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Misconceptions


I stopped writing regularly for this blog a few weeks ago because of work. Many a night now spent at the computer, but not watching hockey. So in the most recent post-game, when I was asked to explain what Pythagorean expectation meant, well I wasn't going to answer. For one, I don't have that much free time to spare these days, and for two, why reinvent the wheel, right? This is the age of Google, after all.

But I can't let a comment like this go:

Good Lord I didn’t realize that a teams projected point total based on their GF and GA was related to the equation for solving the hypotenuse of a right triangle (surely you know the formula a^2 +b^2 = c^2). The Pythagorean Point Equation eh?

Please explain this formula to me R O. You posted the link so you should be willing to back it up. Maybe you could complete a derivation of it so I can gain a better understanding. Or you could complete an uncertainty analysis based on the projected point totals and actual point totals of all thirty teams using their GF and GA (I will explain this if needed) to prove that it isn’t complete bogus

Two things:

1.) My personal belief, which Kent may not share (this is his blog after all), is that commenters who want to move the conversation forward should be ready and expected to do some legwork themselves. There are a ton of resources out there (www.nhl.com, www.behindthenet.ca, www.hockeyanalytics.com, etc. etc.) and Microsoft Excel is also your friend here.

2.) Asking why P% can be predicted by an a^2 + b^2 = c^2 formula is missing the forest for the trees. Again, my belief (and one that Kent may not share as well) is that this whole hockey-stats movement is not about glorifying seemingly meaningless numbers. It's about gaining insight into hockey - measuring what's important, seeing what drives success, and more importantly, what doesn't drive success.

So the right question to ask isn't "why does a^2 + b^2 = c^2 predict points percentage"?? The right question to ask is "is it really possible to predict how often you pick up points simply by looking at how often you score and how often you get scored on"??

Because, folks, if this is possible, then BIG GOALS and BIG SAVES are mirages.

Star-divide

Pythagorean formula: Points% = GF^2 / ( GF^2 + GA^2 )

I was asked what the origins of the Pythagorean point expectation was. I believe its origins are in baseball, and as far as I know it's an empirical equation. Basically some sharp guys decided that they ought to take runs scored for and against and try and map it to wins, and the Pythagorean formula fit best.

Alan Ryder of HockeyAnalytics did some research on a more improved model for goals->wins. Some really crazy stuff, you can check it out here under "Probability Models". The point is not the model though, it's the assumption he made: goals are rare, uniformly random and memory-less.

Read that again.

Goals are rare.

Goals are uniformly random.

Goals are memory-less.

Alan Ryder did a quick check that suggests with a high degree of convincing-ness that the above is true, except for in the first minute of each period and the last minute of the third period. There are some conjectures as to why these exceptions occur (the last minute of the game is empty-net time, the first minute of each period contains less time spent in the zone during PPs) but the exceptions are less interesting than the main result.

For you see, if goals really are rare, and uniformly random, and memory-less, then there's not very much real estate for the BIG GOAL and BIG SAVE to coexist with regular, boring hockey truth. No matter how much a skater might elevate his game to the level of CLUTCH or how well a goalie just makes BIG SAVE after BIG SAVE because he wants to win more, goals keep getting scored and saves made in the same measures, minute after minute after boring minute.

And this ties in to Pythagorean expectation quite well. Because sometimes you look at a team that just keeps on winning, but without the goal differential to back it up. For example, the Avs. And their troll fans will come to you after winning against your team despite playing like garbage, and when you point that out one of the usual retorts is "we found a way to win" or "we scored a big goal and our goalie made some big saves".

And that, friends, is absolute grade-A bullshit. Because point percentage can be modeled by the Pythagorean expectation. And even more important, point percentage can be modeled by randomly distributing goals scored for and against over 82 games.

Observe:

 Gfga_medium

The red dots are true points% for the 120 post-lockout team seasons. By "true points" I mean 2 points for a win (regulation or OT), 0 points for a loss (regulation or OT), 1 point for a tie. Shootouts and loser points are gone, the model doesn't account for that. And before anybody says anything, you can augment the model to account for shootouts by adding a coinflip component in the case of ties. See point 1.) above about legwork.

The blue line is what the Pythagorean expectation predicts. Pretty straight down the middle eh? I didn't calculate R^2 or what not, that doesn't really interest me - more insight can be gained by the green dots.

