There has been a lot of discussion in recent months about the importance of rebounding on the player level. Much of this debate has been in reaction to the high value that Dave Berri’s Wins Produced player rating puts on rebounds. On Berri’s blog there have several posts with long, insightful debates in the comments about the issue (that is, if you ignore the unfortunate mudslinging often directed at those with differing points of view). In particular, I would recommend the comments sections of “The Best One-Two Punch in the Association”, “Chris Paul vs. Deron Williams, Again”, and “How Has Texas Survived the Loss of Kevin Durant?”. There have also been some good debates on the topic in the APBRmetrics threads, “Current season Win Scores/Wins Produced” and “Can some one explain the ‘possession cost’ scheme?”.
These are wide-ranging debates, involving such issues as the relative value of rebounding versus scoring and the apportioning of credit for a defensive stop between the defensive rebounder and his teammates. The issue that I want to pick up on is the extent to which the law of diminishing returns applies to rebounding.
Diminishing returns in basketball
The idea behind diminishing returns in basketball is that in some areas of the game, players aren’t just competing with the five opposing players on the court, they’re also competing with their four teammates. For example, no matter how many shot-creators/high-usage-players/ballhogs a team has, on each play the team can take at most one shot. Assemble a lineup of five players who each typically take 30% of their team’s shots when they’re in the game and something’s got to give.
But diminishing returns aren’t just relevant to the extreme situations when a team pushes up against a fixed ceiling. They are also important in determining the values of various player statistics on the basis of how much they contribute to team wins. Wins Produced was constructed by determining (through regression) the extent to which different team statistics contributed to wins, and then working downwards to the level of player statistics. The translation from the player level to the team level is complicated by diminishing returns. If there are diminishing returns for a stat like rebounds, then for each rebound a player gets he shouldn’t be credited for adding a full rebound to his team’s total because had he not grabbed that rebound, some percent of the time his team would have got it anyway.
How can we measure diminishing returns?
An initial hint that there could be diminishing returns for rebounding comes from the fact that the defense rebounds misses far more frequently than the offense. Last season 73% of missed shots were rebounded by the defense while 27% were rebounded by the offense. One might be tempted to say that there must be diminishing returns for defensive rebounds - even if a defensive player tried and failed to grab a rebound, his teammates still would have got it over 70% of the time. But this actually does not follow, and thus the offense/defense split fails to provide evidence for diminishing returns.
To see why this is so we can construct a simplified (and fictional) model of rebounding where the defense grabs 73% of misses but where there are clearly no diminishing returns. Imagine that for each missed shot, only two players are in position to grab the rebound - the defensive player closest to the spot the ball caroms to, and the offensive player that he was guarding. Because prior to the shot defenders are typically positioned closer to the basket than the man they are guarding, the defensive player will have the advantage, which leads to the defender grabbing the rebound 73% of the time on average (this doesn’t have to be a fixed number - some players may defensive rebound more or less than 73% of the caroms that come to them due to their height, weight and leaping ability relative to the players that they guard). But in this model, when the defender closest to the carom fails to grab the rebound, the offensive player will grab it every single time. Thus there are no diminishing returns - if the defender nearest the carom fails to grab the rebound, there is a 0% chance that one of his teammates will rebound the ball. When a player gets a defensive rebound he should be credited with adding a full rebound to his team’s total.
Obviously this model isn’t what’s really going on in the NBA. Rebounding isn’t just a one-on-one matchup of a defender and the player he was guarding. Sometimes the ball caroms in between two or more teammates, some players are better than others at getting “out-of-area” rebounds, etc. (for a starting point on making a better model of NBA rebounding, see this 82games article). But the point of the model was just to show that the fact that defenders rebound 73% of misses does not imply that had a defender failed to grab a rebound that came his way, most of the time (or even 1% of the time) one of his teammates would have rebounded it anyway. The 73%/27% defense/offense split isn’t evidence for diminishing returns for defensive rebounding.
