Diminishing Returns and Rebounds (Eli W, 2008)
Posted: Fri Apr 22, 2011 9:32 pm
Eli W
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PostPosted: Tue Feb 05, 2008 7:19 pm Post subject: Diminishing Returns and Rebounds Reply with quote
I just put up a long post on my blog about diminishing returns and the value of offensive and defensive rebounds. It was prompted by a lot of the Wages of Wins discussion on this board and on Berri's blog. I'd be interested to hear any thoughts.
Diminishing Returns and the Value of Offensive and Defensive Rebounds
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Harold Almonte
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PostPosted: Tue Feb 05, 2008 9:02 pm Post subject: Reply with quote
It's very strange that the team's Reb% slopes are very diferent than individual players, but very closed to ths SG's slopes. I can't find thwe why, but, can you make some table with the FGMissed (Rebounds chances) produced by those positions or heights?
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Mountain
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PostPosted: Tue Feb 05, 2008 10:19 pm Post subject: Reply with quote
I'd be interested in learning more about how rebounding changed over time by height and position.
I came across the chart on page 10 of "A Starting Point for Analyzing Basketball Statistics" today showing the pretty sharp and steady decline of offensive rebounding over its 26 year period of study. How does that trend breakout to different positions and heights over your 33 year study? I be interested in seeing how your charts and tables for the most recent 5-10 years compare to past periods.
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cherokee_ACB
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PostPosted: Wed Feb 06, 2008 3:42 am Post subject: Reply with quote
Take a look as well to this thread, in particular EdK's graph.
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Eli W
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PostPosted: Wed Feb 06, 2008 8:56 am Post subject: Reply with quote
Harold Almonte wrote:
It's very strange that the team's Reb% slopes are very diferent than individual players, but very closed to ths SG's slopes. I can't find thwe why, but, can you make some table with the FGMissed (Rebounds chances) produced by those positions or heights?
I think that's probably just a coincidence. I wouldn't attach much weight to the specific values at each position given the small sample sizes. I can work on a chart like that.
Mountain wrote:
I'd be interested in learning more about how rebounding changed over time by height and position.
That is worth looking at since while height has increased over time, obviously position has not. I have some ideas about better position estimates for players from seasons past that I may post about.
cherokee_ACB wrote:
Take a look as well to this thread, in particular EdK's graph.
Thanks, I had forgotten about that thread. If Ed's lurking I'd be interested to know how many seasons of data that's based on and how many players were looked at for each position.
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Ed Küpfer
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PostPosted: Wed Feb 06, 2008 10:30 am Post subject: Reply with quote
Eli W wrote:
I'd be interested to know how many seasons of data that's based on and how many players were looked at for each position.
4 seasons. Can't remember anything about the players, but counting the dots in the graph, there appear to be between 80 and 120 players in each group.
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Mountain
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PostPosted: Wed Feb 06, 2008 1:16 pm Post subject: Reply with quote
I wonder if as league's tallest players got on average taller perimeter rebounding fell. And if the 3 pt revolution took the trend much farther.
Taller on average modern perimeter players might have greater ability to rebound more than their shorter predecessors but might not be positioned as well or make as much effort?
Speculation in wait of time period data.
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Guy
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PostPosted: Wed Feb 06, 2008 10:57 pm Post subject: Reply with quote
Nice work, Eli. I had some similar data I was going to post on the old WOW thread, but it seems like a better fit here now. The results are generally quite consistent with yours and Ed's, I think, and show an enormous diminishing returns effect.
I looked at rebounds by position, using 2006-07 data from 82 Games, and compared it both to rebounds at the other 4 positions on the same team and net rebounds for the team. Using positions rather than individual players has some advantages: MP is constant and it largely eliminates the good-rebounders-get-paired-with-weak-rebounders issue you raise. To deal with the underlying position differences, I converted the position values into rebounds above/below average for that position. So I get 150 "X" values, where X reflects the extra/fewer rebounds a team got from a given position.
Looking first at straight rebounds (position-adjusted), we see a negative correlation coefficient of -0.49 between one position's rebounds and the team's other positions. And regression indicates that for each additional rebound at a position, the other four positions lose 0.65 rebounds on average. If we look at the team total, each rebound at the position level translates into .27 team rebounds.
However, this actually understates the diminishing returns, because the shared rebounding opportunities (determined by pace and FG%) will tend to create positive correlations both among the five positions on a team and between a team and its opponents. So let's look at the real benefit to the team, defined as rebounds above average (Reb - .5*(Reb + OppReb)). Now we find that for each additional rebound gained at the position/player level, the team gains only .18 rebounds. In other words, WP and Win Score are crediting rebounds at more than 5 times their actual value.
Following Eli's lead, I also looked at Reb% by position, again normalized by position. Since we're now controlling well for opportunities, we expect to see a stronger relationship between position and team rebounds, and we do. But still, each additional 1% from a position increases team Reb% by only 0.25. (And decreases the Reb% for the other 4 positions by 0.75).
Clearly, this analysis is leaving out two potentially important dimensions: OReb vs. DReb (it seems clear that ORebs result more frequently in a real gain for the team), and differences by position (it may be that player Reb totals are more meaningful at some positions than others). But I think this helps set overall values, which coefficients for specific rebound types or positions should then be consistent with.
Finally, the SD for position-normed Reb% is .014 at the position/player level, and at the team level is just slightly higher at .016. This also tells us that there must be a huge negative correlation among teammates. If each player's rebounding was largely independent from that of his teammates, the team SD would then be sqrt(5*.014^2) = .032, or twice as large as it in fact is. (I think I misstated this gap as being much larger in an earlier post, because I had failed to control for player position, but the inter-dependence point stands).