The green dots, ah the green dots, these are points %'s from simulations that I did for every team season. Basically I took the GF and GA of each team-season and randomly distributed them (with uniform distribution) over 82 games. Did that 500 times per team season, calculated the point %, put it on the plot.

You see? EVERY team season in the post-lockout is well within the range that one would expect from a parallel universe where there was no dispute about BIG GOALS or BIG SAVES - where goals are scored when they are scored and goalies make saves as best they can and we don't ever talk about clutch or elevating your game.

Maybe, just maybe, that parallel universe is this one.

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Love it. Sometimes we tend to forget the human element in the game though. Emotion is not a predictable statistic, and sometimes, as Kent so often has said over the years I’ve been reading his blogs, the bounces just don’t favour the skaters.

by jessnbrown on Dec 17, 2009 2:31 PM PST reply actions  

Very interesting, thanks RO.

Big goals and big saves exist insofar as humans can point to events after a game as pivotal. Ie; they are narrative to nodes to explain the game afterwards. Which is cool, that’s how humans best (most easily) interpret things.

In reality, though, you’re right – almost every save or goal is potentially “big” and there’s probably little to no actual truth to certain teams or certain individuals being more “clutch” or the like (outside some players being just generally better than others, which applies at all times during a game).

You’re also right about the Pythagorean formula coming from basebeall (the work of Bill James).

by Kent Wilson on Dec 17, 2009 2:44 PM PST reply actions  

Based on what I’ve seen of the debate on ‘clutch’ in baseball, however, you’re pissing into the wind trying to convince a large number of people that clutch doesn’t exist as a measurable skill. Many people will see Iggy scoring in the last minute against Anaheim as a measure of his clutch skill, but probably won’t notice that same skill doesn’t exist in the last minute of 20 other games they’re down a goal.

by maimster on Dec 17, 2009 4:15 PM PST reply actions  

“Clutch” is an emotional argument and should be treated as such. Once you start trying to quantify something emotional, you get into trouble.

i.e.: However some may criticize Iginla these days, when he is inducted into the HHOF in about ten years time, every article about it will fawn over his performance in the 2002 gold medal game, the 04 playoffs, and if the hockey gods favour us, the 2010 games and a Flames’ league championship. Great players will always be revered for their greatest moments, and I have little doubt that Iggy will be remembered for being clutch. Cringe all you want at that. ;)

by Resolute on Dec 17, 2009 4:59 PM PST up reply actions  

Agreed, there are moments when someone is ‘clutch’ and we love them for it. Absolutely true. It just isn’t a measurable, repeatable skill, that’s all. I certainly don’t cringe thinking about Gelinas as clutch in 2004, though!

by maimster on Dec 17, 2009 5:36 PM PST up reply actions  

Personally, I find it comical that a simple two word phrase generated such a diatribe. All I can really say is this: Stratomatic hockey is played on a spreadsheet. Ice hockey is played elsewhere. The sabermetric view of hockey is interesting, but I fail to see how the observational and emotional aspects should be treated subservient to it.

Ultimately, my argument in the last thread was this: Curtis McElhinney is not a goaltender you can count on to win you hockey games. I suspect your spreadsheets will tell you the exact same thing my eyes told me.

As far as pythagorean wins goes, I’ve always loved that stat in baseball, and am not surprised it follows in hockey. But honestly, do you really need a spreadsheet and a litany of simulated seasons to prove that teams with greater GF/GA ratios tend to win more? Personally, I think this comes down to a macro view vs. a micro view. A spreadsheet might show you trends and long term expectations. But when it comes down to a single game, there are key moments throughout the game where a BIG SAVE or BIG GOAL, a big hit, a big penalty kill, etc, can make all of the difference.

by Resolute on Dec 17, 2009 4:50 PM PST reply actions  

So, what you’re saying is that the OIlers just need to find clutch goal scorers? :)

Editor of The Copper & Blue, and leader of The Cult Of Hartikainen.

by Derek Zona on Dec 17, 2009 5:24 PM PST reply actions  

RO, you’re spitting into the wind. You can lay all of this out and you can do the math and, much like creationists, your opponents will swipe it all away. They’ll make grand-sweeping statements like “you can’t measure…” insert [heart], [emotion], [big play ability], [clutch play], [mental strength] here. They’ll say this and offer nothing in the way of proof.

You’re better off ignoring the small-minded and going after the people that will at least give you a listen before exposing their ignorance.

