Advanced Stats for Basketball

February 23, 2008

More Diminishing Returns

Posted by Eli in Stat Theory, Studies

Following up on my last post, I’m going to look at the issue of diminishing returns for rebounding from a different angle. The new method I’m going to use has several advantages over the previous one (and some disadvantages). What I like best about it is that it does a great job of presenting the effect of diminishing returns visually, rather than just through a table of numbers.

The approach I will use was first suggested to me by Ben F. from the APBRmetrics forum. But before I got a chance to try it out, another poster, Cherokee_ACB, presented results of his own using a similar method. So this post can be seen as building on the ideas of both of these posters.

Instead of comparing individual players’ rebounding percentages to the rebounding percentages of the lineups they played in, this method takes into account the rebounding of all five players on the court for a team. Instead of just speculating about how well a team would rebound if it put five strong rebounders on the court together (or five poor rebounders), it looks at what has actually happened in such situations in the past.

For each five-man lineup that has played together this season, I created a projected offensive rebounding percentage and a projected defensive rebounding percentage. To get these I simply summed the season offensive (or defensive) rebounding percentages of each of the five players in the lineup. If there are no diminishing returns in rebounding, one would assume that rebounding percentages are simply additive in this way. I then compared each lineup’s projected rebounding percentage to its actual rebounding percentage in the minutes it played together.

Because many lineups only play together for a few minutes, I combined lineups into bins based on their projected rebounding percentages, and looked at the total rebounding percentages for each bin. For instance, I combined all lineups with a projected offensive rebounding percentage between 20%-22%, and calculated the actual overall offensive rebounding percentage of those lineups by summing all of the lineups’ offensive rebounds and dividing by the sum of all the lineups’ offensive rebound opportunities. This controls for the randomness present in lineups with small sample sizes.

Using data through January 31st that I got from Ben F. (unfortunately BasketballValue data doesn’t work great for this kind of study because it includes some team rebounds), I combined lineups based on their projected rebounding percentages into bins with a width of 0.02 (e.g. projected ORB% from 20%-22%, projected DRB% from 74%-76%, etc.). I then plotted the projected vs. actual rebounding percentages for all the bins that contained at least 1000 rebound opportunities (for ORB% this meant I looked at lineups with a projected ORB% from 16%-38%, and for DRB% the range was 60%-86%). Here are the results for offensive rebounding in graphical form:

The blue dots mark the values for each bin (e.g. the dot that aligns with 0.25 on the X-axis represents the 24%-26% bin). The gray dashed line indicates what we would expect if there were no diminishing returns and the rebounding percentage for a lineup was simply the sum of the rebounding percentages of the five individual players in the lineup. Where the blue line is above the gray line, this indicates that those lineups rebounded better than one would have projected by simply summing the players’ rebounding percentages. And where the blue line is below the gray line, the lineups rebounded worse than the projection.

Overall, this picture shows that summing player offensive rebounding does a pretty good job of predicting lineup offensive rebounding. The blue line basically tracks the gray line. However, there does appear to be evidence of a slight impact from diminishing returns. For lower projected ORB% lineups, the actual ORB% is slightly higher than the projection, which suggests that players with low offensive rebounding percentages don’t hurt their team’s offensive rebounding quite as much as would be expected. And for higher projected ORB% lineups, the actual ORB% is slightly lower than the projection, which suggests that players with high offensive rebounding percentages don’t help their team’s offensive rebounding quite as much as would be expected. So on the whole this method suggests there may be some slight diminishing returns effect on offensive rebounding.

Now let’s look at defensive rebounding:

Here we see a much different picture. The projections don’t do nearly as good a job of predicting the actual lineup defensive rebounding percentages. At the bottom, lineups with a projected DRB% from 60%-62% actually had a DRB% of more than 70%. And at the other extreme, lineups with a projected DRB% from 84%-86% actually had a DRB% of just 78%. The range of actual lineup defensive rebounding percentages was much smaller than the range that would be predicted by summing the defensive rebounding percentages of the players making up those lineups. All of this suggests a large diminishing returns effect on defensive rebounding. Even if you put together a lineup of players with a very low combined DRB%, such lineups will typically be only slightly below average on the defensive glass. And lineups composed of players with a very high combined DRB% will be above average on the defensive glass, but not by a huge amount. This is more evidence that the marginal value of a player defensive rebound is much less than one on the team level.