So we need some other methods to weed out diminishing returns. The clever sabermetrician-turned-APBRmetrician Guy did a few small studies to try to get at the heart of the issue, comparing the variances of rebounds at the team and player levels, and looking at the correlation between the rebounding of a team’s top rebounder and his teammates. Some people have suggested looking at how team rebounding levels change when a team replaces a poor rebounder with a good one, and a few cases of this have been examined on an anectodal level. The poster cherokee_ACB looked at some data on how players rebounded when paired in the frontcourt with great rebounders vs. poor rebounders.
These are all good starts, but I think one has to step back and take a larger look to really see what’s going on. So I’m going to try to go through a variety of statistical studies of rebounding, some of which are new and some of which I’ve written about previously. The ultimate goal is to try to get a better understanding of the value of offensive and defensive rebounds.
Regressing team rebounds onto player rebounds
The first study I will look at involves regressing team rebounds onto player rebounds in order to determine the marginal value of a rebound (i.e. how many rebounds each player rebound contributes to the team total). Many posters such as Guy, Jason and Panda Bear have suggested that such a study be done, but the difficulty comes from the fact that it requires data for each player on how many rebounds his team and the opposing team got in the minutes when the player was on the court. Prior to this season, such data wasn’t publicly available without mining through play-by-plays, but a few years ago Ed Küpfer managed to get some data from 82games and performed the kind of regression that people have been asking for. You can see his results here (minus the charts, which are now dead image links). BasketballValue just recently made the needed data available on their site for the current season, so I am going to repeat Ed’s study on that new data set, adding in a few twists of my own. In discussing the results that I obtained I will also compare them to Ed’s results, and tie both into the context of the current debates.
Though technically this study involves linear regression, don’t let that scare you off. Since we’re just looking at one independent variable (player rebounds), it’s pretty easy to understand what’s going on. Basically, we’re making a scatter plot with player rebounding percentage on the X-axis and team rebounding percentage (when that player was on the court) on the Y-axis. Once we plot enough players, we’ll start to see a trend - the dots will generally rise from the lower left of the chart to the upper right. This is because the team lineups with a poor rebounder in them will generally rebound worse than lineups with a good rebounder in them. The question we’re interested in is how steep the rise is from the lower left to upper right. Does a large increase in player rebounding mean a large increase in team rebounding (reflected by a large slope), or just a small one (reflected by a small slope)? The smaller the slope, the greater the impact of diminishing returns on rebounding (lineups with “great” rebounders in them don’t rebound as well as might be expected because those rebounders have seemingly great numbers due to them taking some of their rebounds away from their teammates rather than from their opponents). The actual regression comes into play in calculating the slope of the line that best fits the cluster of points.
Using data through the games of 1/23 (basically half the season), here are the results in chart form (using the 264 players who were on the court for at least 400 offensive rebound opportunities and 400 defensive rebound opportunities).
First, for offensive rebounding:
Next, for defensive rebounding:
The key number in each chart is the slope of the regression line (the coefficient on the independent variable), which I highlighted in red. The slope for offensive rebounding is 0.21 and the slope for defensive rebounding is 0.06 (the standard error for the ORB% coefficient was 0.052 and the standard error for the DRB% coefficient was 0.025). These numbers are similar to those that Ed found in his study - he got 0.25 for offensive rebounding and 0.06 for defensive rebounding.
Suprisingly, both of these slopes are far below 1.0, which is what we might expect were there no diminishing returns (each additional percentage point in player rebounding would add one percentage point to team rebounding). And the slope for defensive rebounding is shockingly low at 0.06, perhaps suggesting that diminishing returns have a large impact in this area and that “great” defensive rebounders don’t really have much of an impact on their team’s defensive rebounding. But there’s more to this story.
Why a marginal value less than 1.0 does not imply diminishing returns
One issue that some have brought up is the possibility that coaches try to balance out their lineups by pairing good rebounders with poor ones, such as Orlando being willing to play undersized players at PF knowing that they will have Dwight Howard alongside them. If this is so, then the impact of a good rebounder may not fully reflected in looking at how the team rebounds when he is in the game, since he is often burdened by playing alongside poor rebounding teammates (and conversely, poor rebounders will look better on the team level because they are paired with good rebounders). In other words, if teams balance out the rebounding in their lineups, then the slope of player rebounding to team rebounding (the marginal value of a rebound) will be depressed below one even if there are no diminishing returns.