Editor of The Copper & Blue, and leader of The Cult Of Hartikainen.

by Derek Zona on Dec 17, 2009 5:35 PM PST reply actions  

I admit I am new to the hockey blogosphere, but I am suprised at the hostility shown towards people who don’t value statistics as highly as others. It is just a game, right?

by Ken Williams on Dec 17, 2009 5:39 PM PST up reply actions  

You misunderstand. It’s not “people who don’t value statistics as highly as others”, rather, it’s people who broadly dismiss the enormous value that statistics bring to understanding the underlying game of hockey.

Editor of The Copper & Blue, and leader of The Cult Of Hartikainen.

by Derek Zona on Dec 17, 2009 6:14 PM PST up reply actions  

R O,
 
 A great piece, you define “clutch”.
 
 Anyways, you do have a very convincing argument, that stats dictate
 the results in hockey. Yes, to a point, this is very true I do not doubt your
 judgment one bit. I am a mechanical engineering student and through
 statistical models we gain a better understanding of many aspects of
 this world including what materials to use in construction, where
 oil/gas might be located, what measuring tool instruments are most
 accurate, etc..
 
 Over an 82 game schedule it is very likely that a team’s results will
 follow your Pythagorean expectation graph based on their GF and GA.
 
<img src="http://assets.sbnation.com/assets/228948/GFGA.jpg <http://assets.sbnation.com/assets/228948/GFGA.jpg> "/>
 
 If you look closely there is that beautiful Pythagorean blue line of best fit. Now if you look even closer, you will see the red dots,
 which dictate the actual results of every team post lockout and the
 green lines are the predicted outcome of every team for this year (R O
 did a better job explaining this above if I was unclear).
 Suppose we look at last game as an example as it is fresh in our
 minds. CuMac (or Tabasco as we like to call him) was heavily criticized
 for his performance in net. Resolute even opted to go down the so-called forbidden road that everyone who is elite enough to comment on this blog would never do- said he couldn’t come up with the almighty
 "BIG SAVE." And sure, we can all agree he had the chance to make that
 BIG SAVE on the TJ Oshie goal. Over an 82 game season, statistically
 speaking it was an insignificant event, and it won’t change R O’s
 Pythagorean graph. But yesterday’s game wasn’t an 82 game season or a
 decade worth of games. It was one game, or one sample. And that’s why
 we watch this game, why we participate in the game threads, read game
 previews and comment on each individual players shifts. It’s because
 we watch for the BIG SAVE or Iggy coming through with a clutch goal. Just like the players say, ‘you gotta take it game-by-game."
 
 I get the feeling that "CLUTCH" and "BIG SAVE" are not possible events
 in the way you analyze the game, R O. And they wouldn’t be in mine either if all I did was
 follow that Pythagorean line of best fit, or so many of these other statistics that more and more bloggers refer to, but have so little significance on the individual 50 second shifts that make up periods, games, seasons and so on. That’s where the red dots
 come into play and which is why those "clichés" are in my mental
 dictionary. It’s these minor events that "skew" the results.
 
 Sure, using a model like that to predict the future or an upcoming game
 for the Flames would probably work quite well and one could gain a
 better understanding of where the team is at at this point of the
 season. But to be bitching people out for using a cliché that
 both of us know is right is just utterly wrong. On that note, was it really necessary of you to tear apart Resolute for using the word ‘big save?’ Dude- it’s hockey. Its’s a bunch of overpaid, undereducated kids skating around trying to swat a piece of rubber into a net. There is absolutely no reason on why you needed to attack Resolute for simply stating that CuMac didn’t make the ‘big save.’ Because you know what? At the end of the day, it’s just a game we enjoy following. Who cares if someone believes that a clutch goal or awesome late game save is the difference between a win and a loss? No reason to paint someone as naive or clueless when they have those beliefs, no reason to label them as a ‘Calgary Puck-ster’ which seems to be quite the heady insult. But the most important thing is that the players themselves refer to weak goals let in late in the third period in a 2-2 game that leads to a loss as backbreaking. Remember when Luongo got hurt in late October, and Willie Mitchell said it was really hard to overcome how the tandem of Schneider and Sanford would let in bad goals in when Bobby Lu was hurt last season? At the end of the day, if its the players who actually play the game who feel that the term ‘clutch’ is critical in winning, then isn’t that all that matters? I mean, come on. Your attempts to make people who don’t value statistics as highly as you do look stupid aren’t necessary. If an outsider with little knowledge of hockey read this article, they would assume the likelihood of winning a hockey game is comparable to tossing a coin. You can’t always depend on stats for this reason! If we did, then we would never consider things like willingness to battle for pucks, for pure skill, for speed, for determination, for any of those ‘intangibles’ that statisticians refuse to acknowledge. Nor am I trying to deflect any attention away from the importance of stats, as an engineer, I have seen enough numbers in my life to appreciate their significance. But to deny the fact that human emotion and such doesn’t play a part in the game is foolish. If that were the case in professional sports, Jean van de Velde should have been just as likely to get a triple bogey on his 45th hole in the 1999 British Open instead of what everyone knows as truth: he choked on the 72nd hole because the pressure got to him.
 
