The Rest of the Story

Phil has a couple of new posts on rebounding.  In his initial post (the first link), he ran a regression using the number of rebounds a team’s centers get per game to predict how many rebounds the rest of the team would get per game.  He found that there is a negative relationship between center rebounds and rest of team rebounds.  I thought I would contribute some more stats.

I got the same data that Phil did, also entered manually.  I reran his rebounding analysis and got some slight differences, but nothing that big; they could be due to a typo here or there.  Here are the coefficients I found if you predict rest of team from each position, with the standard error in parentheses: center-.695 (.15), PF -.677 (.182), SF -.729 (.291), SG -.643 (.34), PG -.874 (.303).  You’ll notice that if you add and subtract twice the standard error to get the usual 95% confidence interval, the estimate for each position contains -1 (except for centers, where it’s very close).  For this reason, I’ll add more data to the analysis by running the regression on every player instead of breaking it out by position.  I get -.968 (.0327).  So in general, each rebound gathered by a position takes a rebound away from the rest of the team.

When I got the data, I also entered the other count variables listed, which were FGA, FTA, assists, blocks, turnovers, and fouls.  Here are the numbers from the same analysis for assists: centers -.557 (.2976), PF .062 (.359), -.968 (.241), SG -1.015 (.252), PG -.692 (.122), all players -.939 (.0452).  You get some differences across positions that I think are mostly noise; I would be surprised if someone can come up with a good reason why power forwards don’t have diminishing returns with their assists but every other position does.  Just presenting the all player numbers, here are FGA -.787 (.083), FTA -.62 (.109), TO -.211 (.15), blocks -.805 (.0797), and fouls -.687 (.0996).

Like Phil, I also looked at the standard deviation for team stats per game and the same stat at the player level.  Phil was surprised that centers had more variability than teams did, although I’ll admit I’m not entirely sure why.  I have the same team data I’ve used before, which covers 10 years.  I find that not only do centers vary in rebounding more than teams do, but this is also true for point guards and assists.

Given the acceptance Phil’s analysis got in his comments, I think these results should be pretty uncontroversial.  We might be able to do a little better, as in Phil’s second post, by using some percentage values instead of per-game values.  The reasoning by Phil and his commenters was that since defensive rebounds have a coefficient of about .7-.8, they should be worth .2-.3 rebounds at the player level; offensive rebounds have a coefficient of maybe .5 (hard to tell with it broken out by position), so they should be worth about .5.  Under this logic, at the player level I would give a player .2 credit for assists, .4 for FGA and blocks, .5 for FTA and fouls, and .8 for turnovers.  For example, since so many assists are just assists stolen from teammates, a player should get very little credit for the ones he gathers in the boxscore.

For the record, let me say that I don’t believe any of this.  I think there’s a better explanation, but I need more data to check it and don’t have the time to enter more seasons by hand.  If anyone wants to help out, let me know.

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38 Responses to The Rest of the Story

  1. StL Reflections says:

    I’m curious what your hypothesis is as to what this evidence of diminishing returns means.

    My thoughts: I wonder about how role/style plays in, and how we tease things out-that is, sure, if your point guard isn’t your primary playmaker, your shooting guard/small forward will be getting more (but probably not quite as many) assists, or that if you’re the guy under the basket trying to block shots, other people on the team won’t have that role-blocks taken away from them, but you probably have the role on the team that you do because you’re good at that role.

    But I do think there is something true about saying turnovers are your fault, while things like assists, FGA and defensive rebounds are more likely to be contributed by someone else on your team if you don’t do it.

  2. Guy says:

    Alex: A couple of suggestions if you’re going to pursue this. First, on offense you need to adjust for usage for any stats that are involved with use of possessions. Everyone understands that when a player uses possessions, that leaves fewer possessions for others. You get a -.787 for FGA because 86% of all FGA end a possession. And so it’s not that surprising that assists are negatively correlated at the position level.

    Second, I think you really need to use position rather than individual players for most defensive stats, and offensive rebounds, or position-adjust your variables. These stats can vary a lot by position, and if you ignore that your correlations will greatly exaggerate the extent of diminishing returns. Blocks is a good example. A lot of blocks means you are looking at a big man, so “rest of team” is smaller players; few blocks means a guard, so now “rest of team” includes all the bigs. You can see how this will create the illusion of diminishing returns, even if they aren’t there. (Assists probably also need a position adjustment.)

    What will be interesting is to see where diminishing returns seems to exist even after controlling for position and/or usage.

    • Alex says:

      If everyone understands that using a possession leaves fewer possessions for everyone else, why do no models that I’m aware of adjust shooting for that fact?