BTW, if anyone wants the dataset, just send me a pm with your email address. I've sent it to both Berri and Jason, but neither have commented on it.
* *
Eli: one thing you might consider is position-adjusting or height-adjusting your data. This gives you much larger samples for your regressions (though at the cost of learning about position/height differences).
Last edited by Guy on Thu Feb 07, 2008 10:22 am; edited 1 time in total
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cherokee_ACB
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PostPosted: Thu Feb 07, 2008 3:09 am Post subject: Reply with quote
Guy wrote:
Using positions rather than individual players has some advantages: MP is constant and it largely eliminates the good-rebounders-get-paired-with-weak-rebounders issue you raise.
I still fail to see why. Could you explain it?
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Guy
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PostPosted: Thu Feb 07, 2008 8:11 am Post subject: Reply with quote
Cherokee: The data I'm using is all rebounds from each position. If a team has a "good rebounding" center and plays "weaker rebounders" at other positions when he's on the floor, then the reverse must be true when backup centers are on the floor. So in that scenario, there should be no relationship between total Rebs at C and total Rebs at other positions -- at the position level, it's all a wash.
That still leaves the issue of team construction: in theory, once a team has 1 or 2 good rebounders they might be more willing to accept poor rebounders at other positions. But while there may be a little truth to this, I can't see how it can possibly account for the huge negative correlations among teammates we observe. It would require GMs not simply to undervalue rebounds (plausible), or even be indifferent to rebounding ability (not plausible), but to systematically seek out terrible rebounders whenever they already had some good rebounders (really not plausible).
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Harold Almonte
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PostPosted: Thu Feb 07, 2008 9:48 am Post subject: Reply with quote
I think an advantage in using positions rather than individual height is, for example: a team has two 6'7" wings and a tweener 6'7" PF, the three with the same height, but defending at different floor position; one of them will have an advantage over the others that is not dependant of his height (let's remember that about 60% of rebounds caroms around the rim), and probably somebody will think that this advantaged player is more skilled (allthough the study shows almost no skill variation around this height, I think if you have a team with 5 6'7" players, one of them will rebound almost like an ordinary center because the floor position).
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cherokee_ACB
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PostPosted: Thu Feb 07, 2008 2:22 pm Post subject: Reply with quote
Guy wrote:
Cherokee: The data I'm using is all rebounds from each position. If a team has a "good rebounding" center and plays "weaker rebounders" at other positions when he's on the floor, then the reverse must be true when backup centers are on the floor.
I'm not sure I understand what you mean. I don't see how backups compensate for starters, since they play less time and, anyways, that scenario only exacerbates the problem: when compared with the backup, it looks as if the "good rebounding" center is stealing rebounds from other positions.
Quote:
It would require GMs not simply to undervalue rebounds (plausible), or even be indifferent to rebounding ability (not plausible), but to systematically seek out terrible rebounders whenever they already had some good rebounders (really not plausible).
Or that weak rebounding teams look for rebounding specialists, or that GMs prefer not to invest in other skills rather than rebounding when this is already taken care by the existing roster, or that coaches tend to go small when the opponent team also does it, etc. Conventional wisdom says that diminishing returns exist with rebounds, and GMs and coaches make decisions based on it (for the record, I believe that conventional wisdom is right here, but the effect is not as big as your data, and Eli's, suggests).
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cherokee_ACB
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PostPosted: Thu Feb 07, 2008 2:33 pm Post subject: Reply with quote
Eli, a couple of questions on the blog post:
- How did you compute player rebound rates?
- Do you include the 'worthy' team rebounds in your data, as BasketballValue does? I assume you do, but then you mention an average 73% DR rate, which comes from ignoring team rebounds (DR rate falls to 70% when team rebounds are added)
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Guy
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PostPosted: Thu Feb 07, 2008 2:52 pm Post subject: Reply with quote
Cherokee: I'm not sure I understand what you don't understand. Here's what I'm doing: I compare how many Rebs each team got at C (for example), to how many Rebs they got at the other 4 positions, or to the team's overall rebounds above average. For every extra reb at C, a team will get about .67 fewer reb's at the other 4 positions. And so on. Eli, in contrast, is looking at individual players, who might be paired with other players based on their respective rebounding ability.
As for the larger issue of team construction, I'm sure there's some truth but I basically don't buy it. I did the same exercise for points per shot (and turnovers), and do not find any diminishing returns there. GMs and coaches clearly don't say "I've got two good scorers, now let's find some guys who can't score to fill out our team." And to the extent they do say "there's no reason to pay for 3 great rebounders," I think they are recognizing the reality of diminishing returns, not creating the illusion of diminishing returns through their player selection.
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Eli W
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PostPosted: Thu Feb 07, 2008 6:39 pm Post subject: Reply with quote
cherokee_ACB wrote:
Eli, a couple of questions on the blog post:
- How did you compute player rebound rates?
- Do you include the 'worthy' team rebounds in your data, as BasketballValue does? I assume you do, but then you mention an average 73% DR rate, which comes from ignoring team rebounds (DR rate falls to 70% when team rebounds are added)
For the regressions, which were just using data from this season, I calculated rebound rates using BasketballValue's data. PlayerORB/(ORebForOnCourt + DRebOppOnCourt) and PlayerDRB/(DRebForOnCourt + ORebOppOnCourt).