 Anyways, consider how each and every person has a different reason on why they watch hockey and cheer for the Flames. I prefer to watch every game as its own, but that is just me.

And sorry if the spacing is all messed, did this all on my phone

by M F on Dec 17, 2009 5:43 PM PST via mobile reply actions  

Damn… Posting all that from your phone is clutch, man. Tremendous heart. ;)

by Resolute on Dec 17, 2009 6:22 PM PST via mobile up reply actions  

I think you make a lot of sense, MF. To deny the existence of big saves etc etc is pretty shallow, despite how Derek Zona, who it appears is an Oilers Blogger, compares people who favor stats to proponents of evolution, and vice versa for creationists. Wow… That is completely irrelevant in this context, so I will ignore it. Stats vs emotion doesn’t compare, that’s just picking a fight for the sake of it, although that seems to be more common than I expected, especially with yourself, R O. What stood out for me was when M F brought up Willie Mitchell’s quote that the Canucks struggled to get anything going last year when Luongo pulled his groin. I recall a few games that the Canucks played where they completely fell apart after some brutal goals were allowed. One vs Nashville stands out.

If players themselves attribute team success based on the fact that their goalie made a clutch save, I agree that that should be taken into consideration. Like it or not, R O, hockey is a sport of emotion that statistics will never be able to delve into. If every game was statistically equal, then wouldn’t a game between Calgary and Carolina monotonously occur with the exact same significance to the players than a rivalry game against Vancouver or a game against a team they immensely dislike due to some past event? Goals may be memoryless, but I’m sure that the next time Washington and Colorado square off, a little more will be on the line simply based on the memory of the Koci hit.

I just think there has to be something to the argument that if the players who are paid millions of dollars to play the sport we all love BELIEVE that big saves, clutch goals, etc all play a role in the sport, then maybe we should just agree to that and throw in the proverbial towel that states all events in a hockey game can be mathematically modeled. I don’t think its becoming of yourself to ignore this and try to make us believe that hockey is nothing more than a “Pythagorean roll of the die.” Regardless of what your intentions for following hockey are, its not up to you to dictate how it should be followed.

Excellent post, MF.

by Ken Williams on Dec 18, 2009 5:41 AM PST via mobile reply actions  

I get the impression that some commenters here might have skipped past the boring-ly methodical middle parts that I wrote (the “evidence”) and jumped right to my conclusions. And then proceed to tell me I’m wrong. It’s akin to flipping straight to the end of a book and then criticizing the story (“I can’t believe that the old crippled man was the bad guy!”). And not that I don’t know people who do that, but… it is to laugh.

This post was not about statistics. It was about hockey truth. Math is a useful tool to help uncover truth. Logic, reason, and a keen eye are other good tools.

I mean, I think the best post in this thread was maimster’s:

Many people will see Iggy scoring in the last minute against Anaheim as a measure of his clutch skill, but probably won’t notice that same skill doesn’t exist in the last minute of 20 other games they’re down a goal.

Well said – he didn’t need to bust out his stats textbook to capture the spirit of my post.

If you want to challenge me on what I propose, with evidence, is part of hockey truth (the big goal is a mirage), then you could probably start by commenting on the evidence.

It took me literally five seconds to come up with two counter-arguments to my position:
1.) Just because the team-seasons fit in the range one expects from purely random goal scoring, doesn’t mean there isn’t a clutch component hidden in the noise.
2.) Showing the range of green dots is nice but surely the dots closer to the centre of the range are more clustered? Meaning they are more likely to happen, and it would be interesting to see how likely a result each of the 120 team-seasons was. If there are too many outliers, maybe clutch does exist.