      • Guy says:

        Alex, what on earth are you talking about? WP debits a player 1 point for every FGA. Other metrics debit players for FGmiss. Evan’s ezPM subtracts the average value of a possession (approx. 1.07) for every possession a player uses, so players only earn positive points to the extent they produce more than 1.07 points per possession. There is some disagreement about how best to measure shooting efficiency, but if there is any metric that completely ignores efficiency I’m not aware of it.

        • Alex says:

          When a player gets a rebound, they get credit for that rebound in WP. But the DR argument is that since any of a number of players could have gotten that rebound, we should take some of it away from him and give it to his teammates. If it’s also true that when someone takes a shot any of his teammates could have taken that shot, should we not take the credit (or debit) for it away from him and give some to his teammates?

      • Guy says:

        Oh my, this is very confused Alex. When metrics take account of efficiency, they ARE taking account of diminishing returns. We care about efficiency because FGAs are a limited resource. If they weren’t (if teams had a 100% oreb%), then we wouldn’t care how efficient shooters were — they would always score eventually. So when you debit players for FGA, or FGmiss, you are taking account of diminishing returns.

        Look, a player could take every single FGA for his team in a game, shoot 50%, and he would still have a WP of only .100 if all his other stats were average. And all of his teammates would also be .100 if their non-shooting stats were average. So how in the world can you say that ignores diminishing returns? You’re getting hung up on the question of exactly how metrics apportion credit to other players — directly or indirectly — and missing the big picture here.

        • Alex says:

          I have to say Guy, if anyone is hung up on how metrics apportion credits to players, it would be you.

          Also, none of the metrics give credit for efficiency directly, it’s only through how they value makes and misses. You have to shoot 50% in WP because it gives equal credit for two pointers and misses; assuming all your two pointers are unassisted you only have to shoot about 40% to break even in Evan’s model. They both come from what they think the value of a make or a miss are, neither have anything to do with diminishing returns.

          Also, you still haven’t addressed how none of the models account for DR when it comes to assists, free throw attempts, blocks, or anywhere else they exist. Shouldn’t we be outraged at all these other ridiculously flawed models?

  3. EvanZ says:

    “Under this logic, at the player level I would give a player .2 credit for assists, .4 for FGA and blocks, .5 for FTA and fouls, and .8 for turnovers. For example, since so many assists are just assists stolen from teammates, a player should get very little credit for the ones he gathers in the boxscore.”

    It depends what you mean by “credit”, of course. In ezPM, for example, on an assisted 2pt field goal, I “credit” the shooter with 70% of the marginal point created, and the passer with 30%. This isn’t too far off your estimate of 0.2. In fact, one of the things that myself and others have noticed is that point guards tend to be slightly favored in the model, which would be consistent with your finding.

    So, it’s still not clear to me where you stand on diminishing returns for rebounding and how it relates to WP. Between your challenge with Guy and now Phil’s work (and your own on top of that), are you now ready to accept that WP gives too much credit for rebounds. If not, why not?

    • Alex says:

      I’ve said before Evan, I think all the models are wrong. WP might be wrong for overweighting rebounds; it probably also has some other things wrong. I still find it the most useful. Do you think your model overweights shooting, since FGA has such a high diminishing return rate? Since you base the value of blocks on the value of a missed shot, do you think it’s overweighted? I haven’t read if you’ve updated the model since the initial post, have you changed any of your values to account for diminishing returns for any statistic?

      • EvanZ says:

        It’s easy to say “all the models are wrong”, isn’t it?

        The question is, if you know that WP is wrong, why not fix it? At least, take one little step forwards. Why are you (apparently) so opposed to that?

        Don’t let the perfect be the enemy of the good.

        • Alex says:

          It’s very easy to say so; in my field, it’s implicitly and explicitly understood. Instead models are evaluated by how useful they are. In my opinion, WP is very useful, and more useful practically speaking than the other models I know of. I’m also not trying to build a basketball model, so it isn’t up to me to change anything. But I will happily change gears and start using another model in my thinking if one proves to be better.

      • EvanZ says:

        BTW, in terms of giving credit for FGA, I’m not opposed to it, but I would want to know what the results are for field goals made and missed, too. I would need to know how much additional credit/debit to give or deduct for these results. Obviously, you need to know how the FGA was completed.

        • Alex says:

          The site the data comes from only lists FGA and eFG%, so I couldn’t tease it all apart. But Berri’s research shows that points scored has as much DR as shots attempted, so I would assume that the effects on makes and misses are similarly high.

      • ilikeflowers says:


        why bother with all the apples and oranges stuff at all? If I wanted to look into wp48 rebounding further here’s what I’d do:

        [1] Grab as big a dataset as possible.
        [2] Split it up by position.
        [3] Split each position up into 3 to 5 rebounding levels (split it into offensive and defensive if you like).
        [4] You now have 15 to 25 populations.
        [5] Run the wow regression on each population.
        [6] Look at the rebounding weights.

