Thursday, February 13, 2003

Rebounding, Part II: Defensive Rebounds

Part I is here. While each offensive rebound is worth The situation is different on defensive rebounds. Anyone who’s watched basketball can remember many situations where the defensive team surrounded the ball with everyone boxed out. Then, which player happened to get rebound is irrelevant, as discussed above. So a defensive rebound is not as valuable as an offensive rebound, since it’s more common, and because if a particular player didn’t get the defensive rebound, there’s still a good chance someone else on his team would.

So, I’m going to use the idea of removing the player who got the defensive rebound from the play. The value of the rebound is determined by how likely it would be for the opposing team to get the offensive rebound if he didn’t. But how to quantify this? I’m going to consider the rebound as a loose ball, with each player in the region having an equal chance of getting the rebound. Assume one offensive rebounder, with a chance (from the league average) of 28.5% of getting the rebounder. Now, if each rebounder has an equal chance of getting the ball, that means there have to be 2.5 defensive rebounders (on average, obviously) in the vicinity, each with a 28.5% chance of getting the board, in order to give the total defensive rebounding probability of 71.5%.

Now, what happens when you take away one of the defensive rebounders, when you remove his chance of getting the rebound. Then, it’s only 1.5 to 1 in favor of the defense, and the chance of the offensive rebound is 40%. So, if you remove the defensive player who actually got the rebound from the play, the offense then has a 40% chance of getting the board. So the value of the defensive rebound is 40% of a possession, or 0.4 points.

This might seem a little arbitrary, but it ends up giving a result which is interesting and, on further consideration, very plausible for a completely independent reason. Let’s use these numbers to figure out what the value is of a player who gets a typical 10 rebounds per game. On average, he would be getting 7.15 defensive rebounds and 2.85 offensive rebounds. The value of the offensive rebounds is easy, 2.85 points. The value of the defensive rebounds is 7.15 * 0.4 which come out to, surprise, 2.85 points. This means that, with this formula, the average team gets the exact same value from its offensive rebounds as it does from its defensive rebounds.

And it wasn’t just a lucky coincidence, either. If you go back and do the algebra (which is left as an exercise for the reader), it’s straightforward to show that the methodology I used above to calculate the value of defensive rebounds will, no matter what the base probabilities for offensive and defensive rebounds, produce the result that the average team (and the average player) gets the exact same value from their offensive as from their defensive rebounds.

Now, in physics, these sorts of strange coincidences come up all the time—you work through lots of seemingly aimless math and end up with a very simple or symmetric result. And when you do that, it’s usually a sign that your theory is correct. Now, I don’t know if God plays dice with basketball or not, but this nice symmetric result gives me more confidence in my methodology. And in the next post, I’ll explain why I think this result makes sense.

Rebounding, Part I: Offensive Rebounds

Next up is considering how valuable rebounds are. Rebounds are valuable because they give you possession—they convert a loose ball, specifically a missed shot, into a possession for your team. For offensive rebounds, this can be analyzed fairly easily. Offensive rebounds are unusual—in the 2000-2001 NBA, they amounted to only 28.5% of all rebounds, just a little over a quarter.

At this point, since our goal is to evaluate individual players, we need to drop a conceptual level from the team to the individual. At the level of the team, all rebounds are equally valuable, since if the team got one less rebound, that means they got one less possession. But looking at what the actual contributions of individual rebounders were, this is not the case, as I hope will become clear in the following discussion.

Since they are rare, we can start out by giving each offensive rebound the value of a possession, 1 point. Now, you might object that in my previous definition, I included in a possession all the subsequent offensive rebounds, until the other team got the ball back. Now, they’ve gotten an offensive rebound, and I’m giving that the value of the entire possession?

Yes, since, once you get the offensive rebound, you’re back in the same situation you were when you first brought the ball over the timeline. You have the ball and a full shot clock, so the expected number of points you will score will be identical in the two cases, as long as the probability of future offensive rebounds is not affected by the fact that you’ve already gotten one. And it shouldn’t be.