When I looked at data from 73-74 to 06-07, I used data that did not include any team rebounds, and I calculated rebound rate by the typical method of estimating rebound opportunities based on what percentage of his team's minutes a player was on the court.
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herokee_ACB
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PostPosted: Fri Feb 08, 2008 8:59 am Post subject: Reply with quote
Guy wrote:
Here's what I'm doing: I compare how many Rebs each team got at C (for example), to how many Rebs they got at the other 4 positions, or to the team's overall rebounds above average.
Ok, I see. But, still not convinced that is the best method to obtain accurate results, I've done my own analysis. Basically, for every rebound opportunity so far this year, I've computed the sum of the individual rebounding ratios for the defense and the offense, and compared that to the outcome. I'm using basketballvalue data for this. The results:
- Linear regression:
Code:
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 69.73663 2.00587 34.77 <2e-16 ***
drt 0.30864 0.02719 11.35 <2e-16 ***
ort -0.80256 0.04006 -20.04 <2e-16 ***
If I only regress against defense or offense, the coefficients are 0.28 and -0.78, which suggests coaches tend to adapt to the opponent rebounding strength (the coefficient of correlation is small, just 0.07).
- Using a logit model:
Coefficients:
Code:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.84266 0.09811 8.589 <2e-16 ***
drt 1.47949 0.13291 11.131 <2e-16 ***
ort -3.85022 0.19571 -19.673 <2e-16 ***
For a 70-30 distribution, this translates into a 0.31 actual team increase for every extra point in the defensive players' sum, and 0.81 for offense. Essentially, same as above.
- If I estimate actual team rates using the above parameters, and then aggregate all observations into 0.1 wide bins for the expected rate, R-squared is 0.6093. Quite good.
- Correlation is even higher (0.6709) if I use a simple rebounding model for the estimation, where:
playerRate = playerAbility / (sum of playerAbilities on the court)
playerAbility is deduced from the actual player rates and his on-court team rebounding rate. This is similar to the correction MikeG uses in his numbers. Let's plot it (only for bins with more than 100 rebound opportunities):
Conclusions? Yes, I conclude my post here.
Last edited by cherokee_ACB on Wed Feb 13, 2008 4:14 am; edited 1 time in total
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Westy
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PostPosted: Tue Feb 12, 2008 11:10 am Post subject: Bottom line Reply with quote
So to summarize, what credit should any individual player get for garnering a defensive or offensive rebound, if the baseline for a possession is 1.0?
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Eli W
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PostPosted: Tue Feb 12, 2008 4:48 pm Post subject: Re: Bottom line Reply with quote
Westy wrote:
So to summarize, what credit should any individual player get for garnering a defensive or offensive rebound, if the baseline for a possession is 1.0?
Both less than 1, defensive rebounds less than offensive rebounds, defensive rebounds much less than 1. Possibly differing numbers for players at different positions. More work needs to be done in order to pin down more specific values.
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Eli W
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PostPosted: Sat Feb 23, 2008 2:05 pm Post subject: Reply with quote
I just put up another post on diminishing returns which uses a method suggested to me by Ben and similar to the one Cherokee_ACB used. I think it does a very good job of visually presenting the impact of diminishing returns on rebounding.
http://www.countthebasket.com/blog/2008 ... g-returns/
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cherokee_ACB
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PostPosted: Sun Feb 24, 2008 6:19 am Post subject: Reply with quote
Quote:
There is another issue with this technique that may lead to it underestimating the impact of diminishing returns
True. But, probably, the lowest (highest) projected rebounding units are the result of teams going small (big). If we accept that position affects rebounding percentages then, in those cases, it's normal that the projection underestimates (overestimates) the actual ratios. It's another issue worth looking at.
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Mike G
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PostPosted: Sun Feb 24, 2008 6:32 am Post subject: Reply with quote
I haven't followed this thread very closely, sorry. Eli, in your latest blog post, there's a strikingly weak correlation between (a lineup's) total player DReb% and team DReb%. You conclude that this indicates a diminishing return: Even if you stack your lineup with rebounders, you don't get a lot more (% of) rebounds. I think this is partly true.
What about the opponent's rebounders? If they decide to 'go small', what do you do? You could say, '"We will kill them on the boards"; but you might try to do better. You might go smaller -- which is to say, you'll be playing (with and against) lesser rebounders. Two teams always total 100% of the rebounds.
As small as sample sizes were for team's running with <65% total DReb'ers, it's probably a lot smaller for when the opponent was >70%. I think that to come up with an actual 'marginal value' for the DReb, you have to factor in the opponent's lineup.
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Eli W
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PostPosted: Sun Feb 24, 2008 1:56 pm Post subject: Reply with quote
You're right that that the opponent's lineup should be factored in as well, Mike. Cherokee did that in his study, but mine did not. I will try to look into that more soon.
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Guy
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PostPosted: Sun Feb 24, 2008 2:22 pm Post subject: Reply with quote
Cherokee: I don't follow your argument. Why does Eli need to worry about position when he's looking at 5-man lineups? Seems like he has that covered.
The opposing player point is a good one, though I'll be surprised if it changes the DRB results very much.
Another way this method may understate diminishing returns is by looking at players only as they function in a single season, which usually means on a single team. For example, if Lineup X is projected at 16% ORB%, these guys clearly aren't expected by this coach to do a lot of offensive rebounding. Thrown together for 400 minutes over the season, maybe they come in at 17-18%. But is that really the best these 5 guys could do, if they became a starting five? I'd guess not. Both are valid ways to look at the question, but I think the latter scenario is closer to the question we're usually interested in answering: when a high/low rebounder is added to a team, how much does that increase/reduce team rebounds? One way to get at this might be to use career rebound rates (prior to season being analyzed) to make your projections.
