for the guy now playing heavy minutes at PF and presumably doing poorly at it.

APM is a nice top down approach that can assign team performance to individual players. The central weakness of APM is that it has trouble assigning value to duplicative and non-duplicative output that is a function of roster limitation vs. duplicative and non-duplicative output that is a function of a player being really, really bad or good. It can be too roster specific, but it is very good at picking up Battieresque things you can't see in a box score.

SPM has an extra layer of problems. It is a bottom up approach that attempts to count things that lead to pts/poss on a league average. A guy efficiently getting 20 ppg on a good offensive team will have a nice SPM value. The problem arises if there is excess offensive firepower on the team and they don't miss a beat without him. Or maybe one player is the default rebounder to start fast breaks on 50-50 defensive rebounds that fall between teammates. Those defensive rebounds make the designated rebounder look good, but they don't add any incremental value to the team. Going back to the 20 ppg scorer. Assume for a second he got a lot of his baskets off open looks when the team's all-star center was double teamed. Next year he is traded to a team where he can't get open looks and his scoring drops to 15 ppg. So the team context problem still pops up, but in a slightly different way. There is also a problem with the weights used in the linear regression model. Everything is based upon league average, but the value of a rebound, pt, steal, etc could vary materially across teams due to the same types of roster limitations described above, so in this respect, it ignores team context.

This is a very good point....for different teams, different things are more important. However, you could deal with this to some extent by using different linear weights for each team. Maybe for the Pistons, a defensive rebound is worth X, while for say the Raps it is worth Y. Of course, you'd have to verify that this results in an improved model...something I'm not really sure is necessarily true. If there are 10 boxscore weights you want learn for the entire league, you probably have enough data to learn that to a high degree of accuracy. But if you do it per team, that now becomes 300. So you really have to be sure that you have enough data to learn all these new parameters (I'm not saying you won't, you probably will, but it is just something that needs to be kept in mind.)
BTW, while I like the idea of APM and SPM, I don't think they are the final word. There are some ways to improve both models that result in something quite a bit more powerful.