If we fit some regression lines to these charts, we get slopes of 0.77 for offensive rebounding and 0.29 for defensive rebounding (for ORB% the R^2 is 0.98 and the SE for the coefficient is 0.036; for DRB% the R^2 is 0.93 and the SE is 0.023). These suggest that each player offensive rebound contributes around 0.8 offensive rebounds to the team total, and each player defensive rebound contributes around 0.3 defensive rebounds to the team total (these figures are very close to those that Cherokee_ACB arrived at in his study). These aren’t definitive numbers for a variety of reasons, but again they provide strong evidence for a large diminishing returns effect on defensive rebounding.

Advantages and disadvantages of this methodology

A big advantage to this method is that you don’t have to worry about lineup balancing issues. Even if coaches tend to pair good rebounders with poor rebounders, this won’t skew the results because we’re taking into account the season rebounding percentages of all five players in each lineup.

A downside of this method is that by looking at lineups as a whole, we can’t isolate the different positions, and there is some evidence that the marginal value of a rebound varies by position. However, one could make some adjustments to this method to break things down by position (instead of one independent variable summing the season rebounding percentages of the five players in each lineup, split each lineup into five independent variables with one being the PG’s season rebounding percentage, one the SG’s season rebounding percentage, and so on). That’s something I may take a look at in the future.

There is another issue with this technique that may lead to it underestimating the impact of diminishing returns. If players always played with the same four teammates, then projected lineup rebounding percentages would exactly predict actual lineup rebounding percentages (since each player’s season rebounding percentage would have been accumulated solely in the context of the lineup we were projecting). Of course, in reality, players play in a lot of different lineups over the course of the season, and that diminishes this effect. But to the extent that players do tend to play with many of the same teammates, the projected rebounding percentages will be artificially pushed closer to the actual rebounding percentages. So it’s possible that the estimated marginal values of 0.77 for offensive rebounds and 0.29 for defensive rebounds may actually be too high, and diminishing returns may have a larger effect on both offensive and defensive rebounding. There are ways to try to control for this issue, but I’m not going to attempt any of them now.


In some ways I think this study provides stronger evidence for the impact of diminishing returns on defensive rebounding than my previous post. The charts allow one to easily see the effects of diminishing returns, and by looking at the rebounding of all the players in each lineup, the issues brought up by coaches potentially pairing good rebounders with poor rebounders are largely eliminated.

The specific marginal values found of 0.8 for offensive rebounds and 0.3 for defensive rebounds are also interesting. These match closely with how John Hollinger’s PER weights offensive rebounds relative to defensive rebounds (ORB are weighted by the league DRB%, which is around 0.7, and DRB are weighted by the league ORB%, which is around 0.3). And again, these values suggest that Dave Berri’s Wins Produced greatly overvalues players with high defensive rebounding percentages and undervalues players with low defensive rebounding percentages because the system assumes that each player DRB contributes a full DRB on the team level. Alternative Win Score (or AWS), the variation on Wins Produced suggested by Dan Rosenbaum in his paper, “The Pot Calling the Kettle Black”, weights ORB at 0.7 and DRB at 0.3. While these values are based on an assumption and not backed by evidence (just like Berri’s assumption that both should be weighted at 1 is not backed by any evidence), the evidence from the study I have done here (and Cherokee_ACB’s study) suggests that AWS (and PER) may be a lot closer to the mark on rebounding than Wins Produced.


  1. I hate to do this, I really really do, but I am psychologically incapable of preventing myself from nitpicking about graphs. Those look good (you put both graphs on the same scale, x- and y-axes — excellent), but you should really get rid of the grid lines, which are completely unnecessary (they almost alway are). Also, you should consider showing each of those graphs as residual plots — ie make that dashed diagonal line the horizontal zero line.

    More later.

    Comment by edkupfer — February 23, 2008

  2. Fair enough. I put in the grid lines to make it easier to eyeball exactly what the actual value was for each projected lineup, especially since I didn’t also present the numbers in a table. And I considered residual plots, but I thought these charts would make it easier to see the impact of diminishing returns for those who have no clue what a “residual” is.