That said, I think this is excellent work by Eli. Keep it coming. (And it would be great if Eli could post the numbers for each of his buckets: total minutes, projected reb%, and actual reb%.)
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cherokee_ACB
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PostPosted: Sun Feb 24, 2008 2:53 pm Post subject: Reply with quote
Guy wrote:
Cherokee: I don't follow your argument. Why does Eli need to worry about position when he's looking at 5-man lineups? Seems like he has that covered.
Take Marion, for instance. He averages 3 rebounds more as a PF. Part of that it's because his teammates are worse rebounders, but I believe position has a bigger impact. The regression assumes we should expect his contribution to team rebounding, in absolute terms, to be the same in both positions, and roughly the average of his SF and PF rebound ratins. This is not entirely realistic. It shouldn't come as a surprise then if lineups with Marion at SF underperform in this analysis, and overperform with him as a PF.
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Guy
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PostPosted: Sun Feb 24, 2008 3:40 pm Post subject: Reply with quote
cherokee_ACB wrote:
It shouldn't come as a surprise then if lineups with Marion at SF underperform in this analysis, and overperform with him as a PF.
OK. But in Eli's overall analysis, won't that be a wash?
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cherokee_ACB
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PostPosted: Sun Feb 24, 2008 4:59 pm Post subject: Reply with quote
Guy wrote:
cherokee_ACB wrote:
It shouldn't come as a surprise then if lineups with Marion at SF underperform in this analysis, and overperform with him as a PF.
OK. But in Eli's overall analysis, won't that be a wash?
Don't think so. The effect is likely to go in the same direction in all cases.
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Eli W
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PostPosted: Mon Feb 25, 2008 5:00 pm Post subject: Reply with quote
I've uploaded a spreadsheet containing all the data I used for the recent study that I did. It includes some formulas so you can see how everything was calculated. The file is around 8mb. Thanks again to Ben for providing the raw data.
http://rapidshare.com/files/94915270/Re ... y.xls.html
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Guy
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PostPosted: Mon Feb 25, 2008 5:05 pm Post subject: Reply with quote
Cherokee: I follow you now. Multi-position players will tend to cause this method to understate diminishing returns, except where two players effectively "share" two positions and often play together (in which case the overperformer will be offset by an underperformer in the same lineup). Not clear how big an impact this will have.
Going back to our earlier exchange, it is hard to square the coefficients you and Eli are getting with my position-based analysis, which seems to show a larger diminishing returns effect. To recap, when I look at positions rather than players, each additional 1% of Reb% is associated with a loss of .75 rebounds at the other 4 positions, and thus a total team gain of just .25%. The advantages of the position approach are that it shouldn't be affected by good rebounders being paired on the floor with bad rebounders. It also doesn't have the "Marion" problem you just raised, and avoids the issue of using year X data to "predict" year X outcomes. Do you see problems with my method that are avoided by your and Eli's studies? (Good faith question -- I'm not sure my approach is any better, just trying to understand why we're getting such different results.)
Eli: would welcome your thoughts as well.
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Harold Almonte
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PostPosted: Mon Feb 25, 2008 6:58 pm Post subject: Reply with quote
This study didn't mention that the WP's regressed DRebs already account for shot defense, then that's the "why" about the bigger weight. If they decide to rate rebounding, they can make OR*0.7, but they couldn't make regressedDReb*0.3, they would need to do it to the real rebounding portion only. Or they, and any other metric, need to find another way to account for shot defense, and to make DReb much less than the VOP. I think for a metric that confessed it doesn't attempt to account shot defense, PER is overrating DRebs as a whole possession and underrating shot defense action, applying the rebounding rating to this action too.
PER is right about the reb. rating, but WP has some alibi when you account for shot defense. In the end WP is not overrating its rebounds too much more than PER is overrating its owns, they just aren't rating it, and adding shot defense value inside, as a product of a team win regression. But the PER notion that DReb is VOP is not right: Advantage WP.
Another thing about DRebs is that it's true that players with position advantage produce diminishing return on teammates when it comes to account rebounds grabbed, but they also produce increasing return (negative slope in a negative quadrant) when it comes to account opponents's rebounds-allowed. I'm surprised the slope of DR%proyected vs. actual isn't more plane.
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cherokee_ACB
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PostPosted: Tue Feb 26, 2008 10:49 am Post subject: Reply with quote
Guy wrote:
Cherokee: I follow you now. Multi-position players will tend to cause this method to understate diminishing returns ...
Do you mean "to overstate"?
Quote:
The advantages of the position approach are that it shouldn't be affected by good rebounders being paired on the floor with bad rebounders.
I already disagreed with this. Pairs like DHoward-Rashard are the biggest problem with your approach. And it's a big problem. If you want to argue for diminishing returns, you shouldn't use data with a bias in favor of your position. Well, at least I'd try not to do that.
Going back to Eli's 2nd issue in his blog post ("If players always played with the same four teammates ..."), I've ran the regressions separately for each team. This way, I focus on the rebounding ratios differences within a roster. Nothing changed. Of course, standard errors are higher and coefficients vary across teams, but averages stay at 0.3 and 0.8.
