## Thursday, January 13, 2011

### 2010-11 NBA rebounding correlations

My last post showed that, in the NBA, players generally "steal" 2/3 of their rebounds away from their teammates. That is, when a player grabs a rebound, there's almost a 70% chance that someone else on their own team would have got it if they hadn't.

That study was based on the 2008-09 season, and I didn't have the data broken down into offensive rebounds and defensive rebounds. In addition, I used raw numbers of rebounds, which means my analysis would underestimate the amount of stealing (for reasons given in the "gravity" argument in the previous post).

But, now, commenter DSMok1 solved both problems. He's provided data for the current 2010-11 season (up to last night, I assume?), broken down, and his numbers are in rebound percentage, rather than raw numbers.

And the results are even stronger than before.

For defensive rebounds:

-- PG: every extra DREB% reduces his teammates' DREB% by 0.79.
-- SG: every extra DREB% reduces his teammates' DREB% by 0.92.
-- SF: every extra DREB% reduces his teammates' DREB% by 0.52.
-- PF: every extra DREB% reduces his teammates' DREB% by 0.94.
--- C: every extra DREB% reduces his teammates' DREB% by 0.77.

It looks like the "stealing" rate is about 0.8, on average, for defensive rebounds.

Having said that, however, I should say that the standard errors of these estimates are pretty high. Here are those estimates again, along with the SE:

PG -- 0.79 +/- 0.35
SG -- 0.92 +/- 0.21
SF -- 0.52 +/- 0.20
PF -- 0.94 +/- 0.13
C --- 0.77 +/- 0.14

Here's the same chart for offensive rebounds. This time, I'll put the SEs in brackets at the end.

-- PG: every extra OREB% reduces his teammates' OREB% by 1.18 (+/- 0.55).
-- SG: every extra OREB% reduces his teammates' OREB% by 1.06 (+/- 0.46).
-- SF: every extra OREB% reduces his teammates' OREB% by 0.15 (+/- 0.40).
-- PF: every extra OREB% reduces his teammates' OREB% by 0.13 (+/- 0.16).
--- C: every extra OREB% reduces his teammates' OREB% by 0.38 (+/- 0.27).

A lot different. As expert basketball sabermetricians have said, diminishing returns are at a much lower level for offensive rebounds. The PG and SG numbers are probably enhanced by random errors -- it would be hard to come up with an explanation of why a shooting guard's OREB would cost his teammates *more than one* OREB in exchange. [UPDATE: not true! See comments. I now agree a coefficient more extreme than -1 could actually be correct.]

Even though you could argue that the bottom three estimates are not statistically significantly different from zero (they're all less than 2 SDs), I think our best guess is still the point estimates we have here (since there's no prior reason to believe the diminishing returns rate should be zero).

Overall, other analysts have suggested an average of somewhere between .2 and .3, which seems very reasonable looking at the above table.

Finally, overall rebounds:

-- PG: every extra REB% reduces his teammates' REB% by 0.95 (+/- 0.15).
-- SG: every extra REB% reduces his teammates' REB% by 1.05 (+/- 0.21).
-- SF: every extra REB% reduces his teammates' REB% by 0.99 (+/- 0.26).
-- PF: every extra REB% reduces his teammates' REB% by 0.75 (+/- 0.12).
--- C: every extra REB% reduces his teammates' REB% by 0.71 (+/- 0.14).

All five of these seem to be about the average of OREB and DREB, except for the SF. Not sure why the SF is so different.

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So, I think this roughly confirms what certain basketball analysts are doing: assigning only about 0.3 of a defensive rebound to the player, and about 0.7 of an offensive rebound. [UPDATE: Guy makes an excellent argument for why my estimate of 0.3 could be too low. See comments.]

Thanks again to DSMok1 for the data.

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P.S. one other finding: overall, there is a negative correlation between a team's *overall* OREB% and its overall DREB%.

For every one percentage point extra on a team's OREB%, its DREB goes down by 0.3 percentage points. It's not statistically significant -- about 1.5 SDs -- but I thought I'd mention it anyway.

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At Friday, January 14, 2011 12:17:00 AM,  Guy said...

Nice analysis, Phil. But I think you understate your findings a bit. I think they suggest the net value of an oreb is about .5, while the average value of a dreb is .2 or even less. That would also be consistent with Eli Witus' regressions in his first rebounding article: http://www.countthebasket.com/blog/2008/02/05/diminishing-returns-and-the-value-of-offensive-and-defensive-rebounds/ (see regressions near end of post).

Why do you think teams gain 70% of an offensive rebound?

At Friday, January 14, 2011 12:41:00 AM,  Guy said...

Nice analysis, Phil. But I think you understate your findings a bit. I think they suggest the net value of an oreb is about .5, while the average value of a dreb is .2 or even less. That would also be consistent with Eli Witus' regressions in his first rebounding article: http://www.countthebasket.com/blog/2008/02/05/diminishing-returns-and-the-value-of-offensive-and-defensive-rebounds/ (see regressions near end of post).

Why do you think teams gain 70% of an offensive rebound?