And another five seconds for myself to come up with responses to the counter-arguments:
1.) That’s fair, but I would question the value of a skill that has so little effect that it is drowned out by noise.
2.) Very interesting point. I have laid the groundwork out for this analysis, if you want you can have the data and figure out how likely each team-season was under the assumption of purely random and uniformly distributed scoring. Then get back to me, and we’ll discuss it some more.

And in both cases the conversation moves forward and everyone learns something new.

But do I have to converse with myself for this to happen? There’d be no point in blogging, I could just stare at my belly button and come up with this shit in my dimly lit basement whilst eating Cheetos. Some might ask how that is different from blogging. Nonetheless, it seems less fun.

I welcome dissenting opninons. But you have to back it up with evidence and logical reasoning. I see I’m being accused of arrogance but the height of arrogance imo is telling someone they’re wrong without telling them why they’re wrong.

by R O on Dec 18, 2009 7:47 AM PST reply actions  

First, I did read your entire article, and I did consider your evidence supporting how goals for and against, when spread out over an entire season, are good indicators on why a team may or may not be successful. I’m not denying that. How could I? Goals are what wins games, right? The bigger the difference between goals for and against are obviously a very good indicator of whether or not a team is any good. And when you analyze every goal scored throughout the season by looking at the stat column, but without considering the situation in which the goals were scored in or when that big save that prevented a goal was, it’s hard to come up with explanations on why Team A that has huge goal differentials in their favor has a worse win-loss record than Team B who is barely breaking even in goal differential. Maybe, just maybe, a clutch save was in there at the end of the game for Team B, when the nearly-drained clock raised the stakes of the situation above what they were ten minutes into the first, giving Team A, a statistically superior team, the win.

At the end of the day, its pretty difficult for those who support clutch play to defend ourselves with statistics. Here is His Holiness Bill James’ opinion of clutch-ness:

Since this elusive "clutch play" has no particular statistical dimension, it has become popular within the discussion as a bullshit dump. All discussions have bullshit dumps; we need them. Our logic, whatever it is that we are talking about, can never be completely worked out; all subjects worthy or discussion are too complicated to be fully encased in logic. Thus, in all discussions, the least precise areas become bullshit dumps, elements of the discussion which are used to reconcile our formal logic to our intuitive sense of right or wrong, justice or injustice, accuracy or inaccuracy, reason or madness, moderation or extremity.

You know what? I’m ok with that. There isn’t really a way to quantify ‘clutch play’ through statistics. The ‘big goal’ or ‘big save’ that you won’t acknowledge are, as you said, ordinary numbers that don’t look any different from any other when just looking at the post game stat sheet. But can you honestly say that’s fair? Can you truly say that in a game of high magnitude, a late goal to win a game or tie it up to force overtime can’t be considered ‘clutch?’ I’m not going to find stats to back it up. I suppose there are ways to defend myself. There are numbers out there that are easily accessible. Maybe by looking at regular season vs playoff scoring totals or save percentages? Regular season or playoff overtime goals scored/allowed? I could do it, but quite frankly, I believe that the judgment of the viewer takes precedent over the numbers when determining whether or not a goal is clutch or a save is huge.

Look at a mediocre player like Chris Drury, who rarely has statistically good regular seasons. Is it fair to say that he may step it up in big moments if he can somehow score 3 game winning goals as well as a critical game tying goal with 7 seconds left in game 5 vs NYR in the 07 playoffs for Buffalo? What about Mike Vernon’s save (sorry, no video) on Stan Smyl in the 89 playoffs? I’d have to say that this save was probably more significant than the first shot in the game Fleury faced in game 7. What about this whole video? You get the idea.

I could drone on all day, but that’s beside the point. All I am trying to say is that to most hockey fans, the clutch play is there. I know that all of the talking heads on TV and news media get ostracized by the stat heads for using terms like ‘big time players make big time plays’ or ‘that guy is a clutch player’ or what not, but they have a point. When the stakes are the highest, a key play is going to be the difference in a game. If you can’t acknowledge that some element of clutch exists in pro hockey, I guess this is just going to be an endless argument. That’s my opinion, as well as the vast majority of hockey fans out there. To the select few who refuse to admit this, well, enjoy reading behind the net or whatever sites you use to analyze a game. Personally, I’d rather watch the games.

by Ken Williams on Dec 18, 2009 12:51 PM PST reply actions  

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