A second question would be why the rarity of offensive rebounds should affect their value. Isn’t a rebound a rebound? Yes and no. The thing is, the more common an offensive rebound is, the more likely it is that some other player on your team will get it if you don’t. The value of the rebound is proportional to its scarcity, which is why our intuition is that offensive rebounds are more valuable than defensive ones. To understand this, consider the extreme case where the defensive team gets 100% of the rebounds. If this is the case, it doesn’t matter at all which player on the team actually gets the rebound. If there’s no chance the opposing team will get the ball, then the individual value of a rebound is nothing—the team would not be hurt at all if a specific player didn’t get the rebound.

Most offensive rebounds involve one offensive player who’s managed to sneak in to get position, or who simply beats the defenders to the ball. So it’s not unreasonable to assume that, if the player who got the offensive rebound didn’t get it, one of the defenders would. So the value of the offensive rebound is an entire possession, 1 point.

Wednesday, February 12, 2003

What’s a possession worth?

Before tackling the worth of rebounds, I first need to look at the value of possessions. The best and simplest way to analyze the value of rebounds (and steals and turnovers) is by considering them as adding or taking away possessions from your team. And on each possession, you can expect to score a certain number of points.

So, there are two jobs here. The first is to define exactly what a possession is, and the second is then to determine what the average number of points you can expect to score on each one. First, the way I’ll define a possession will be the time your team gets the ball, all the way up until the other team gets it back. Pretty standard and straightforward, but this means that, at least for the purpose of this number this number, offensive rebounds don’t add possessions. This makes sense, since to determine the cost of a turnover, for instance, we want to know what the total chance there was that you’d score some points before the other team got the ball back. That includes getting offensive rebounds, since the turnover eliminates the possibility of getting second chance points just as it did first chance points.

So, the total number of possessions a team has can be calculated as

(FGM + Opp Def. Rebounds + Turnovers + FTA/1.9)

The 1.9 factor on free throws is a rough estimate. Since, in the NBA, there is no 1 and 1, it’s easier, since the only time you’ll shoot a single FT will be on a 3-point play attempt. I’m estimating that happens about 10% as often as a regular shooting foul or bonus attempt. This could be off a bit, and ideally I’d have the data giving me this number. But I don’t, and regardless the effect of a small error here won’t be great.

Then, you just take the total points scored by the team, divide by this number of possessions, and you get the points scored per possession. In 2000-2001, this number was 0.986 points per possession. Obviously, it will vary somewhat from year to year. To keep the formulas simple, I’ll approximate this as 1 point per possession.
What is the value of an assist? Conclusion (for now)

Scroll down to see the previous two installments. So, I’ve started on the effort I mentioned in the previous post, to try and plot team stats game by game, to see what effect, if any, a high assist rate has on shooting percentages. Unfortunately, this was incredibly tedious. has splits for the performance of each team against each other team, at least for the current season. While not ideal, this at least gives me 28 data points. But it took a long time to cut and paste all the relevant data over into Excel to try and use it. I looked at two teams, Atlanta and Dallas, and it probably took me an hour or more. This wouldn’t be too bad, if the data was worthwhile. But it was just as useless as the data presented in the previous posts. I didn’t feel like doing a lot of work for nothing, so I stopped on that.

However, I believe the underlying concept is still good. The effect of a good pass is to increase the probability of making a basket, so the value of an assist is increasing the points your team will score by increasing their FG%. But by how much? Well, there’s another way to attack the problem.

I have the complete season stats from 2000-2001, which gives me the FGM, FGA, and assists for the “average” team in the NBA that year. Per game, the numbers are (35.7, 80.6, and 21.08) Now, assume a simple model, where all shots with a good pass have the same FG%, and all shots without a good pass have the same, but lower FG%. But we know the number of shots taken, and the number made with (and hence without) good passes. Since everything has to be self consistent, if you guess what the FG% ought to be without a good pass, then you can calculate the FG% with a good pass.

As it turns out, one such possible pair is 39% FG% without a good pass, and 50% with one, which matches up pretty well both with my gut feeling (players playing one-on-one probably shoot a little below 40%) and with my guess about a data fit from below. So, in the absence of better data, I’ll take this as a starting point for further analysis. I’ll keep looking to see if I can find new data, and will let you know if I do.