At Friday, January 14, 2011 12:47:00 AM,  Phil Birnbaum said...

Guy,

You may be right. I was thinking something along the lines of:

The five positions show 1.18, 1.06, .15, .13, and .38. Obviously the 1.18 and 1.06 can't be right -- so let's suppose they're really .8 or something. Then the five numbers average out to 0.45. Figure the top two numbers belong to the positions with the fewest OREBs, and you lower that a bit.

However: the five estimates aren't independent. If the top two really did go down to .8 with a different sample, the other three might move higher to meet them. I can't really explain why I think that would happen, but I think it would.

In that case, .5 might indeed be more realistic. I'll check out Eli's article when I'm not so tired.

At Friday, January 14, 2011 10:39:00 AM,  parinella said...

Here's a possible explanation of why a guard OREB might cost his teammates more than 1 OREB: the number of offensive players going for a rebound is relatively fixed, so when a guard is one of those, the better rebounders clear out.

At Friday, January 14, 2011 11:12:00 AM,  Phil Birnbaum said...

Parinella:

Hmmm ... so what you're saying is, suppose that the PG goes for two rebounds and gets one of them. If he hadn't, the C would have gone for both, and got 1.2 of them. So every rebound the PG gets substitutes for 1.2 rebounds the C would have got.

Yeah, I guess that could happen. Good call. Anyone want to comment on how plausible that is?

At Friday, January 14, 2011 11:39:00 AM,  Guy said...

"Obviously the 1.18 and 1.06 can't be right -- so let's suppose they're really .8 or something. Then the five numbers average out to 0.45. Figure the top two numbers belong to the positions with the fewest OREBs, and you lower that a bit."

I think the important question is the amount of variance at each position, not the mean -- right? Looking at Eli's data the variance seems to be similar across positions. So I don't think we should count the guard correlations any less.

If one had to pick values based on your correlations and Eli's results, I think it would be about .5 of oreb and .1 for dreb. In Eli's data the marginal value of drebs was very low, and for big men was essentially zero. But maybe there are good reasons to think that overstates the degree of diminishing returns?

At Friday, January 14, 2011 11:43:00 AM,  Phil Birnbaum said...

Guy, I'm not sure what you mean about the variance. If a rebound position A reduces other rebounds by .8, and a rebound at position B reduces other rebounds by .4, wouldn't .6 be a reasonable estimate of the combined value?

And if A grabs three times as many rebounds as B, wouldn't .7 be better?

I don't see where variance figures into it ...

I still have to read Eli's studies. You may be right about the best estimates based on Eli's better evidence.

At Friday, January 14, 2011 2:38:00 PM,  Anonymous said...

I'm sort of shocked by the result showing that PGs are taking rebounds away from the rest of the team more often than at other positions.

In my day to day observation I see many occasions where the PG gets "long" rebounds off missed 3 pointers and other long jumpers and none of the bigs were in a position to get it. So it looks like a clear cut plus. I rarely see PGs batting inside for rebounds against bigs.

I always assumed that the value of rebounds by PGs was greater than for Cs.

At Friday, January 14, 2011 3:33:00 PM,  EvanZ said...

So, this is not too far off the coefficients I am using, which are simply the league average ORR (26%) and DRR (76%).

One question: Don't the coefficients have to add up to 1, due to symmetry? Or am I crazy?

At Friday, January 14, 2011 3:36:00 PM,  EvanZ said...

One more thing...was this a logistic regression, since the dependent vaiable is a proportion?

At Friday, January 14, 2011 8:56:00 PM,  Guy said...

Evan: I think the apparent similarity between the rate of diminishing returns on each type of rebound and the respective reb%s is just a coincidence. The fact that drebs are "easier" to get probably is related to the fact that that position drebs have a weaker correlation to team drebs (compared to orebs). But I don't think there's any reason the diminishing return rate has to be function of oreb% and dreb%.

And the correlations really don't point toward coefficients of .74 and .26. It seems like a player oreb yields about .5 team oreb on average, while drebs are worth only .1 to .2.

Phil: Since the correlations are effectively looking at the marginal effect of "extra" rebounds, my theory was that positions should be weighted by the variance rather than the mean. That is, what matters is how many "extra" (or fewer) rebounds come from each position, since those are the only ones we're valuing.

At Friday, January 14, 2011 9:28:00 PM,  Phil Birnbaum said...

Evan: agree with Guy on the coincidence thing.

On the logistic regression thing, I don't think it's appropriate here ... in any case, the Xs and Ys are so close that the logs of the odds ratios would be very close to proportional to the original values. That means the logistic regression would give you almost the same results.

At Friday, January 14, 2011 9:40:00 PM,  Phil Birnbaum said...

Guy,

>"Since the correlations are effectively looking at the marginal effect of "extra" rebounds, my theory was that positions should be weighted by the variance rather than the mean. That is, what matters is how many "extra" (or fewer) rebounds come from each position, since those are the only ones we're valuing."

That's brilliant! I think you're right!

