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.