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 $(1-p) \times r \times (1-t)$.

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.

1. 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.
2. 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.
3. 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:

1. 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.

2. 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.
3. 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.
25 Comments leave one →
January 5, 2016 6:21 pm

Nice analysis. And right about the value of calculus (fn 3). But you don’t need calculus to sum up that infinite series.

January 5, 2016 7:10 pm

I guess I first learned about infinite series in calculus, so in my mind it’s part of it.

2. January 5, 2016 6:53 pm

Wow! Sabermetrics comes to basketball!😄

January 5, 2016 7:12 pm

It’s been here for almost a decade now! But with the counterpart acronym being the unpronounceable “apbrmetrics”, it never caught on in the vernacular.

3. January 5, 2016 7:06 pm

In your analysis, free throws are not mentioned. On the possession tree, shouldn’t there be a branch that deals with the possibility of a foul and free throws? I know this adds a significant level of complexity because the PPP on FTs is different from shots and because the OR% on FTs is significantly lower than on a regular shot–I believe ~15 percentage points lower than a typical shot or roughly 15%. I should note that the OR% numbers you use include FTs as a potential source of rebounds, hence the 23% average for the league. This also indicates that there might be a minor data mismatch in your VIOR calculation since you are using FG% for p, and using OR%, which includes chances from FTs, as r.

Were free throws deliberately omitted for simplicity, or am I missing something here? If FTs were omitted for simplicity, how do you think they would impact your VIOR calculations, if they to be included? I don’t know the exact stats, but I believe the literature indicates that free throws are more valuable, on average, than a shot, even accounting for the lower chance of an offensive rebound.

One more thing, I think there is an inconsistency in how VIOR are calculated. You incorporated FG% and 3 Point Attempt % but these two numbers do not account for the fact that 3 pointers and 2 pointers are made at a different rate. I believe that your VIORs implicitly assume that the overall FG% of a team is the rate that they shoot on both 2s and 3s. To account for the difference you would need to include 3p% and 2p% separately. I ran the numbers quickly for the Pistons and was able to replicate your .672 calculation initially. I then split up 2s and 3s then added the VIORs from each (I think this is mathematically sound) the VIOR for the Pistons dropped to .660. The impact is small but I think this makes sense logically as 3s were being slightly overweighted previously. Let me know if I have misunderstood what you did.

Regardless, really enjoyed the post. Solid analysis and a good follow up on Lowe’s work, who I too think is the best in the business.

January 5, 2016 7:14 pm

Hi Ezra,

Thanks for your comments. There are a number of limitations here, including some that you pointed out, so I think that this analysis is best read as a roadmap for how one can figure out the value of offensive rebounding. I find it very likely that the exact definition of VIOR will have to be tweaked a little bit before it has real predictive power.

How to incorporate free throws is kind of a tricky problem. You’re right that the most precise way would be to make it its own branch in the tree of possibilities. Another, more approximate way would be to just lump free throws in with every other scoring attempt, which is what I was trying to do.

I was also pretty approximate with my treatment of 2’s and 3’s. I lumped them together, and tried to adjust for their different value by having the variable v, which is between 2 and 3. But you could again make 2’s and 3’s as separate branches on the tree of possibilities.

If you end up looking more deeply into this kind of refinement to the VIOR concept, feel free to write up your results and post the link here in the comments. I’ll be glad to read about it.

• January 5, 2016 7:30 pm

Thanks for the reply Brian. That makes sense as a basic roadmap. I think an adjustment for 2s and 3s would be pretty simple and could make the numbers slightly more precise. However, without FTs it could be false precision. And I completely agree that free throws are very tricky–as a first pass I think simplifying to lump them in with other scoring attempts makes sense. I would have to think longer about how they would be included in a more complex model.

I should reiterate that I really like what you did and I may post something building off your work if I get around to it. Thanks again for posting.

January 5, 2016 7:44 pm

Thanks. It’s kind of you to say so.

• January 5, 2016 9:41 pm

Hi everyone.
Sorry if my English isn’t very good but it’s not my first language. I believe a good solution for this would be using TS% instead of FG% since it accounts for both, two point shots and threes, but also it includes free throws.
Really interesting and great article! Keep up the good work!

January 5, 2016 9:53 pm

Hi R,

Your English is fine, so don’t worry about that!

The tricky thing about using TS% is that if you want to know the probability of getting a rebound, then you need to know the probability that the shot will be missed. This is just (1 minus the) normal FG%, and not TS%.

But I agree that, in general, there is probably a smart way of incorporating the difference between FTs, 2s, and 3s into a single statistic.

January 6, 2016 2:20 am

Just curious, as a math dilettante, was the “little calculus” you mentioned just the a/1-r formula?

Great post, by the way.

January 6, 2016 9:30 am

Yes, that’s it. Of course, there’s some calculus in taking the derivative dF/dr also.

January 6, 2016 3:27 am

Brian,

This is great and the way I always thought of this but never had enough of an inclination to pursue further.

One problem I have with this is FG% is not independent from where you start (i.e. a team is generally more likely to make a FG after an offensive rebound)

This twist makes it a while lot more difficult from a maths’ perspective

KB

January 6, 2016 9:32 am

Yeah, a number of people have pointed that out. I see two potential ways to handle this. One is to just increase the value of p for shots following the offensive rebound. The other way would be to divide the offensive rebound into two branches: simple “putback” shots and longer rebounds that lead to a complete reset of the offense.

