Bonner & Stats: Fun with the Binomial Distribution

[Drz]

Let's have some fun with stats and look at Bonner's numbers from a statistical point of view. For this exercise, we are assuming Matt Bonner's shots follow a Binomial distribution (http://en.wikipedia.org/wiki/Binomial_distribution). Is that assumption reasonable? Feel free to debate below.

Career 3Ps: 641 of 1545. --> Binomial parameters are n = 1545, p = 0.4149.
Playoff 3Ps: 28 of 85.

We can then use basic binomial distribution calculations to answer a number of questions.:


What is the probability that p <= 0.3294, given n = 85 shots?
Using an online Binomial CDF calculator gives the answer: 6.7%.


What is the probability he makes exactly 28 3-pointers?
That is 2.5%. Astute readers may think, hmm, that number is closer to the 6.7% than I expected, does that really mean P(X<28) = 4.2%? Yes it does. Even one less made shot takes away over 1/3 of the 6.7%. This demonstrates the large degree of variability in 85 shots.


Based on Bonner's playoff three point performance, what's the range of 3P% he'd make over an infinite number of shots?
This last one isn't really appropriate since we're assuming the Binomial distribution and p are known, but it's interesting so I'll put it in the post for completeness. According to the binomial distribution, given only his 28 for 85 playoff performance, we can be 95% confident that Matt Bonner's true 3P% lies between 23.1% and 44.0%. Does it? Indeed it does.

If we widen our alpha and use a 90% confidence interval, the range becomes 24.5% to 42.3%, which still contains Bonner's career number.

___

Tune in next week when, if I'm not lazy, I run a hypothesis test with the null hypothesis that Matt Bonner's true shooting percentage is 0.4149, and I see if his playoff performance is enough for us to reject that null hypothesis.

[timvp]

Drz is the most effective Bonner Defender but it's an impossible battle. If it were only his missed threes that made him a playoff choker, that'd be one thing. But it's the deterioration of everything that Bonner does in the regular season once the bright lights are shining.

Bonner being a playoff choker is about as much as a fact as something can be in basketball without literally being a fact, tbh.

[Fabbs]

Drz is the most effective Bonner Defender but it's an impossible battle. If it were only his missed threes that made him a playoff choker, that'd be one thing. But it's the deterioration of everything that Bonner does in the regular season once the bright lights are shining.

Bonner being a playoff choker is about as much as a fact as something can be in basketball without literally being a fact, tbh.
So true.
I am continuing to offer money for .gif(s) of Bonbons "Essence of Bonner" defense during last nights playoff Game 3 vs OKC.

http://spurstalk.com/forums/showthread.php?p=5934441#post5934441

[therealtruth]

I doubt Horry's career 3pt percentage comes close to Bonner but who would you rather have shooting 3's in a playoff game? So you can throw out the stats. Bonner just doesn't have the ability to hit important shots.

[ElNono]

Drz is the most effective Bonner Defender but it's an impossible battle. If it were only his missed threes that made him a playoff choker, that'd be one thing. But it's the deterioration of everything that Bonner does in the regular season once the bright lights are shining.

Bonner being a playoff choker is about as much as a fact as something can be in basketball without literally being a fact, tbh.

The extrapolation is where it all goes wrong. There's no such thing as "infinite number of shots"... Time is a very, very important part of the equation. All players have to perform within a specific time frame.

The whole "if we extrapolate X to 1,000,000 samples" is fantasy. Matt Bonner won't take more than 2-4 3s a game, and that's the chance he has to make an impact. And it isn't just him. Every player has a window of opportunity to produce. This is further subdivided into regular-season (less important) and playoffs (more important).

Take this series. He's currently 1-7 (14%). Let's say he finishes the series 2-10 (20%)... If the Spurs season is over, it really doesn't matter if he statistically evens out the next regular next season over the first 5 games or so... we needed his production in this series.

[ElNono]

I doubt Horry's career 3pt percentage comes close to Bonner but who would you rather have shooting 3's in a playoff game? So you can throw out the stats. Bonner just doesn't have the ability to hit important shots.

Heck, take Stephen Jackson... 30% in the regular season, currently 55% in the playoffs... small sample? Possibly. But he's shooting that now, when we need it.

[ViceCity86]

Bonner hit two crucial back to back 3s in game one of the Grizzlies series last year.He's got some gnads.I hope that Bonner shows up for next game.

[Russ]

How appropriate the only defense of the Spurs' resident geek (Bonner) is an equation. :toast

(Just kidding Matt.)

[DMC]

Bonner won't hit 3s unless the Spurs are up big. He will never hit a clutch 3. He could hit 45 out of 50 in practice, but cannot hit once the game starts.