    Comment by EliFebruary 23, 2008

  3. BRAVO, Eli! I posted a bunch of times on the Wages of Wins site a few months ago ragging on how Berri completely overvalues rebounding and you have certainly corroborated my (and MANY others) feelings. Great work.

    I especially like the point you made above that a group of bad rebounders will gain almost the average amount of rebounds when they play together. This certainly implies my long-standing belief (and numerous posts at WoW) that rebounding skill is just not a scarce commodity. Berri (at WoW) just can’t get it through his head that just because players are consistent in their rebounding from period to period does NOT mean that this is not an easily substitutable resource… You can (without great difficulty) substitute one player for another to grab the rebounds.

    At any rate, congrats on the great work. I left three posts complementing and citing you at the Wages of Wins site… but Berri deleted them all.

    Comment by TG Randini — February 24, 2008

  4. Eli what do you think about using a similar appraoch for team FG%?
    I’d be interested in that at the aggregrate league level, team level and with particular big name “shooters” and “scorers”.

    Comment by Mountain — February 24, 2008

  5. Since a gained offensive rebound means a lost defensive rebound, I’m not seeing intuitively how one can be more variable than the other. It’s possible that I’m missing something, but I do wonder if the exclusion of team rebounds may be contributing to the results.

    And could you explain why you discounted data including team rebounds? Since many of these are real changes of possession (I suspect most are simply missed shots that go out of bounds without being touched–effectively a defensive rebound and effectively the prevention of an offensive rebound) I would think that these would be important to consider when looking at rebounding rates.

    Comment by Jason — February 25, 2008

  6. Mountain, that’s something I want to look at in the future. This method could be used for a variety of stats. The tricky part with shooting is that lineup FG% is a combination of both individual player FG percentages and individual player shot attempts.

    Comment by EliFebruary 25, 2008

  7. Jason, regarding the differing variances of two seemingly equal but opposite stats, I discussed that more in this previous post:

    The basic idea is that while the league means of two stats like FT% and FT% allowed must be the same, their variances need not be identical or even close. That of course is the team level, but the same kind of thing applies on the player level.

    As for team rebounds, they were not included because I was trying to predict lineup rebounding based on player rebounding, and player rebounding by definition doesn’t include any team rebounds. So all the projections would come out low, as none of the five players’ rebounding percentages would capture the team rebounds present in the lineup totals. I did try the study with team rebounds included, and the result was just that the blue lines were shifted up slightly, but with the same slopes.

    Comment by EliFebruary 25, 2008

  8. Maybe the individual player FG percentages and individual player shot attempts could be made into a product and cast in terms of shooting points created above or below league average. That could be added up for a hypothetical lineup and for a real lineup.

    Comment by Mountain — February 25, 2008

  9. If it is done at team level then seeing the results with and without the star would be an interesting split below the team level story.

    At league level does the inclusion of a + whatever level individual on shooting points created above average see increasing returns or decreasing returns for rest of team and total team? What is the tipping point? Looking at the list of such players do any characteristics correlate highly with increasing or diminishing returns? Position, superior or inferior to league average FG%, FGA level, experience, presence of a superior passer, superior passing by this particular passer, etc. some acummulation of positive factors?

    Comment by Mountain — February 25, 2008

  10. Jason, I posted this on Berri’s site but for some reason it’s not showing up:

    Here’s another way of explaining it that may be easier to understand.

    Imagine a league where coaches had vastly differing strategies on crashing the offensive glass. Some sent all five players to the boards, while some sent zero players and had everyone get back on defense instead. This variation in strategies would lead to a large team-to-team variation in ORB%, with teams that sent five players to the offensive glass having much higher offensive rebounding percentages than those that sent zero. On the other hand, suppose that at the same time, all coaches used the same strategy when it came to defensive rebounding - send three players to the glass and have the other two leak out down the court for potential fast break opportunities. Because all teams used the same strategy, there would be little variation in DRB%. The end result would be greater variation between teams in ORB% than DRB% - there’s nothing contradictory about that.

    The key is that for each team, their ORB% is independent of their DRB% - they happen at different ends of the court. It’s true that a team’s ORB% is identical to (one minus) their opponent’s DRB%, but that’s just like how a team’s FT% is identical to their opponent’s FT% allowed.