It seems to me that this would make things more complicated, but not actually much more difficult.

January 6, 2016 10:17 am

Nice analysis. I wonder, though, whether the only tradeoff to consider is diminished transition defense. Couldn’t it also be true that teams would pay some price in diminished offensive efficiency if they positioned (or selected) players with an eye toward maximizing offensive boards?

7. January 6, 2016 4:24 pm

Reblogged this on InsideStoop and commented:

January 6, 2016 4:41 pm

Very nice article and thank you for reminding me why I enjoyed calculus so much way back when. Here’s a stab at all the variables the branches could expand to (motivated mostly by the fact FTs weren’t in the model), that is all the possible endings to a possession. An offense can:
– miss a 2-pointer (no foul)
– miss a 2-pointer (shooting foul) (then tack on the 3 FT possibilities, make 0, 1 or 2) (and also tack on the possibility of rebounding the second FT if missed)
– make a 2-pointer (no foul)
– make a 2-pointer (shooting foul) (then tack on the FT make/miss possibilities and rebound opportunity if missed)
– miss a 3 (no foul)
– miss a 3 (foul, with all the FT possibilities)
– make a 3 (no foul)
– make a 3 (foul, with the FT possibilities)

Each of these need to be multiplied by the rate at which they occur within the given team’s offense of course.

Further, as if all those variables weren’t enough, they could be broken down much further, adding incredibly more complexity. Two-pointers span the difficulty spectrum from contested long jumpers to uncontested dunks. Each type of two-point shot along that spectrum would need 5 variables [how often attempted, how often missed (no foul), how often missed (foul), how often made (no foul) and how often made (foul)]. And the same with 3s (corner or not, contested or not).

You could also have many offensive rebound rates (off of close shots, long jumpers and FTs) in the model. Different turnovers rates (transition, half-court, after off rebound), too.

Then take all those variables/outcomes and quadruple them because they could be different for different situations, namely 1) initial half court possession, 2) after offensive rebound half court possession, 3) fast break possession and 4) after missed fast-break possession. That is, the rate at which you take a 3 and miss it and get the rebound, as one example, is certainly different if it’s a transition 3 vs a half-court 3.

And, to this point, I’ve been talking about team rates. What about for a given player? This two-man line-up, that 3-man line-up, small-ball 5-men, “traditional” 5-men, etc? This would to answer questions such as, if Duncan and Aldridge are on the floor together, what’s our best offensive rebound strategy? Is that strategy solely based on their presence or also on the presence of their 3 floor mates? That is, a) D&A with good shooters around them means go for offensive rebounds while b) D&A with bad shooters, means get back on D. These are the insights such a model might produce, which would then lead to winning strategies.

Now, are some of these rates materially different enough to warrant such precision (as if all those numbers could be known precisely in the first place)? Well, the answer is that’s why you start with the basic model (which this article has) and expand it seeing how your model improves with the extra complexity. Maybe you find the complexity adds no further value. Maybe you find you scrap all that complexity and just use an aggregate like TS% and it gets you 98% of the way to something insightful/meaningful, which is certainly sufficient for the simplicity gained.

Lastly, I’ll also reiterate Guy’s sentiment. If you increase your emphasis on offensive rebounding, naturally you de-emphasize other areas including shooting percentages. That is, if you model your team to increase r (by player selection, not by strategy), expect a decrease in p. That trade off would be a whole other analysis as well. But the point is you don’t just change r and have p stay constant.

Have we crossed into the realm of differential equations yet?

January 6, 2016 5:20 pm

Yikes. Reading through your post, my reaction is that this is why it’s so easy to get scared when you start trying to describe basketball (or just about anything, for that matter) with math: it seems like there are a million things that can matter, and everything depends on everything else.

So that when you finally hit on a description that really works, it can feel like more luck than great insight.

But those helpful, simplified descriptions do exist. So there’s value to this kind of “excessive theorizing”, even if it never seems satisfying until it works.

Anyway, thanks for your comment. Here’s hoping that the time for differential equations is still not at hand. 🙂

9. January 7, 2016 4:29 pm

I think the value of an offensive rebound here is understated. I’d be willing to bet that the ratio of 2s attempted to 3s is much higher after an offensive rebound than after a defensive rebound or score, and I also think that the FG% on 2s and 3s is also likelier higher after an OReb. This has the effect of increasing the value of v, which in turn increases the offensive value of an offensive rebound.

Still, this is a great first step, and very interesting!

10. April 13, 2016 3:10 pm

Some past studies by 82games didn’t really show much of an improvement in shooting % after an offensive rebound, which was surprising. I think to register the impact, using the league average offensive rating expressed and adjusted per 100 possessions serves the purpose of illustrating the impact. The other team is getting more shots – that’s an automatic offensive improvement, and it’ll kill any teams D-rating.

Milwaukee Bucks this season are a good example — dead last in defensive rebounding % all season long but for a few weeks the Rockets have led. If they were simply average defensive rebounders, it works out that they’d be 12th in league def-rating as opposed to where they finished — 26th.