YMCA is calling him.

[SA210]

The extrapolation is where it all goes wrong. There's no such thing as "infinite number of shots"... Time is a very, very important part of the equation. All players have to perform within a specific time frame.

The whole "if we extrapolate X to 1,000,000 samples" is fantasy. Matt Bonner won't take more than 2-4 3s a game, and that's the chance he has to make an impact. And it isn't just him. Every player has a window of opportunity to produce. This is further subdivided into regular-season (less important) and playoffs (more important).

Take this series. He's currently 1-7 (14%). Let's say he finishes the series 2-10 (20%)... If the Spurs season is over, it really doesn't matter if he statistically evens out the next regular next season over the first 5 games or so... we needed his production in this series.


Heck, take Stephen Jackson... 30% in the regular season, currently 55% in the playoffs... small sample? Possibly. But he's shooting that now, when we need it.

Excellent posts :tu

[slick'81]

lol per 48 matt bonner

[Assman]

Here's a fun stat: Bonner has the same amount of points in this series as James Anderson.

[ShoogarBear]

This is another variant on the "is there such a thing as clutch hitting?" argument in baseball, which has never been proven statistically (and probably never will be proven due to there always being small sample size), but I would argue that the fundamental question posed by the OP is not the right one. The OP is asking whether the results of the entire 85 total shots taken during the playoffs is statistically different from what you would expect given then 1545 shots taken during the regular season. I think that is the wrong question, and because of the large difference in sample sizes between regular season and playoffs, the answer for almost *any* comparison of anybody's playoff vs. regular season stats is going to be "it's within expected limits".

The real question is whether the performance in each individual playoff season is significantly worse than the comparable regular season, which is in essence a repeated measures comparison within each season (with the null hypothesis being that the playoff percentage is no different from that of the regular season).

Here are Bonner's numbers for the years he was in the playoffs (R=regular season, P=playoffs)
2007: R=.383 (n=94) P=.250 (n=4) delta = -0.133
2008: R=.336 (137) P n/a (n=0) delta n/a
2009: R=.440 (268) P=.231 (n=13) delta = -0.209
2010: R=.390 (231) P=.370 (n=27) delta = -0.020
2011: R=.457 (230) P=.333 (n=18) delta = -0.124
2012: R=.420 (250) P=.348 (n=23) delta = -0.072

The first thing to notice is that in every single season for which there is data, the percentage drops in the playoffs. If one was trying to argue that the overall playoff performance is no different than the regular season, then there should be some years when it's better and some when it's worse, not worse every single year. So the statistical question then becomes what is the probability that this drop in three-point percentage every year is just due to chance?

If the probability distribution was normal, this would be a straightforward repeated-measures (or "paired") T-test. For the binomial distribution, the analagous test would be McNemar's test for any single year, but to determine the significance of the delta over multiple years I believe would require a generalized linear model.

No matter what you do, you're still going to bump against the issue of small sample size in the playoffs, which should always bias the tests *against* finding a significant statistical difference.

It's valid to argue that because the quality of the playoff opponent is always better than the average regular season opponent, that a drop in statistical performance is to be expected. The question then is how much of a drop is acceptable. For that we would need to know what is the playoff delta for everyone else in the playoffs.

[Man In Black]

Reg Season Bonner>Playoff Bonner

Anything else?

[Spur|n|Austin]

The extrapolation is where it all goes wrong. There's no such thing as "infinite number of shots"... Time is a very, very important part of the equation. All players have to perform within a specific time frame.

The whole "if we extrapolate X to 1,000,000 samples" is fantasy. Matt Bonner won't take more than 2-4 3s a game, and that's the chance he has to make an impact. And it isn't just him. Every player has a window of opportunity to produce. This is further subdivided into regular-season (less important) and playoffs (more important).

Take this series. He's currently 1-7 (14%). Let's say he finishes the series 2-10 (20%)... If the Spurs season is over, it really doesn't matter if he statistically evens out the next regular next season over the first 5 games or so... we needed his production in this series.

Good points! :tu

[iManu]

Dear Concept Of Phenomenology In Architecture As Developed By The Norwegian Theorist Christian Norberg-Schulz,

Matt Bonner has been sucking. I want to figure out why he plays. I know that he is a ginger and doesn't have a soul, but has he taken Coach Pop's soul, as well? I know he's an awesome guy and I think that searching for good sandwiches is cool, but I wish he could shoot threes. Last year he did, I guess, what I'm trying to say is, is it possible that Bonner will come out of this slump and perform during these playoffs? It's something that I'd really like to know, and if we hadn't won 20 games in a row before this, I would be much more upset. I'm afraid I'll lose sleep if the thunder even the series. God, he's being such a faggot.