    Or going back to the FT% example, imagine a league where those where made baskets didn’t count for any points, and you could only get points by making FTs. In that league teams would vary fairly widely in their points scored (because some teams have better free throw shooters than others), but vary little in their points allowed (because all teams are about equal in free throw “defense”). This would be the case in spite of the fact that for any game, one team’s points scored would equal the other team’s points allowed, just as for any game, one team’s ORB% equals (one minus) the other team’s DRB%.

    There was a post on Berri’s site last night by Guy giving some actual standard deviations from last season, but unfortunately it looks like it was deleted. So I went ahead and put together a quick spreadsheet with the data just from last year. The SD’s are close but not the same because they are really not the same. You can try it yourself for any other season - there’s no hiccup in the raw data that’s causing it.

    Anyways, I don’t want to stray too far off topic. This issue doesn’t really affect the issue of diminishing returns in rebounding, where the theory and data both show differing effects on the offensive and defensive glass.

    Comment by EliFebruary 25, 2008

  11. Here’s a simple simulation to illustrate the point.

    Three team league, two games in the season.

    A vs. B: A has 5 ORB, B has 5 DRB, A has 7 DRB, B has 3 ORB
    A vs. C: A has 5 ORB, C has 5 DRB, A has 9 DRB, C has 1 ORB
    B vs. C: B has 3 ORB, C has 7 DRB, B has 8 DRB, C has 2 ORB

    A: 10 ORB in 20 chances, 50% ORB
    B: 6 ORB in 20 chances, 30% ORB
    C: 3 ORB in 20 chances, 15% ORB
    League: 19 ORB in 60 chances, 32%

    A: 16 DRB in 20 chances, 80% DRB
    B: 13 DRB in 20 chances, 65% DRB
    C: 12 DRB in 20 chances, 60% DRB
    League: 41 DRB in 60 chances, 68%

    SD(ORB%) = 0.18
    SD(DRB%) = 0.10

    Comment by EliFebruary 25, 2008

  12. Despite what edkupfer said, as a professional statistician, I believe for the audience this article is intended for your choice in characteristics of the graphs is optimal. Nice analysis.

    Comment by Jimbo — February 25, 2008

  13. I’ve uploaded a spreadsheet containing all the data I used for the study. 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.

    Comment by EliFebruary 25, 2008

  14. In some defense of David Berri, most of his argument for the valuation of a rebound is that rebounds are very valuable to winning. Now, if marginal rebounds only contribute a bit to winning, then with this analysis, he greatly overvalues them. But if they contribute a lot to winning, even adding a small chance of an additional rebound is very valuable. I mean, it’s theoretically possible that he undervalues them if it turns out that rebounds are the end all and be all to winning. (clearly they are not, just making an extreme case argument.)

    The fact that more rebounding talent has diminishing marginal returns (which I think even Berri would agree with, especially when presented with this data) doesn’t address the question of how valuable that marginal rebound is.

    Echoing a previous commenter I would be curious to see similar work for offensive efficiency metrics. I would imagine that you would see similar diminishing marginal returns. But again that doesn’t address of the value of adding at the margins. And the answer is, probably a lot.

    Comment by Rashad — February 25, 2008

  15. Interesting post. However, this jumps out at me:

    “The key is that for each team, their ORB% is independent of their DRB% - they happen at different ends of the court. It’s true that a team’s ORB% is identical to (one minus) their opponent’s DRB%, but that’s just like how a team’s FT% is identical to their opponent’s FT% allowed.”

    This is simply not true, and frankly displays a pretty glaring lack of insight into how two variables might co-relate, unless you believe that grabbing defensive rebounds is a skill-less endeavor.

    That is, a team that boxes out well will rebound well defensively, and thus prevent offensive rebounds. Therefore, a teams DRB% directly influences their opponents’ ORB% (unless you believe that DRB% isn’t a consequence of boxing out well) — how much and how to quantify this is a difficult question to answer, but if you truly believe that rebounding well defensively has no impact on offensive rebounding, then I suggest that you haven’t really thought this true. The reverse is true as well, of course, teams that crash the boards a lot will lower their opponents DRB%. BUT, this effect should be smaller when playing a good defensive rebounding team and larger when playing a poor defensive rebounding team

    With ft% allowed, there is absolutely no (legal) way for any given team to influence the other teams FT%, other than perhaps some minor effects, like being very smart about not fouling Chauncey Billups, or by passing out “thundersticks” that may (or may not) distract opposing FT shooters.