That would also support Parinella's hypothesis (comment above). All the rebounds that the guard goes for that he shouldn't are at the margin, so that's why the coefficient comes out negative and with an "value" of less than -1.

It's absolutely right that the regression knows only about the margin. You can easily make up an example to show how that might happen.

Suppose every kid's parents buy their kid 100 marbles at 20 cents each, and hand them \$2. Then the kids start trading amongst themselves, at a market price of 10 cents each. If you do a regression on cash vs. marbles, you get 10 cents with a correlation of 100%. But only the marginal marbles cost 10 cents ... the original parental purchase, which is the overwhelming majority of marbles, was still 20 cents.

Brilliant!

At Saturday, January 15, 2011 1:56:00 AM,  Anonymous said...

At some level, aren't teams building diversity of skill/talent.

In other words, if they have a great rebounder at SG (Landry Fields) then they might be less concerned about having a really good rebounder at SF and PF (Gallinari and Chandler) as long as those other players have required skills that Fields may not have in high quantities.

If so, some of what appears to be stealing rebounds is actually just balanced team building.

I say this for several reasons.

1. When I see an excellent rebounder on a poor rebounding team move to a team with other very good rebounders, there will usually only be a marginal reduction in that player's rebounds and his new team will improve overall.

2. When I visually observe games and take notes I don't see near as many instances of multiple players in position to get a rebound as suggested here. Obviously it happens, but not a huge percentage of the time.

3. Even though there are diminishing returns, when 1 player commits to rebounding he is not in as good a position to do other things (like score or get an assist on the break). So perhaps even though others "could have" gotten the rebound, the fact that he did should give him close to full value for it.

4. I simply can't agree that most PG rebounds are stolen. With few exceptions, most PG rebounds come off missed 3 pointers and other long jump shots that few other players are in a position to get. There are exceptions, but not many.

At Saturday, January 15, 2011 10:22:00 AM,  DSMok1 said...

Well said, Anonymous. I think this is absolutely the case, which is why Eli Witus' research (looking at lineups where players were expected to sum well above or below 100% based on other lineup numbers) may be the more accurate way to do this.

At Saturday, January 15, 2011 4:56:00 PM,  Guy said...

I think there is likely some truth in what anonymous says, but it is overstated.

1) When high-reb players change teams, it is true they tend to maintain their reb%. But what you are missing is they tend to REDUCE the reb% of their new teammates (assuming they have replaced a lower reb% player). Because that reduction is spread over 10 players, it is less obvious. But the cumulative cost offsets much of the gain you thought you were getting. Team reb% will NOT increase as much as the difference between the new and old player would predict.

2) This effect is not only a result of plays where multiple players could grab the rebound. It's also, perhaps mainly, a function of positioning -- that some players are positioned to get more rebounds, as a function of the team's scheme. The fact that it appears to your eyes that only one player could have reached a rebound doesn't mean DR was not at work.

3) Maybe there is some benefit to the team, but the burden is on someone to prove that. And there is no reason whatsoever to think this benefit is so large that players should be given full credit for a Dreb if that only produces only .2 or .3 actual possessions.

DSMok1: Eli's work does suggest a higher payoff from oreb than Phil's results indicate, and that may be right. However, his data is limited to 5-man units within a given team-season. Let's assume that differences in oreb% reflect a mix of rebounding talent and different opportunities based on team offensive scheme (plus sampling error). And say we're comparing two units, one with a .12 oreb% center and one with a .08 oreb% center (other players constant). On this particular team, a lot of that .04 difference between the centers is likely to reflect skill (since the other players and team strategy are relatively constant). So when Eli compares the two units, he sees a big change in team oreb% (about 70% of the projected gap). However, it does NOT follow that if we look at two random centers in the NBA with those same oreb% rates, skill will again explain so much of that gap. For those two players, it might be that 50% or 70% of the difference was in opportunities. So it could be true that in general a player only delivers 50% of the expected oreb, but Eli's finding is also valid within any given team. (I think.)

At Friday, January 21, 2011 8:46:00 AM,  parinella said...

This might also be related to shot distribution. http://www.82games.com/rebounds.htm has some stats on shot and rebound location. Summarizing, longer shots result in more long rebounds at a lower offensive rebounding position.

ShotType Long% OReb%
3 pt 7% 31%
2 jump 4% 24%
paint 2% 37%

So a team that puts up a bunch of threes will get more long rebounds but fewer rebounds overall. Isn't this consistent with the stealing theory? So maybe if you added in 3 point attempts to the inputs, that would explain it.

At Tuesday, February 15, 2011 1:03:00 PM,  Anonymous said...

guy,

I do realize there are diminishing returns, I am simply arguing that based on observation and manual charting, rebounding studies seem to overstate the effect.

While it is true that part of that is schemes, the players assigned to "rebounding" are at a disadvantage in other ways. For example, the rebounder is not in a position leak out and get an assist or finish on the break.

To not give the rebounder a lot of credit for that rebound is to punish him for his role.

You'd be giving partial credit for the rebound to the leakers (who theoretically could have gotten it also) while simultaneously giving them credit for assists and scoring.