Last edited by cherokee_ACB on Tue Feb 26, 2008 11:44 am; edited 1 time in tota
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PostPosted: Tue Feb 05, 2008 7:19 pm Post subject: Diminishing Returns and Rebounds Reply with quote
I just put up a long post on my blog about diminishing returns and the value of offensive and defensive rebounds. It was prompted by a lot of the Wages of Wins discussion on this board and on Berri's blog. I'd be interested to hear any thoughts.
Diminishing Returns and the Value of Offensive and Defensive Rebounds
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Harold Almonte
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PostPosted: Tue Feb 05, 2008 9:02 pm Post subject: Reply with quote
It's very strange that the team's Reb% slopes are very diferent than individual players, but very closed to ths SG's slopes. I can't find thwe why, but, can you make some table with the FGMissed (Rebounds chances) produced by those positions or heights?
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Mountain
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PostPosted: Tue Feb 05, 2008 10:19 pm Post subject: Reply with quote
I'd be interested in learning more about how rebounding changed over time by height and position.
I came across the chart on page 10 of "A Starting Point for Analyzing Basketball Statistics" today showing the pretty sharp and steady decline of offensive rebounding over its 26 year period of study. How does that trend breakout to different positions and heights over your 33 year study? I be interested in seeing how your charts and tables for the most recent 5-10 years compare to past periods.
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cherokee_ACB
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PostPosted: Wed Feb 06, 2008 3:42 am Post subject: Reply with quote
Take a look as well to this thread, in particular EdK's graph.
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Eli W
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PostPosted: Wed Feb 06, 2008 8:56 am Post subject: Reply with quote
Harold Almonte wrote:
It's very strange that the team's Reb% slopes are very diferent than individual players, but very closed to ths SG's slopes. I can't find thwe why, but, can you make some table with the FGMissed (Rebounds chances) produced by those positions or heights?
I think that's probably just a coincidence. I wouldn't attach much weight to the specific values at each position given the small sample sizes. I can work on a chart like that.
Mountain wrote:
I'd be interested in learning more about how rebounding changed over time by height and position.
That is worth looking at since while height has increased over time, obviously position has not. I have some ideas about better position estimates for players from seasons past that I may post about.
cherokee_ACB wrote:
Take a look as well to this thread, in particular EdK's graph.
Thanks, I had forgotten about that thread. If Ed's lurking I'd be interested to know how many seasons of data that's based on and how many players were looked at for each position.
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Ed Küpfer
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PostPosted: Wed Feb 06, 2008 10:30 am Post subject: Reply with quote
Eli W wrote:
I'd be interested to know how many seasons of data that's based on and how many players were looked at for each position.
4 seasons. Can't remember anything about the players, but counting the dots in the graph, there appear to be between 80 and 120 players in each group.
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Mountain
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PostPosted: Wed Feb 06, 2008 1:16 pm Post subject: Reply with quote
I wonder if as league's tallest players got on average taller perimeter rebounding fell. And if the 3 pt revolution took the trend much farther.
Taller on average modern perimeter players might have greater ability to rebound more than their shorter predecessors but might not be positioned as well or make as much effort?
Speculation in wait of time period data.
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Guy
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PostPosted: Wed Feb 06, 2008 10:57 pm Post subject: Reply with quote
Nice work, Eli. I had some similar data I was going to post on the old WOW thread, but it seems like a better fit here now. The results are generally quite consistent with yours and Ed's, I think, and show an enormous diminishing returns effect.
I looked at rebounds by position, using 2006-07 data from 82 Games, and compared it both to rebounds at the other 4 positions on the same team and net rebounds for the team. Using positions rather than individual players has some advantages: MP is constant and it largely eliminates the good-rebounders-get-paired-with-weak-rebounders issue you raise. To deal with the underlying position differences, I converted the position values into rebounds above/below average for that position. So I get 150 "X" values, where X reflects the extra/fewer rebounds a team got from a given position.
Looking first at straight rebounds (position-adjusted), we see a negative correlation coefficient of -0.49 between one position's rebounds and the team's other positions. And regression indicates that for each additional rebound at a position, the other four positions lose 0.65 rebounds on average. If we look at the team total, each rebound at the position level translates into .27 team rebounds.
However, this actually understates the diminishing returns, because the shared rebounding opportunities (determined by pace and FG%) will tend to create positive correlations both among the five positions on a team and between a team and its opponents. So let's look at the real benefit to the team, defined as rebounds above average (Reb - .5*(Reb + OppReb)). Now we find that for each additional rebound gained at the position/player level, the team gains only .18 rebounds. In other words, WP and Win Score are crediting rebounds at more than 5 times their actual value.
Following Eli's lead, I also looked at Reb% by position, again normalized by position. Since we're now controlling well for opportunities, we expect to see a stronger relationship between position and team rebounds, and we do. But still, each additional 1% from a position increases team Reb% by only 0.25. (And decreases the Reb% for the other 4 positions by 0.75).
Clearly, this analysis is leaving out two potentially important dimensions: OReb vs. DReb (it seems clear that ORebs result more frequently in a real gain for the team), and differences by position (it may be that player Reb totals are more meaningful at some positions than others). But I think this helps set overall values, which coefficients for specific rebound types or positions should then be consistent with.
Finally, the SD for position-normed Reb% is .014 at the position/player level, and at the team level is just slightly higher at .016. This also tells us that there must be a huge negative correlation among teammates. If each player's rebounding was largely independent from that of his teammates, the team SD would then be sqrt(5*.014^2) = .032, or twice as large as it in fact is. (I think I misstated this gap as being much larger in an earlier post, because I had failed to control for player position, but the inter-dependence point stands).