One might think that this isn’t that big of a deal since coaches take into account a lot more than just rebounding in deciding who to play together. In addition, some coaches might go the other direction and pair up good rebounders to have an even greater advantage on the glass. And because of factors such as foul trouble, injuries, and roster limitations, ultimately teams are going to give minutes to all variety of lineup combinations. All of these rationales would seem to mitigate the impact of lineup choices on the marginal value of a rebound, but in reality the impact is large and pervasive because of an obvious but overlooked way in which teams balance their lineups.
In every lineup that every team puts out on the court, the coach balances out good rebounders with poor rebounders. How so? By including in each lineup both big men and guards.
Consider the case of two players. One has an offensive rebounding percentage of 9%, the other has an offensive rebounding percentage of 3%. If we assume a simplified model with no diminishing returns we would expect the team’s lineups with the good offensive rebounder to have a team ORB% that is 6 percentage points (9% - 3%) greater than the team ORB% for the lineups with the bad offensive rebounder (e.g. using 6% as an approximation of the average player ORB%: 9% + 6% + 6% + 6% + 6% = 33% tmORB% vs. 3% + 6% + 6% + 6% + 6% = 27% tmORB%).
But what if the 9% ORB% player is a center and the 3% ORB% player is a point guard? Consider the four teammates that each will typically be paired with. Both will play alongside a SG, SF & PF, but the C will have a poor rebounding PG rounding out his four courtmates, while the PG will have a good rebounding C instead. In other words, PGs will almost always be playing with four players who collectively rebound better than the four players that Cs typically play with. This will reduce the slope of player rebounding to team rebounding. Instead of a 6 percentage point difference in team ORB%, something more like this could happen: 9% + 4% (SG) + 6% (SF) + 8% (PF) + 3% (PG) = 30% tmORB% vs. 3% + 4% (SG) + 6% (SF) + 8% (PF) + 9% (C) = 30% tmORB%. In this extreme case, the marginal value of an offensive rebound would be reduced all the way to zero. The team lineups with the 9% offensive rebounder rebound no better than the lineups with the 3% offensive rebounder - there is no increasing trend. But even if this were the case, this would not be evidence of diminishing returns nor would it be reason to give no credit to players for offensive rebounds.
Here’s how this could look graphically, using defensive rebounds for this example:
Here we can see that team lineups with a PG that grabs 1% of defensive rebounds rebound no worse than lineups with a C that grabs 21% of defensive rebounds (both are at 63%). This is because despite the 20 percentage point difference, each of those players rebounds the same relative to their position, and we’re assuming that players are teamed up with four teammates who are each average rebounders for their position. Unfortunately, the regression line doesn’t account for this, and thus we get a very small slope of 0.04. However, if we look within each position, we can see that the slope of those diagonals is 1.0 - each percentage point increase in PG DRB% increases the team DRB% by one percentage point, and the same goes for SGs, SFs, PFs, and Cs. In this situation, the slope of the overall regression line misleads us into thinking there is a huge effect from diminishing returns, when in fact the marginal value of a defensive rebound is really one and there are no diminishing returns. This confound arises from teams balancing their lineups with good and bad rebounders by matching big men with guards in each lineup.
Is the lineup balancing issue affecting our results?
The first question to address is whether this theoretical issue of lineup balancing is actually affecting the data we used to find that the marginal value for offensive rebounding percentage was 0.21 and the marginal value for defensive rebounding percentage was 0.06. For it to come into play, two things have to be true - lineups have to be composed of a mix of positions, and rebounding has to vary by position. The first seems obviously true - some teams play bigger or smaller lineups than others, but basically every team uses lineups with two guards, a small forward, and two big men (rather than putting five PF/Cs in the game, or five PG/SGs). The second condition also seems obviously true, but it’s worth quantifying.
