Sunday, January 23, 2011

NBA: does taking more shots lead to lower accuracy?

One of the hypotheses about why you can't take a player's FG% at face value, is that good players will be asked to take a lot of worse shots, like desperation FG attempts with the shot clock running out. That will, perhaps misleadingly, lower their percentage.

While I had my 2008-09 data typed in, I'd thought I'd give that a quick test. I'm sure this has been done before, but I figured I'd post it anyway.

I tried to predict a team/position's eFG% using (a) his position; (b) the rest of the team's FG% (averaged by position); and (c) the percentage of his team's FGA he took. There were 30 rows in the regression, one for each 2008-09 NBA team.

The resulting equation:

FG% equals:

+ .181
+ .022 if he's a SG
+ .016 if he's a SF
+ .025 if he's a PF
+ .043 if he's a C
+ .643 * the eFG% of the rest of the team
- .114 * the percentage of the team's FGA he takes.

The hypothesis seems to check out. The more shot attempts, the lower the overall percentage. For instance, suppose a player takes 20% of this team's attempts, and shoots .500. If he took only 19% of his team's attempts, he'd shoot .5114. If he took 21% of his team's attempts, he'd shot .4886.

That's bigger than it looks. A .500 percentage over 20 shots is 10 FG. A .5114 percentage over 19 shots is 9.727 FG. So, the extra shot nets only 0.283 FG. That means that, at the margin, the player shoots only .283 on those extra shots.

Second, note the high correlation between a position and his teammates. For every percentage point the teammates shoot better than average, the individual player will shoot 0.643 points better than average.

Everything was statistically significant at the 1% level, except the -.114, which was significant at only 7.2%. Its standard error was 63 points, so we definitely need at least another year's data before we can say we have a true understanding of the size of the "shoots more, therefore shoots less accurately" effect.

Also, if certain positions are asked to take desperation shots more than others, the regression might benefit from interaction terms. I'll leave that to you guys. My dataset is available on request if you want to play with it a bit.

Looking forward to any comments you basketball guys may have.

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At Sunday, January 23, 2011 8:59:00 PM, Anonymous Ryan J. Parker said...

"That means that, at the margin, the player shoots only .283 on those extra shots."

Why would you interpret that this way? I mean, we don't expect these guys to shoot 28.3% on those shots, so I'm probably not understanding your logic here.

Why not instead compare 0.5114 over 20 shots as well and see that the difference is 10.228-10=0.228? What benefit is there to even looking at that sort of comparison?

At Sunday, January 23, 2011 9:02:00 PM, Blogger Don Coffin said...

I think I've seen similar results elsewhere, but I can't remember where.

Of course, as an economist, I have to suggest that this isn't surprising. Suppose you have a team of 5 shooters, each taking 20% of the shots, but with shooting percentages that look like this:

A: 54%
B: 50%
C: 46%
D: 42%
E: 38%

Then it's obviously rational to shift shots upward. So E and D (at least) reduce their shots taken, and, we might assume, by not taking the more difficult shots. A and B take more shots, and the additional shots they take are likely to be more difficult than the shots they are already taking.

The (sort-of) equilibrium is that all players wind up with the same shooting percentages *at the margin*--it's not that their average shooting percentages converge to 46%, it's that their shooting percentages on additional shots converge.

One result is, of course, that the team shooting percentage will be higher than the 46% it started out at.

Note that coaches can influence this by designing plays that emphasize more shooting by better shooters, so it's also a "managerial" decision-making option.

At Sunday, January 23, 2011 11:01:00 PM, Blogger Phil Birnbaum said...

Ryan, the regression says that the more shots you take, the lower your percentage. So you if you shoot .500 over 20 shots, you only get to be .5114 if you reduce your shooting to 19 shots.

So you're .5114 over the first 19 shots, and .5000 over the first 20. The only way that can happen is if you're .283 on the 20th shot.

At Sunday, January 23, 2011 11:03:00 PM, Blogger Phil Birnbaum said...


Doc: agreed. One quibble: you might not necessarily "give" D and E the easiest shots. Presumably the defense will be concentrating on A, B, and C, so that D and E will wind up with easier opportunities because of the opposition's defensive strategy.

Do you know of any way where you can show this is what's happening? I can't think of any...

At Monday, January 24, 2011 1:04:00 AM, Anonymous kds said...

Phil, could you look at shots only in the last 3 seconds of the 24 second clock or the period? Before that there is always a possibility of another pass, so who takes the shot is somewhat a matter of choice. With the defense trying to restrict your choices.

At Monday, January 24, 2011 12:01:00 PM, Anonymous Ryan J. Parker said...

Ok, I see where you're coming from, but I wouldn't expect a player to shoot 0.283 on any one of those shots.

I mean, what if you instead looked at 200 vs 190 shots. We wouldn't expect a player to shoot 0.283 over those 10 shots. Even if that's something we can do mathematically, it's not something that makes basketball sense.

Or am I still not following your logic properly?

At Monday, January 24, 2011 12:45:00 PM, Blogger Phil Birnbaum said...

It might just be random error in the coefficient. Bump it up an SE or so, and you get a different number that's less extreme than .283. That's why we need more than one season's worth of data ...

At Saturday, January 29, 2011 2:03:00 AM, Anonymous Crow said...

- .114 is the best fit average slope for change in the percentage of the team's FGA a player takes across the entire range right?

But couldn't the marginal actual slope between 2 adjacent percentages could be different in the middle of the range than at the extremes? That is the slope between 19 and 20% could be different than between 1 and 2% or 35 and 36%. And -.114 is the average slope and not necessarily real close across each part of the range as it may not be a linear function. I think there is work showing that is not. I'll look around a bit for it. If someone knows where the best charts are chime in. One of Eli's charts at team level appears to have different marginal slope at the extremes than in the middle. At player level this could be more dramatic and it will vary by player as the skill curves can show.

At Saturday, January 29, 2011 2:20:00 AM, Anonymous Crow said...

The skill curves shown in Chapter 19 on Basketball On Paper clearly show different slopes in the middle vs the extremes at player level for several examples. On page 233-4 Oliver says as much.

At Saturday, January 29, 2011 2:43:00 AM, Anonymous Crow said...

Skill curves can vary by age.

And may vary on average by player / role / position too, in part because it depends on what kind of shot location they use / can get. Inside shots and 3 point shots may be constrained and extra shots may end up being mid-range more often and worse for that reason, which is a usage effect but the effect will not be even along the entire range or between players and teams. Assists or potential assists to a specific player are probably constrained too.

At Saturday, January 29, 2011 2:53:00 AM, Anonymous Crow said...

Defensive changes probably aren't going to take affect with 1% changes but they may occur with modest changes and affect the results.


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