    Comment by Patrick MintonFebruary 25, 2008

  16. All of that is exactly right, and when I wrote that I thought about offering a qualification. My point was just that for the purposes of Jason’s contention about differing variances, a team’s ORB% and its DRB% are basically independent. Meaning that they don’t have the same lock-step relationship as team ORB% and opponent’s DRB%.

    I didn’t at all mean to imply that there wasn’t a connection between a team’s ORB% and its DRB%. In a previous post I discussed that relationship and how both share some common factors, finding a stong correlation of 0.77 between the two:

    Obviously many of the skills involved in being a good or bad offensive rebounder are the same skills that make one a good or bad defensive rebounder. Though what I think is most interesting are the differences that make some players (and teams) better at one than the other.

    Comment by EliFebruary 25, 2008

  17. I do think that on the team level, offensive rebounding has a greater impact than defensive rebounding though. I discussed that more here:

    Comment by EliFebruary 25, 2008

  18. “I didn’t at all mean to imply that there wasn’t a connection between a team’s ORB% and its DRB%. In a previous post I discussed that relationship and how both share some common factors, finding a stong correlation of 0.77 between the two.”

    Eli: that r of .77 is at the player level, right? As best I can tell, there is little or no correlation at the team level (.05 last season). Which is itself an interesting issue: how can the two be correlated at the player level, but not at the team level? The only answer I can see is that the player stats have a very weak relationship with team totals, i.e. diminishing returns.

    Rashad: I have looked at shooting efficiency a bit, and see no sign of diminishing returns there. In fact, there appears to be a “multiplier effect,” in which greater efficiency from position X is associated with greater efficiency at the teams’ other 4 positions. However, this could reflect team construction as opposed to an actual player interaction effect.

    Comment by Guy — February 25, 2008

  19. Yeah Guy, that’s the player level. And of course a lot of that correlation comes from the fact that big men are better rebounders than guards on both the offensive and defensive glass. That factor is diminished on the team level, which is one reason the correlation isn’t as high. But diminishing returns is also likely a big factor as you mention.

    Comment by EliFebruary 25, 2008

  20. Guy: I agree with you on the shooting efficiency “multiplier effect”. On the Feb 22/23 Kidd WoW blog, I had posted “The value component of Win Score that is most susceptible to diminishing returns is rebounds. When the ball is clanging off the rim, there is only one ball and (up to) five team members engaged in acquiring it. This is not the case with efficient shooting. Heck, the more good shooters you have, the better, because the other team can’t key on just one or two players. There may even be a synergy with putting several good shooters together… where their shooting %’s as a team INCREASES… When looking at trade values and incremental improvements… you have to look at the components of win score to see which components will translate at 100% and which ones will most likely be diminished.”

    (By the way, that was my next-to-last post on that site before Berri started deleting my posts complimenting Eli’s work. Obiviously, I’ll never post there again.)

    Great work, Eli. When Berri referred to a comment about “putting the nail in the coffin” regarding his (rebounding) work, that was one of my (deleted) comments citing your excellent work.

    Comment by TG Randini — February 26, 2008

  21. Eli, just out of curiosity, would you be willing to compute/share the Jason Kidd numbers by this method in NJ and Dallas? Are diminishing returns with him higher or lower than average for all players or just PGs?

    Across the league for the cases where returns vs projected vary by more than some amount (positive or negative) are any type of players present or absent more often than others or than usual for that type? I’m thinking of criteria like younng vs vet, especially tall or short, presence of big “hops” or “great strength”, the great box out guy vs out of area rebounder (does he rob his teammates more often?). the effects probably will vary on offensive and defensive glass. Just some possibilities to continue to push further.

    Comment by Mountain — February 27, 2008

  22. I don’t have continual access to this data throughout the season, but after the season I could try to go back and look at Kidd specifically.

    Comment by EliFebruary 29, 2008

  23. Ok, just an option that would probably draw a fair amount of reader interest.

    Analyzing who / which type player affects the diminishing returns and what direction would seem to a necessary step to get to the point of helping or evaluating team construction choices.

    Comment by Mountain — February 29, 2008

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