DatabaseBasketball classifies all players as guards, forwards, or centers. Here is how rebounding breaks down by position for all player-seasons from the 73-74 to 06-07 seasons:
ORB% DRB% ---- ----- Guards 3.2% 8.7% Forwards 7.8% 16.0% Centers 9.6% 20.7%
If we’re willing to use height as a proxy for position, I’ve previously discussed the high correlation between height and rebounding. Using the same group of players from 73-74 to 06-07, the correlation of height to ORB% is 0.69 and the correlation of height to DRB% is 0.74. Here is a chart breaking down rebounding percentages by player height:
So, as suspected, rebounding varies by position. One way of reformulating the lineup balancing issue is to think about it in terms of positional averages. PGs have a lower rebounding percentage than SGs, who have a lower rebounding percentage than SFs, etc. Thus PGs will dominate the points on the far left side of the chart of player rebounding vs. team rebounding, and Cs will dominate the points on the far right. But as we move from left to right (low player rebounding percentage to high), will the Y-values (team rebounding percentage) increase? Do lineups with PGs in them have a lower team rebounding percentage than lineups with SGs in them, which have a lower team rebounding percentage than lineups with SFs in them, etc.? Obviously not - lineups with PGs in them have the same team rebounding percentage as lineups with Cs in them. This will make the trend of the chart flat rather than increasing from lower left to upper right. Lineup balancing results in little “rise” in team rebounding percentage over the “run” of player rebounding percentage, leading to a smaller slope even in the absence of diminishing returns.
How can we work around the lineup balancing issue?
One way to try to get at the effect of diminishing returns is to control for position and compare centers to other centers and point guards to other point guards. Another way is to compare the marginal value of offensive rebounds to the marginal value of defensive rebounds, assuming that both are equally affected by the lineup balancing issue (and thus that the effect would be cancelled out).
The problem with the first method is that the lineup data I have from BasketballValue is already a small sample of just half of one season, and by dividing that into five the potential for small sample size issues multiplies. Nevertheless, it’s worth looking at.
Instead of using height or position designations, I separated players by their average height ordering using the method I discussed in a previous post. A height ordering of 1.0 means the player played all of his minutes as the shortest player on the court for his team. 2.0 means the player was typically the second shortest on the court for his team, and so on. This method results in a number that’s a hybrid of height and position. I grouped players into those with mean height orderings from 1.0 to 1.5 (”point guards”), 1.5 to 2.5 (”shooting guards”), 2.5 to 3.5 (”small forwards”), 3.5 to 4.5 (”power forwards”), and 4.5 to 5.0 (”centers”). These groupings divided up the players very evenly with each of the five containing between 19 and 21% of all players. I’ll also list the results when dividing players by height, though it’s difficult to get five equal-size groupings that way (which is what we want when we’re trying to approximate positions). As in the original regression, the sample is limited to players who were on the court for at least 400 offensive rebound opportunities and 400 defensive rebound opportunities (”SE” is the standard error for the coefficient).
Regressing team rebounds onto player rebounds by position: Mean Ht ORB% DRB% ORB% DRB% ORB% DRB% # of Ordering Slope Slope R^2 R^2 SE SE Plyrs --------- ----- ----- ----- ----- ---- ---- ----- 1.0 - 1.5 0.56 0.27 0.025 0.081 0.49 0.12 54 1.5 - 2.5 0.21 0.10 0.015 0.017 0.24 0.11 52 2.5 - 3.5 0.56 -0.01 0.173 0.000 0.17 0.10 51 3.5 - 4.5 0.47 0.12 0.192 0.053 0.13 0.07 56 4.5 - 5.0 0.49 0.02 0.182 0.001 0.15 0.07 51 Height ORB% DRB% ORB% DRB% ORB% DRB% # of (in) Slope Slope R^2 R^2 SE SE Plyrs ------- ----- ----- ----- ----- ---- ---- ----- <= 75 0.43 0.22 0.013 0.027 0.49 0.17 62 76 - 78 0.24 0.17 0.026 0.072 0.22 0.09 48 79 - 80 0.70 -0.05 0.308 0.007 0.14 0.08 56 81 - 82 0.33 -0.03 0.114 0.004 0.13 0.07 55 >= 83 0.36 0.05 0.124 0.010 0.15 0.07 43
Remember, the original slopes that we got without separating by position were 0.21 for ORB% and 0.06 for DRB%. Most of the positional ORB% slopes are significantly higher than 0.21, suggesting that that figure may have been artificially depressed by the lineup balancing issue and that the role of diminishing returns may be less than originally suspected. But for DRB%, even when broken down by position, the slopes are still very low (except for guards), and in some cases even negative. This suggests that diminishing returns may play a large role in defensive rebounding for big men, as lineups with good defensive rebounding bigs aren’t significantly better on the defensive glass than lineups with poor defensive rebounding bigs.