—Troubled in Toronto

:wakeup

Dear Troubled,

Many may find themselves curious as to why we "need" a comprehensive theory of architecture as a phenomenological concept. Of course, the intellectual abstractions of theory cannot—nor should they—ever replace the direct sense-experience of architecture in our daily lives. The philosophy pioneered by Christian Norberg-Schulz—considered "one of the most impressive intellectual edifices any architect ever produced"—was never intended to compete with perception. However, by organically combining such materials as Gestalt psychology, information theory, the work of Martin Heidegger, linguistic analysis, and semiotics, we can move toward a more correct and profound experience of architecture, with a renewed sense of the "meaning" of structures, both in the buildings we construct and the loci within which we live. Only when examining what the form represents as a manifestation of higher objects can we be said to be discussing a real architectural experience. Even when examining purely formal (aesthetic) properties, there can be no sound basis for the discussion of structural forms without theoretical insight, because the very concept of architecture transcends the formal aspect.



The Concept Of Phenomenology In Architecture As Developed By The Norwegian Theorist Christian Norberg-Schulz is a nationally syndicated advice column that appears in more than 250 papers nationwide.

[temujin]

No less, Bonner and binomial distribution!

:lol:lol:lol

Waiting anxiously for Bonner' 3 points % in
Month of June.
Saturdays, even days.
Saturdays, first week of the month.
Second day of the month, with Uranus in the constellation of Cancer.

[Solid D]

I think it we should look at Bonner's regular season and playoff performance results assuming Matt Bonner's shots follow a Bipolar Inevitabilty model. Is that assumption reasonable?

The Biplor Inevitabilty Model is based on including a multiplier of .1 to all Playoff Production formulae.

Feel free to debate below.

[DarrinS]

I now have more confidence in Tony hitting a 3 than Bonner -- that's how bad it is.

[E-RockWill]

I love you guys...

[Marcus Bryant]

He doesn't hit his shots when they're needed in the postseason.

The limitation of the probabilistic argument above is the demonstrated consistent decline in made 3 pt % between the regular season and playoffs, season after season.

[Russ]

Here's a question for the stats guys.

Rather than comparing clutch 3-pointers to non-clutch 3s (a comparison everyone seems to find some fault with), why not compare clutch free-throw shooting with non-clutch.

The conditions are more controlled, less variables arise from one free throw to another. Therefore it might be a more reliable measure of performance slippage (or increase) when the pressure's really on and the subject is in the perfect situation to allow it to affect him. The problem with using that criteria on Bonner, however, is that he's not that good at hitting clutch or non-clutch free throws.

[picnroll]

The great mathmetician Rasheed Wallace said it all when it comes to Bonner and the playoffs.

[therealtruth]

Here's a question for the stats guys.

Rather than comparing clutch 3-pointers to non-clutch 3s (a comparison everyone seems to find some fault with), why not compare clutch free-throw shooting with non-clutch.

The conditions are more controlled, less variables arise from one free throw to another. Therefore it might be a more reliable measure of performance slippage (or increase) when the pressure's really on and the subject is in the perfect situation to allow it to affect him. The problem with using that criteria on Bonner, however, is that he's not that good at hitting clutch or non-clutch free throws.

Good point. Pressure definitely affects free throw shooting.

[ShoogarBear]

For the binomial distribution, the analagous test would be McNemar's test for any single year, but to determine the significance of the delta over multiple years I believe would require a generalized linear model.

After having slept on it, I think the way to do it is using binomial logistic regression for repeated measures comparing playoff vs. regular season stats for each year. However, this will require some thought to run.



Tune in next week when, if I'm not lazy, I run a hypothesis test with the null hypothesis that Matt Bonner's true shooting percentage is 0.4149, and I see if his playoff performance is enough for us to reject that null hypothesis.

I just did it a one-sample binomial test and the p value is 0.068, so it's not strong enough to reject it, but again I argue that lumping all the playoff vs. regular season data it's not the correct test.

Just for kicks, I ran McNemar's test on each individual year, and with the exception of 2007, in every other year the playoff data was statistically different from the regular season data.


Here's a question for the stats guys.

Rather than comparing clutch 3-pointers to non-clutch 3s (a comparison everyone seems to find some fault with), why not compare clutch free-throw shooting with non-clutch.

The conditions are more controlled, less variables arise from one free throw to another. Therefore it might be a more reliable measure of performance slippage (or increase) when the pressure's really on and the subject is in the perfect situation to allow it to affect him. The problem with using that criteria on Bonner, however, is that he's not that good at hitting clutch or non-clutch free throws.