        Wouldn’t this be definitive? Isn’t this apples to apples?

      • Guy says:

        Alex, you say “WP might be wrong for overweighting rebounds.”

        I think that’s a settled question. Aside from all the evidence provided by Witus, Birnbaum, and others, in his FAQ Berri writes this: “Recently WP48 was re-estimated, but this time a defensive rebound would only be worth half as much as points, field goal attempts, offensive rebounds….turnovers, and steals. This adjustment follows the diminishing returns results reported in Stumbling on Wins for defensive rebounds.” While the writing is a bit muddled here, it seems to me Berri is reporting a finding of 50% diminishing returns. So I think that’s now the minimum estimate of DR on the table.

        What’s interesting is that Berri hasn’t revised WP to reflect this new knowledge. Nor have Arturo, Dre, and the rest of the WOW network in their WP estimates. Doesn’t that seem odd, to not fix this problem in the metric when one’s own research has confirmed it?

        • Alex says:

          When the research shows that evaluations don’t change a lot, I don’t think it’s necessary (although I know you think it does lead to large changes, at least in a sample of 20 players or so who are most likely to have their evaluation change).

          I’ll continue to say, I think what’s interesting is that no one has revised their model to reflect the knowledge that a number of stats suffer from DR. Don’t you find that odd?

      • Guy says:

        On your last point — that no one has addressed DR in their model — I’m afraid you’re still not getting it. Diminishing returns on assists, or FGs, or FTs are all a function of limited possessions: if you use up possessions, your teammates have fewer left. WP and ezPM both address this in multiple ways. In ezPM, for example, an assisted FG is worth about 1 point (divided btwn 2 player). Why isn’t it worth 2 points? To account for the possession consumed. You’ve obviously gotten it in your head that this is somehow different from incorporating “diminishing returns.” But it isn’t. You just need to think harder about this….

        On DR on dreb, it simply isn’t a matter of opinion whether reducing the coefficient for drebs by 50% changes things “a lot.” When a metric purports to measure player productivity to 1/1000 of a win per game played, its creator cannot then in good faith continue deliberately making a mistake that increases the value of many key players by 20% or more. Twenty percent! If your view is that 20% doesn’t matter, well then nothing matters. Why even write or think about these issues at all? Just pick a metric out of a hat and use it, or use points per game. I find it inconceivable that if Hollinger or anyone else said that they had discovered an error of this magnitude in their metric, but said it didn’t matter and wasn’t worth addressing, you would not be very critical (appropriately) of that decision.

        • Alex says:

          So if WP does take diminishing returns into account by giving value according to what they gain/lose a team in terms of possessions, why is it wrong about rebounds? The same method is used there. That is, it figured out the weight for making and missing shots according to possessions, which you say incorporates DR, and it did the same thing to figure out the weight for rebounds. Did the regression somehow treat shooting and rebounding differently? Or are you saying that DR for rebounds is greater than for shots taken?

      • Guy says:

        Alex: Try controlling some of the offensive stats for possessions used. I think you will quickly see that the diminishing returns either disappears or is greatly reduced. I think that will clarify things a lot.

        • Alex says:

          The interpretation of which would be, if you use a lot of possessions and then use one more it doesn’t actually take that possession away from your teammates?

  4. ilikeflowers says:

    Also, wouldn’t this be useful for every stat?

  5. EvanZ says:

    Alex, in your reply to Guy above, you said, “You have to shoot 50% in WP because it gives equal credit for two pointers and misses; assuming all your two pointers are unassisted you only have to shoot about 40% to break even in Evan’s model.”

    Alex, you’re wrong about the 40% value (it’s closer to 50%), but I want you to figure this out for yourself. Try the following. Simulate a game where both teams are exactly average. Average offensive/defensive rebounding, average FTA/FGA, and average TOV. To make it simple, assume that all shots are 2-pointers and there are no assists. What is the FG% that each team would have to shoot to have a WP = 0.500? Is it 50%?

    • Alex says:

      Let’s say a player takes two shots; he makes one (a two pointer) and misses one. In WP, that’s +.032 and -.032 for a net of zero; you break even. After that things might change because of the position adjustment, but in terms of raw productivity you have to shoot 50%. With your weights, the same player would get +1 and -.7 for a net of .3. If he made seven shots and missed ten, he would break even shooting 41.2%. Is there some way that the other variables interact with shooting in your model that gets those efficiencies closer together?

      • EvanZ says:

        I’ve said before (to you even, I think) that it’s not +1 and -0.7.