But for now, let’s use these numbers. What these mean is that approximately 1/4 of the shots that were made with an assist would not have been made without the assist. Thus, each assist is worth 1/4 of 2 points, or half a point. Including three pointers would modify this result slightly, and I know the 1/4 number is slightly off, but the half a point value is nice an convenient. Given the inherent sloppiness of these calculations, I don't think it's worth worrying about.

While this result is not all that surprising to me, given the widespread much higher valuation of assists, perhaps it is surprising to some. Next up, we'll tackle rebounding, and we can see whether the numbers say 10 rebounds per game is more valuable than 10 assists per game. I'm guessing it is, but we'll see.

Tuesday, February 11, 2003

What is the value of an assist? Part II

So, in the previous piece, here, I argued that an assist is worth less, maybe significantly less than 2 points. What it actually does is increase the chances of making a shot. So, how can we figure out from the stats what the actual effect is? A first idea is to simply look at the number of assists a team has and their shooting percentage. More assists should produce a higher shooting percentage. So you plot those, look for correlation, and voila!

However, it’s not that simple, because there’s also correlation the other way. That is, a high shooting percentage will result in more assists. Since your team is making more shots, there will be more opportunities for assists, and all else being equal, there will then be more assists. So just plotting those two variables, it’s impossible to tell how much correlation is from assists leading to high shooting %, and how much is high shooting % leading to assists.

So, we need some way to isolate the effect that the assists are having on the shooting percentage. To do this, I plotted the shooting percentage by team versus the number of assists per made basket. So, the x axis goes from a low of zero (no assists) to a possible maximum of 1 (an assist on every single made basket.) If the data is clean enough, this plot should give us exactly what we want. The intercept at x=0 is the baseline shooting percentage without any assists. The intercept at x=1 is the average shooting percentage with an assist. The difference between these two values is then the value added by the assist—how much it increases the probability of making a shot.

While this is a good idea in theory, unfortunately the data don’t look so good. If this works, below you should see a plot of the data as I describe above, where each point represents one team’s results for the 2000-2001 season. And it basically looks like noise; there’s not a clear trend that could be fitted. The biggest reason is that the data is too tightly confined. Almost all the teams shot between 42 and 47% from the field, and they mostly fell between 0.55 and 0.7 assists per made basket. (The outlier in the top right is, as you might have guessed, Utah, with a FG% of 47% and 0.71 assists per made basket.)

What this plot tells you is that the effect of assists on the FG% is swamped by the team-to-team differences in base shooting percentage. That is, if you have a bunch of bad shooters on your team, then getting a good point guard isn’t going to magically turn you into a good shooting team. The spread in the data can also give a rough upper bound on the effect of assists. If we ask how big an effect would be noticeable on this plot, even given the underlying noise, then we’ll know the actual effect of assists has to be less than that, since no such effect is seen.

To do this, let’s plot it out on a wider axis. Looking at this, you could maybe convince me that the data should be plotted on a line from 38% to 48%. Now, If I were getting paid for this, I’d actually sit down and do the real work to figure out how big the effect would have to be to be really noticeable, but I’m lazy, so I’m just going to guess that the data shown wouldn’t be consistent with a jump from 30 to 60—it’s not doubling your FG%.

So, this idea didn’t turn out so well, although it wasn’t completely uninteresting. The next step is to try and remove the intrinsic spread due to differing teams shooting a different percentage. To do this, what I’d like to do is to plot out, just as in the above figures, FG% vs. assists/made basket, but do it for each team individually, plotting each game. This removes biases that might be caused both by individual team shooting as well as by offensive styles. (Different types of offenses might tend to produce more or less assists.) It will also hopefully give points spread across a larger range on the x-axis, reducing the errors caused by trying to fit a line to several points that are very close together.

Incidentally, it occurred to me that the above data might be skewed by the three pointer, since teams that shoot more three pointers would have lower shooting percentages. So I pulled out the 3-point data and looked just at 2-pointers, and it looked pretty much the same.