Two problems with this:
a) the issue is not really how well the variables are controlled, it's the small sample size. Think of it this way: you can only use statistics to prove that two things are different, not that two things are the same, because if you don't see a difference, the reason could always be that you just don't have a large enough sample size. If Bonner had just as many playoff shots as regular season shots with the same percentages, then it would be easy to show that this was a real difference.
b) nobody really cares if Bonner hits his free throws in the playoffs if he's still missing his three-pointers.

[Russ]

Two problems with this:
a) the issue is not really how well the variables are controlled, it's the small sample size.

You're right, Matt never gets to the line in big game situations. When he gets the ball, the opposing defender flies away from him quicker than Leeds United in an offside trap. The defender actually looks at Matt and smiles, making shooting motions.

Sample size = 0.

[Dr. John R. Brinkley]

That confidence interval with an expected range of 24 to 42% for his 3 point shooting is hilarious...as if that gives us anything useful. That range would cover the vast majority of NBA players.

Anyone trying to prove Bonner is somehow clutch is already showing bias on their part.

Bonner is a choker. We all know this.

[Drz]

Sorry, hadn't come back to this thread since posting it, but here's my brief summary of replies.


Regarding Bonner's playoff 3P% being worse in all 5 years, if we assume it's a 50% chance that someone performs worse (it has to be close to that), it's going to happen to 1 out of 32 players. Is that 1 out of 32 a choker? Or is he the person who just happened to be the 1 out of 32? I argue it's the latter. For excellent books on this topic, I recommend Fooled By Randomness (Nassim Taleb) and Outliers (Malcolm Gladwell).
.
ShoogarBear with the goods! Thanks for running the hypothesis test, I tried to and it was taking more effort than I thought it would. I'll follow up to your posts later.
.
ElNono, "Make over an infinite number of shots" is simply a proxy for expected 3P%. I chose that wording to try to make the question sound more mainstream, but it looks like I made it more confusing instead. Yes, players must perform within a specific timeframe, but that is irrelevant to the point I was trying to make.
.
Dr. John R. Brinkley, "That confidence interval with an expected range of 24 to 42% for his 3 point shooting is hilarious...as if that gives us anything useful. That range would cover the vast majority of NBA players." -- Yes, that's kind of the point. That looking at playoff 3P% will rarely give you anything useful.

[Mugen]

Sorry, hadn't come back to this thread since posting it, but here's my brief summary of replies.


Regarding Bonner's playoff 3P% being worse in all 5 years, if we assume it's a 50% chance that someone performs worse (it has to be close to that), it's going to happen to 1 out of 32 players. Is that 1 out of 32 a choker? Or is he the person who just happened to be the 1 out of 32? I argue it's the latter. For excellent books on this topic, I recommend Fooled By Randomness (Nassim Taleb) and Outliers (Malcolm Gladwell).
.
ShoogarBear with the goods! Thanks for running the hypothesis test, I tried to and it was taking more effort than I thought it would. I'll follow up to your posts later.
.
ElNono, "Make over an infinite number of shots" is simply a proxy for expected 3P%. I chose that wording to try to make the question sound more mainstream, but it looks like I made it more confusing instead. Yes, players must perform within a specific timeframe, but that is irrelevant to the point I was trying to make.
.
Dr. John R. Brinkley, "That confidence interval with an expected range of 24 to 42% for his 3 point shooting is hilarious...as if that gives us anything useful. That range would cover the vast majority of NBA players." -- Yes, that's kind of the point. That looking at playoff 3P% will rarely give you anything useful.



Kill yourself.

[slick'81]

any 1 who uses stats to argue bonners worth should die tbh

[Paranoid Pop]

Drz is a certified relative of Bonner, probably his mom who is a teacher.

[diego]

Or is he the person who just happened to be the 1 out of 32? I argue it's the latter.

looking at playoff 3P% will rarely give you anything useful.


looking at it that way, with a 50% chance of being better or worse, that means 16 other players shoot better in the PO. Wouldnt it be better to have one of those 16, than one of the other 16? remember the objective is to win in the PO


and for PO 3p%, it gives something very useful, the 3p% in the PO. it is factual information. i would argue that the opposite is true- very rarely does it not reflect a player's ability, ESPECIALLY if that player is a 3 pt specialist.

[baseline bum]

Ginger faggot passing up shots again. The Spurs should just start running back on defense every time redhead puts the ball on the floor.

[pjjrfan]

Bonner is the guy who we kept instead of Scola
Pisses me off I think that 90 % of our posters take it for granted that bonner is going to choke in the playoffs why don't the coaches see this?

[O.J. Simpson]

Great thread.