The value of improved offensive rebounding

In case you haven’t been paying attention to this sort of thing, the best NBA writer right now is Zach Lowe. (Even if he has been forced by the tragic Grantland shutdown to post his columns on the super-obnoxious ESPN main site).

In a column today, Zach considered the following question: How valuable is it to try and get offensive rebounds?

Obviously, getting an offensive rebound has great value, since it effectively gives your team another chance to score. But if you send too many players to “crash the boards” in pursuit of a rebound, you leave yourself wide open to a fast-break opportunity by the opposing team.

So there is some optimization to be done here. One needs to weigh the benefit of increased offensive rebounding against the cost of worse transition defense.

In recent years, the consensus opinion in the NBA seems to have shifted away from a focus on offensive rebounding and towards playing it safe against fast-breaks. In his analysis, however, Zach toys with the idea that some teams might find a strategic advantage to pursuing an opposite strategy, and putting a lot of resources into offensive rebounding.

This might very well be true, but my suspicions were raised when Zach made the following comments:

There may be real danger in banking too much on offensive rebounds. And that may be especially true for the best teams. Good teams have good offenses, and good offenses make almost half their shots. If the first shot is a decent bet to go in, perhaps the risk-reward calculus favors getting back on defense. This probably plays some role in explaining why good teams appear to avoid the offensive glass: because they’re good, not because offensive rebounding is on its face a bad thing.

…

Bad teams have even more incentive to crash hard; they miss more often than good teams!

Zach is right, of course, that a team with a high shooting percentage is less likely to get an offensive rebound. But it is also true that offensive rebounds are more valuable for teams that score more effectively. For example, if your team scores 1.2 points per possession on average, then an offensive rebound is more valuable to you than it is to a team that only scores 0.9 points per possession, since the rebound is effectively granting you one extra possession.

Put another way: both the team that shoots 100% and the team that shoots 0% have no incentive to improve their offensive rebounding. The first has no rebounds to collect, and the second has nothing to gain by grabbing them. I might have naively expected that a team making half its shots, as Zach mentions in his comment above, has the very most incentive to improve its offensive rebounding!

So let’s put some math to the problem, and try to answer the question: how much do you stand to gain by improving your offensive rebounding? By the end of this post I’ll present a formula to answer this question, along with some preliminary statistical results.

The starting point is to map out the possible outcomes of an offensive possession. For a given shot attempt, there are two possibilities: make or miss. If the shot misses, there are also two possibilities: a defensive rebound, or an offensive rebound. If the team gets the offensive rebound, then they get another shot attempt, as long as they can avoid committing a turnover before the attempt. Let’s say that a team’s shooting percentage is p, their offensive rebound rate is r, and the turnover rate is t.

Mapping out all these possibilities in graphical form gives a diagram like this:

The paths through this tree (left to right) that end in x’s result in zero points. The path that ends in o results in some number of points. Let’s call that number v (it should be between 2 and 3, depending on how often your team shoots 3’s). The figures written in italics at each branch represent the probabilities of following the given branch. So, for example, the probability that you will miss the first shot and then get another attempt is .

Of course, once you take another shot, the whole tree of possibilities is repeated. So the full diagram of possible outcomes is something like this:

Now there are many possible sequences of outcomes. If you want to know the expected number of points scored, which I’ll call F, you just need to sum up the probability of ending at a green circle and multiply by v. This is

(A little calculus was used to get that last line.)

So now if you want to know how much you stand to gain by improving your offensive rebounding, you just need to look at how quickly the expected number of points scored, F, increases with the offensive rebound rate, r. This is the derivative dF/dr, which I’ll call the “Value of Improved Offensive Rebounding”, or VIOR (since basketball nerds love to make up acr0nyms for their “advanced stats”). It looks like this:

Here’s how to interpret this stat: VIOR is the number of points per 100 possessions by which your scoring will increase for every percent improvement of the offensive rebound rate r.

Of course, VIOR only tells you how much the offense improves, and thus it cannot by itself tell you whether it’s worthwhile to improve your offensive rebounding. For that you need to understand how much your defense suffers for each incremental improvement in offensive rebounding rate. That’s a problem for another time.

But still, for curiosity’s sake, we can take a stab at estimating which current NBA teams would benefit the most, offensively, from improving their offensive rebounding. Taking some data from basketball-reference, I get the following table:

In this table the teams are sorted by their VIOR score (i.e., by how much they would benefit from improved offensive rebounding). Columns 2-5 list the relevant statistics for calculating VIOR.

The ordering of teams seems a bit scattered, with good teams near the top and the bottom of the list, but there are a few trends that come out if you stare at the numbers long enough.

First, teams that have a lower turnover rate tend to have a higher VIOR. This seems somewhat obvious: rebounds are only valuable if you don’t turn the ball over right after getting them. Teams that shoot more 3’s also tend to have a higher VIOR. This is presumably because shooting more 3’s allows you to maintain a high scoring efficiency (so that an additional possession is valuable) while still having lots of missed shots out there for you to collect. The teams that would most benefit from improved offensive rebounding are generally the teams that are already the best at offensive rebounding. This seems counterintuitive, but it comes out quite clearly from the logic above. If your team is already good at offensive rebounding, then grabbing one more offensive rebound buys you more than one additional shot attempt, on average.

(Of course, it is also true that a team with a high offensive rebounding rate might find it especially difficult to improve that rate.)

Looking at the league-average level, the takeaway is this: an NBA team generally improves on offense by about 0.62 points per 100 possessions for each percentage point increase in its offensive rebound rate. This means that if NBA teams were to improve their offensive rebounding from 23% (where it is now) to 30% (where it was a few years ago), they would generally score about 4.3 points more per 100 possessions.

So now the remaining question is this: are teams saving more than 4.3 points per 100 possessions by virtue of their improved transition defense?

Footnotes:

There are, of course, plenty of ways that you can poke holes in the logic above. For example: is the shooting percentage p really a constant, independent of whether you are shooting after an offensive rebound or before? Is the turnover rate t a constant? The offensive rebound rate r? If you want to allow for all of these things to vary situationally, then you’ll need to draw a much bigger tree. I’m not saying that these kinds of considerations aren’t important (this is a very preliminary analysis, after all), only that I haven’t thought deeply about them. Much of the logic of this post was first laid out by Brian Tung after watching game 7 of the 2010 NBA finals, where the Lakers shot 32.5% but rebounded 42% of their own misses. I know I’ve made this point before, but I will never get over how useful Calculus is. I use it every day of my life, and it helps me think about essentially every topic.