BTW, if anyone wants the dataset, just send me a pm with your email address. I've sent it to both Berri and Jason, but neither have commented on it.
* *
Eli: one thing you might consider is position-adjusting or height-adjusting your data. This gives you much larger samples for your regressions (though at the cost of learning about position/height differences).
Last edited by Guy on Thu Feb 07, 2008 10:22 am; edited 1 time in total
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cherokee_ACB
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PostPosted: Thu Feb 07, 2008 3:09 am Post subject: Reply with quote
Guy wrote:
Using positions rather than individual players has some advantages: MP is constant and it largely eliminates the good-rebounders-get-paired-with-weak-rebounders issue you raise.
I still fail to see why. Could you explain it?
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Guy
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PostPosted: Thu Feb 07, 2008 8:11 am Post subject: Reply with quote
Cherokee: The data I'm using is all rebounds from each position. If a team has a "good rebounding" center and plays "weaker rebounders" at other positions when he's on the floor, then the reverse must be true when backup centers are on the floor. So in that scenario, there should be no relationship between total Rebs at C and total Rebs at other positions -- at the position level, it's all a wash.
That still leaves the issue of team construction: in theory, once a team has 1 or 2 good rebounders they might be more willing to accept poor rebounders at other positions. But while there may be a little truth to this, I can't see how it can possibly account for the huge negative correlations among teammates we observe. It would require GMs not simply to undervalue rebounds (plausible), or even be indifferent to rebounding ability (not plausible), but to systematically seek out terrible rebounders whenever they already had some good rebounders (really not plausible).
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Harold Almonte
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PostPosted: Thu Feb 07, 2008 9:48 am Post subject: Reply with quote
I think an advantage in using positions rather than individual height is, for example: a team has two 6'7" wings and a tweener 6'7" PF, the three with the same height, but defending at different floor position; one of them will have an advantage over the others that is not dependant of his height (let's remember that about 60% of rebounds caroms around the rim), and probably somebody will think that this advantaged player is more skilled (allthough the study shows almost no skill variation around this height, I think if you have a team with 5 6'7" players, one of them will rebound almost like an ordinary center because the floor position).
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cherokee_ACB
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PostPosted: Thu Feb 07, 2008 2:22 pm Post subject: Reply with quote
Guy wrote:
Cherokee: The data I'm using is all rebounds from each position. If a team has a "good rebounding" center and plays "weaker rebounders" at other positions when he's on the floor, then the reverse must be true when backup centers are on the floor.
I'm not sure I understand what you mean. I don't see how backups compensate for starters, since they play less time and, anyways, that scenario only exacerbates the problem: when compared with the backup, it looks as if the "good rebounding" center is stealing rebounds from other positions.
Quote:
It would require GMs not simply to undervalue rebounds (plausible), or even be indifferent to rebounding ability (not plausible), but to systematically seek out terrible rebounders whenever they already had some good rebounders (really not plausible).
Or that weak rebounding teams look for rebounding specialists, or that GMs prefer not to invest in other skills rather than rebounding when this is already taken care by the existing roster, or that coaches tend to go small when the opponent team also does it, etc. Conventional wisdom says that diminishing returns exist with rebounds, and GMs and coaches make decisions based on it (for the record, I believe that conventional wisdom is right here, but the effect is not as big as your data, and Eli's, suggests).
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cherokee_ACB
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PostPosted: Thu Feb 07, 2008 2:33 pm Post subject: Reply with quote
Eli, a couple of questions on the blog post:
- How did you compute player rebound rates?
- Do you include the 'worthy' team rebounds in your data, as BasketballValue does? I assume you do, but then you mention an average 73% DR rate, which comes from ignoring team rebounds (DR rate falls to 70% when team rebounds are added)
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Guy
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PostPosted: Thu Feb 07, 2008 2:52 pm Post subject: Reply with quote
Cherokee: I'm not sure I understand what you don't understand. Here's what I'm doing: I compare how many Rebs each team got at C (for example), to how many Rebs they got at the other 4 positions, or to the team's overall rebounds above average. For every extra reb at C, a team will get about .67 fewer reb's at the other 4 positions. And so on. Eli, in contrast, is looking at individual players, who might be paired with other players based on their respective rebounding ability.
As for the larger issue of team construction, I'm sure there's some truth but I basically don't buy it. I did the same exercise for points per shot (and turnovers), and do not find any diminishing returns there. GMs and coaches clearly don't say "I've got two good scorers, now let's find some guys who can't score to fill out our team." And to the extent they do say "there's no reason to pay for 3 great rebounders," I think they are recognizing the reality of diminishing returns, not creating the illusion of diminishing returns through their player selection.
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Eli W
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PostPosted: Thu Feb 07, 2008 6:39 pm Post subject: Reply with quote
cherokee_ACB wrote:
Eli, a couple of questions on the blog post:
- How did you compute player rebound rates?
- Do you include the 'worthy' team rebounds in your data, as BasketballValue does? I assume you do, but then you mention an average 73% DR rate, which comes from ignoring team rebounds (DR rate falls to 70% when team rebounds are added)
For the regressions, which were just using data from this season, I calculated rebound rates using BasketballValue's data. PlayerORB/(ORebForOnCourt + DRebOppOnCourt) and PlayerDRB/(DRebForOnCourt + ORebOppOnCourt).