        A made 2pt FG is (2-1.06) = 0.94 pts
        A miss is 1.06*0.074 = 0.7844

        Also, you have to factor in FTA and TOV. The league average FTA/FGA is about 30%. The league average TOV rate is about 13%. Taking all this into account, you will find the break even point is closer to 50%.

        In fact, at the team level, if you assume 80 FGA, the break even point is almost exactly the league average eFG% of 50%. I made a spreadsheet last night to show this. I can e-mail it to you, if you like.

      • EvanZ says:

        Sorry about that, I can’t always remember who I’ve told about what.

        See this more recent post where I listed the weights (they are simply league-average PPP and ORR, so each season needs to be adjusted on its own):

  6. EvanZ says:

    “I’ll continue to say, I think what’s interesting is that no one has revised their model to reflect the knowledge that a number of stats suffer from DR. Don’t you find that odd?”

    If you think DR are significant for these other stats, I assume you will report your findings to Berri. Let us know what he says, and whether he is willing to improve WP by taking into account your findings.

    • Alex says:

      Berri’s already written about it. I think he’s aware, and doesn’t think that it makes a big difference. Besides the rebounding study in his FAQ, he also looked at the effect of changing teams and it doesn’t seem to be a large change.

      • EvanZ says:

        He knows it makes a big difference, but will lose face, if he changes the model. That’s the real reason.

        He doesn’t want to have to admit that he was wrong about Marcus Camby, Troy Murphy, David Lee, etc.

        The key here is not what Berri says, but what you believe. If you don’t think it makes a “big difference”, well, I’d say you aren’t reading your own stuff. However, if you *do* believe it makes a big difference, tell Berri about it! Just like Guy and myself and others have tried to do.

        What if you found that you had a disagreement with your advisor, and you were sure that you were right. Wouldn’t you do everything you could to convince him or her of it? I would certainly hope that if one of my students thought I was in error, they would try to correct me. The students who aren’t willing to or are afraid to question authority are not the ones I would want to have in my lab.

        • Alex says:

          Were I inclined to tell him to change his model (which I’m not), I’m sure Guy and some dude and the other people who have been trying to tell Berri’s he’s wrong for the past five years or so would tell you that there isn’t much of a point to it.

          Speaking of advisors, to answer your question from your site: I’m in a psychology program. I was a math major as an undergrad (it was easy to get because of the requirements for the physics program through the school of engineering; my degree is in physics), and have taken 7 stats classes at the grad level on top of some programming/modeling stuff and the usual experimental methods stuff. I would say my advisor and I have very reasonable discussions; there isn’t a lot of ego involved either way. Of course, I usually frame my questions or comments as questions or opinions instead of running to his office and yelling that he’s wrong.

  7. EvanZ says:

    That’s cool that you do so much stat work in psychology.

    • Alex says:

      It isn’t required, but I like putting more tools in the toolbox. Don’t tell Guy though! He thinks anyone who doesn’t agree with him is a statistical illiterate.

  8. Guy says:

    “So if WP does take diminishing returns into account by giving value according to what they gain/lose a team in terms of possessions, why is it wrong about rebounds? The same method is used there.”

    Of course it’s not the same. There is no equivalent statistic to FGA (or FGmiss) for rebounds. There is no clear and accepted measure of “rebound opportunities used” at the player level. So it’s rather clear how to account for diminishing returns on shooting, much harder on rebounds.

    Or maybe you’re right. Maybe it somehow escaped the attention of all the people who have been studying these issues for years, and designing metrics, that a possession used by one player cannot also be used by another. Until you figured it out yesterday, that is. You have made a paradigm-changing discovery, and are a veritable Columbus of basketball statistics. Yeah, that’s probably the answer…..

    • Alex says:

      I’m sure you’ve been spreading news of how to account for missed assists then, right? We have some reasonable numbers on those. Accounting for diminishing returns, at least at a league-average level, should be cake and everyone should be doing it.

      • Guy says:

        Sorry, you’ve lost me. What’s your point on assists? And how does this support your claim that that “no models” adjust for the fact that using a possession leaves fewer possessions for teammates?

  9. some dude says:

    Alex, or maybe Berri will be more inclined to listen to you because you’ve been a follower. Sometimes people are more willing to listen when those closer to them (figuratively, of course) try and correct them.

    I don’t understand Berri’s position because you’d think he’d want to get it right (especially as an academic). If he did, then he’d get much more acclaim among the stats community and nba fans. Right now, most people think he’s off the wall with his findings (this is not a personal opinion, this is how it appears the overwhelming majority of advanced stats gurus and fans seem to take him). Sure, he could be Galileo, but chances are he’s really just in need of listening to the evidence.

  10. Pingback: Arturo's Silly Little Stats

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