However, given the small sample sizes, these are very preliminary conclusions. Ed had more data in his study, and when he broke players down by height (he used four groupings instead of five) he also found much higher slopes for ORB% than DRB%, but his specific values for different groupings differ from mine. In the future I hope to obtain lineup data from past seasons and repeat the methods I’ve used on a larger sample.
A point of comparison - regressing player vs. team points scored per possession
I wanted to get a sense of whether these slopes of far below one were out of the ordinary when regressing a team stat onto a player stat, so I ran another regression using the BasketballValue data. The data available didn’t allow me to look at blocks, which I thought would be a good comparison for a variety of reasons, so I looked at points scored per possession.
Player points scored per team possession is not a stat that you see very often (if ever), but it’s pretty simple and kind of interesting. It combines usage (how often a player shoots) and efficiency (how well they shoot) into one stat, as does the traditional stat of points per game. But unlike points per game, it controls for playing time and team pace. And most importantly for our purposes, the sum of the points per possession for each player on the court equals the team points per possession for that lineup (just like the sum of each player’s rebound rate equals the lineup’s rebound rate).
For all players who were on the court for at least 1000 offensive possessions this season (240 players), I ran a regression with team points scored per possession (while the player was on the court) as the dependent variable and player points scored per possession as the independent variable. The marginal value (slope) of player points scored per possession was 0.19, the R^2 was 0.069, and the standard error for the coefficient was 0.046. So here again we see a slope of much less than 1.0. There are almost definitely some lineup balancing issues going on here, though they’re probably not tied closely to position as in the case of rebounding. It’s often assumed that coaches balance out lineups with some better scorers and some worse scorers. And to some extent we know that there are diminishing returns in this area, since usage (in terms of percent of team shots taken) has a fixed upper bound on the team level. I’m not going to try to analyze this any further right now, but I thought I’d put the data out there for a comparison to the rebounding results.
Some other ways of trying to get at diminishing returns for rebounds
Another method for determining the extent of diminishing returns in rebounding was suggested by Guy in one of his posts. This involved comparing the standard deviation of rebounding on the team level to the standard deviation of rebounding on the player level. He suggested that a smaller variance between teams than between players provided evidence for diminishing returns (great individual rebounders don’t lead to great rebounding teams because the great rebounders are taking rebounds from their teammates). I think this method is worth looking at, but I don’t think the conclusion on diminishing returns can be drawn quite so easily.
Again, the problem has to do with positions/height. There is large variance between players in height, and rebounding is strongly correlated with height, thus there will be a large variance between players in rebounding. But on the team level, rosters and lineups aren’t a random combination of players of all heights. Teams impose similar structures in their lineups of player heights (short PG, average-height wings, tall bigs), which also means they impose similar structures of player rebounding (poor-rebounding guards, good-rebounding bigs). In essence, teams structure their lineups in a way that decreases rebounding variance on the team level. Teams could play five centers at once to dominate on the glass, but because of other considerations they don’t want to do that, and thus they end up playing a lineup with a mix of poor and good rebounders that will usually rebound on a similar level to the opposing lineups they face. Thus the lower variance in team rebounding could be caused by something other than diminishing returns.