When I looked at data from 73-74 to 06-07, I used data that did not include any team rebounds, and I calculated rebound rate by the typical method of estimating rebound opportunities based on what percentage of his team's minutes a player was on the court.
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herokee_ACB
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PostPosted: Fri Feb 08, 2008 8:59 am Post subject: Reply with quote
Guy wrote:
Here's what I'm doing: I compare how many Rebs each team got at C (for example), to how many Rebs they got at the other 4 positions, or to the team's overall rebounds above average.
Ok, I see. But, still not convinced that is the best method to obtain accurate results, I've done my own analysis. Basically, for every rebound opportunity so far this year, I've computed the sum of the individual rebounding ratios for the defense and the offense, and compared that to the outcome. I'm using basketballvalue data for this. The results:
- Linear regression:
Code:
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 69.73663 2.00587 34.77 <2e-16 ***
drt 0.30864 0.02719 11.35 <2e-16 ***
ort -0.80256 0.04006 -20.04 <2e-16 ***
If I only regress against defense or offense, the coefficients are 0.28 and -0.78, which suggests coaches tend to adapt to the opponent rebounding strength (the coefficient of correlation is small, just 0.07).
- Using a logit model:
Coefficients:
Code:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.84266 0.09811 8.589 <2e-16 ***
drt 1.47949 0.13291 11.131 <2e-16 ***
ort -3.85022 0.19571 -19.673 <2e-16 ***
For a 70-30 distribution, this translates into a 0.31 actual team increase for every extra point in the defensive players' sum, and 0.81 for offense. Essentially, same as above.
- If I estimate actual team rates using the above parameters, and then aggregate all observations into 0.1 wide bins for the expected rate, R-squared is 0.6093. Quite good.
- Correlation is even higher (0.6709) if I use a simple rebounding model for the estimation, where:
playerRate = playerAbility / (sum of playerAbilities on the court)
playerAbility is deduced from the actual player rates and his on-court team rebounding rate. This is similar to the correction MikeG uses in his numbers. Let's plot it (only for bins with more than 100 rebound opportunities):
Conclusions? Yes, I conclude my post here.
Last edited by cherokee_ACB on Wed Feb 13, 2008 4:14 am; edited 1 time in total
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Westy
Joined: 15 Nov 2007
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Location: Chicago, IL
PostPosted: Tue Feb 12, 2008 11:10 am Post subject: Bottom line Reply with quote
So to summarize, what credit should any individual player get for garnering a defensive or offensive rebound, if the baseline for a possession is 1.0?
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Eli W
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PostPosted: Tue Feb 12, 2008 4:48 pm Post subject: Re: Bottom line Reply with quote
Westy wrote:
So to summarize, what credit should any individual player get for garnering a defensive or offensive rebound, if the baseline for a possession is 1.0?
Both less than 1, defensive rebounds less than offensive rebounds, defensive rebounds much less than 1. Possibly differing numbers for players at different positions. More work needs to be done in order to pin down more specific values.
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Eli W
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PostPosted: Sat Feb 23, 2008 2:05 pm Post subject: Reply with quote
I just put up another post on diminishing returns which uses a method suggested to me by Ben and similar to the one Cherokee_ACB used. I think it does a very good job of visually presenting the impact of diminishing returns on rebounding.
http://www.countthebasket.com/blog/2008 ... g-returns/
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cherokee_ACB
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PostPosted: Sun Feb 24, 2008 6:19 am Post subject: Reply with quote
Quote:
There is another issue with this technique that may lead to it underestimating the impact of diminishing returns
True. But, probably, the lowest (highest) projected rebounding units are the result of teams going small (big). If we accept that position affects rebounding percentages then, in those cases, it's normal that the projection underestimates (overestimates) the actual ratios. It's another issue worth looking at.
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Mike G
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PostPosted: Sun Feb 24, 2008 6:32 am Post subject: Reply with quote
I haven't followed this thread very closely, sorry. Eli, in your latest blog post, there's a strikingly weak correlation between (a lineup's) total player DReb% and team DReb%. You conclude that this indicates a diminishing return: Even if you stack your lineup with rebounders, you don't get a lot more (% of) rebounds. I think this is partly true.
What about the opponent's rebounders? If they decide to 'go small', what do you do? You could say, '"We will kill them on the boards"; but you might try to do better. You might go smaller -- which is to say, you'll be playing (with and against) lesser rebounders. Two teams always total 100% of the rebounds.
As small as sample sizes were for team's running with <65% total DReb'ers, it's probably a lot smaller for when the opponent was >70%. I think that to come up with an actual 'marginal value' for the DReb, you have to factor in the opponent's lineup.
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Eli W
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PostPosted: Sun Feb 24, 2008 1:56 pm Post subject: Reply with quote
You're right that that the opponent's lineup should be factored in as well, Mike. Cherokee did that in his study, but mine did not. I will try to look into that more soon.
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Guy
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PostPosted: Sun Feb 24, 2008 2:22 pm Post subject: Reply with quote
Cherokee: I don't follow your argument. Why does Eli need to worry about position when he's looking at 5-man lineups? Seems like he has that covered.
The opposing player point is a good one, though I'll be surprised if it changes the DRB results very much.
Another way this method may understate diminishing returns is by looking at players only as they function in a single season, which usually means on a single team. For example, if Lineup X is projected at 16% ORB%, these guys clearly aren't expected by this coach to do a lot of offensive rebounding. Thrown together for 400 minutes over the season, maybe they come in at 17-18%. But is that really the best these 5 guys could do, if they became a starting five? I'd guess not. Both are valid ways to look at the question, but I think the latter scenario is closer to the question we're usually interested in answering: when a high/low rebounder is added to a team, how much does that increase/reduce team rebounds? One way to get at this might be to use career rebound rates (prior to season being analyzed) to make your projections.
That said, I think this is excellent work by Eli. Keep it coming. (And it would be great if Eli could post the numbers for each of his buckets: total minutes, projected reb%, and actual reb%.)
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cherokee_ACB
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PostPosted: Sun Feb 24, 2008 2:53 pm Post subject: Reply with quote
Guy wrote:
Cherokee: I don't follow your argument. Why does Eli need to worry about position when he's looking at 5-man lineups? Seems like he has that covered.
Take Marion, for instance. He averages 3 rebounds more as a PF. Part of that it's because his teammates are worse rebounders, but I believe position has a bigger impact. The regression assumes we should expect his contribution to team rebounding, in absolute terms, to be the same in both positions, and roughly the average of his SF and PF rebound ratins. This is not entirely realistic. It shouldn't come as a surprise then if lineups with Marion at SF underperform in this analysis, and overperform with him as a PF.
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Guy
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PostPosted: Sun Feb 24, 2008 3:40 pm Post subject: Reply with quote
cherokee_ACB wrote:
It shouldn't come as a surprise then if lineups with Marion at SF underperform in this analysis, and overperform with him as a PF.
OK. But in Eli's overall analysis, won't that be a wash?
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cherokee_ACB
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PostPosted: Sun Feb 24, 2008 4:59 pm Post subject: Reply with quote
Guy wrote:
cherokee_ACB wrote:
It shouldn't come as a surprise then if lineups with Marion at SF underperform in this analysis, and overperform with him as a PF.
OK. But in Eli's overall analysis, won't that be a wash?
Don't think so. The effect is likely to go in the same direction in all cases.
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Eli W
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PostPosted: Mon Feb 25, 2008 5:00 pm Post subject: Reply with quote
I've uploaded a spreadsheet containing all the data I used for the recent study that I did. It includes some formulas so you can see how everything was calculated. The file is around 8mb. Thanks again to Ben for providing the raw data.
http://rapidshare.com/files/94915270/Re ... y.xls.html
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Guy
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PostPosted: Mon Feb 25, 2008 5:05 pm Post subject: Reply with quote
Cherokee: I follow you now. Multi-position players will tend to cause this method to understate diminishing returns, except where two players effectively "share" two positions and often play together (in which case the overperformer will be offset by an underperformer in the same lineup). Not clear how big an impact this will have.
Going back to our earlier exchange, it is hard to square the coefficients you and Eli are getting with my position-based analysis, which seems to show a larger diminishing returns effect. To recap, when I look at positions rather than players, each additional 1% of Reb% is associated with a loss of .75 rebounds at the other 4 positions, and thus a total team gain of just .25%. The advantages of the position approach are that it shouldn't be affected by good rebounders being paired on the floor with bad rebounders. It also doesn't have the "Marion" problem you just raised, and avoids the issue of using year X data to "predict" year X outcomes. Do you see problems with my method that are avoided by your and Eli's studies? (Good faith question -- I'm not sure my approach is any better, just trying to understand why we're getting such different results.)
Eli: would welcome your thoughts as well.
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Harold Almonte
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PostPosted: Mon Feb 25, 2008 6:58 pm Post subject: Reply with quote
This study didn't mention that the WP's regressed DRebs already account for shot defense, then that's the "why" about the bigger weight. If they decide to rate rebounding, they can make OR*0.7, but they couldn't make regressedDReb*0.3, they would need to do it to the real rebounding portion only. Or they, and any other metric, need to find another way to account for shot defense, and to make DReb much less than the VOP. I think for a metric that confessed it doesn't attempt to account shot defense, PER is overrating DRebs as a whole possession and underrating shot defense action, applying the rebounding rating to this action too.
PER is right about the reb. rating, but WP has some alibi when you account for shot defense. In the end WP is not overrating its rebounds too much more than PER is overrating its owns, they just aren't rating it, and adding shot defense value inside, as a product of a team win regression. But the PER notion that DReb is VOP is not right: Advantage WP.
Another thing about DRebs is that it's true that players with position advantage produce diminishing return on teammates when it comes to account rebounds grabbed, but they also produce increasing return (negative slope in a negative quadrant) when it comes to account opponents's rebounds-allowed. I'm surprised the slope of DR%proyected vs. actual isn't more plane.
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cherokee_ACB
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PostPosted: Tue Feb 26, 2008 10:49 am Post subject: Reply with quote
Guy wrote:
Cherokee: I follow you now. Multi-position players will tend to cause this method to understate diminishing returns ...
Do you mean "to overstate"?
Quote:
The advantages of the position approach are that it shouldn't be affected by good rebounders being paired on the floor with bad rebounders.
I already disagreed with this. Pairs like DHoward-Rashard are the biggest problem with your approach. And it's a big problem. If you want to argue for diminishing returns, you shouldn't use data with a bias in favor of your position. Well, at least I'd try not to do that.
Going back to Eli's 2nd issue in his blog post ("If players always played with the same four teammates ..."), I've ran the regressions separately for each team. This way, I focus on the rebounding ratios differences within a roster. Nothing changed. Of course, standard errors are higher and coefficients vary across teams, but averages stay at 0.3 and 0.8.
Last edited by cherokee_ACB on Tue Feb 26, 2008 11:44 am; edited 1 time in tota