With that caveat in mind, here is some data on the variation in player rebounding broken down by height, for all player-seasons from 73-74 to 06-07 with at least 400 offensive rebound opportunities and 400 defensive rebound opportunities (”SD” is standard deviation, “CV” is coefficient of variation):
Height ORB% DRB% ORB% DRB% # of (in) ORB% DRB% SD SD CV CV Plyrs ------- ----- ----- ----- ----- ---- ---- ----- <= 75 0.027 0.078 0.013 0.020 0.49 0.26 2084 76 - 78 0.045 0.104 0.024 0.036 0.54 0.35 2091 79 - 80 0.071 0.145 0.033 0.047 0.46 0.32 1939 81 - 82 0.087 0.186 0.027 0.040 0.31 0.22 2040 >= 83 0.094 0.208 0.025 0.040 0.27 0.19 1380
As far as the team level goes, in my previous post on Offensive Control Rates I presented data showing that there was greater variation in team offensive rebounding percentages than team defensive rebounding percentages. Averaging each season from 73-74 to 06-07, the average standard deviation of team ORB% was 0.024, and the average standard deviation of team ORB% allowed (the equivalent of team DRB%, but with the same mean as team ORB%) was 0.019. At this point in time I’m not going to try to try to connect these numbers to the player numbers to see what they tell us about diminishing returns, but I may do so in a future post.
What’s the upshot of all this research? Are diminishing returns a factor in rebounding? If they are, how large a factor are they? Do player offensive rebounds contribute more to team offensive rebound totals than player defensive rebounds contribute to team defensive rebound totals? Should players from some positions be credited more for their rebounds than players from other positions? How should rebounding we weighted relative to scoring and other aspects of the game?
I think we can come to some preliminary conclusions for some of these questions. Given the data showing that the slopes for defensive rebounding are consistently lower than the slopes for offensive rebounding, and that defensive rebounding slopes are far below one even when controlling for position, it does look like diminishing returns have a significant effect on defensive rebounding. I also think it’s definitely worth considering the possibility that rebounds from different positions are worth different amounts on the team level, though one would need to look at a larger sample of data than I did here to establish the different values.
The idea that there are diminishing returns for defensive rebounds is not a new one, and some player rating systems try to take that into account. It’s one of the rationales for why offensive rebounds are given over twice the weight of defensive rebounds in John Hollinger’s PER. Mike Goodman uses an alternate rebound rate in which rebounds by teammates aren’t considered rebound opportunities for a player. Dan Rosenbaum’s statistical plus/minus considers player offensive rebounds to be almost twice as important to team offensive efficiency as player defensive rebounds are to team defensive efficiency, though these weights were obtained through regressing adjusted plus/minus onto boxscore stats rather than through theoretical concerns about diminishing returns.
On the other hand, Dave Berri’s Wins Produced weights offensive rebounds basically identically to defensive rebounds, and treats player defensive rebounds as if each one contributed a full team defensive rebound (and thus a full team possession acquired). As a result, being a good or poor rebounder makes a much larger difference in Wins Produced than it does in other player rating systems (this can be seen in Table 3 of Dan Rosenbaum’s recent paper, “The Pot Calling the Kettle Black”). In Chapter 7 of The Wages of Wins, Berri discusses a study showing that diminishing returns do have an impact in basketball, but he doesn’t discuss diminishing returns in the specific context of rebounding, and his Wins Produced rating system does not make any adjustment for diminishing returns in rebounding. By failing to account for diminishing returns for defensive rebounds, Wins Produced is likely overvaluing good defensive rebounders and undervaluing poor defensive rebounders.
I think there is a lot more work to be done in this area. More data is needed, as are more clever methods of looking at the data we have. There are a lot of questions worth investigating. Why are the slopes for offensive rebounding, even when controlling for position, significantly less than one? What’s the connection between player rebounding variance at different positions and diminishing returns? How do diminishing returns play out among individual players - are some players consistently rebound thieves and others consistently rebound bystanders? Are diminishing returns important in areas of the game other than rebounding? And so on.