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Monday, July 27, 2009

The Book Blog: mgl: Most people still don’t understand the concept of regression toward the mean…

Dig, it!

There is a thread on BTF about Sabathia’s “numbers,” particularly his BB and K rates, being down this year, as compared to last year, although he is still pitching very well of course.

While the quality of the posts on BTF is nowhere near that of this blog (although they beat us easily in quantity), there are some reasonably intelligent regulars on that site (if anyone interprets that as a dig, it is).

...The thing that people don’t understand (actually one of the things) about regression toward the mean in baseball is that the reason any above or below average player will always regress, on the average, towards average, is that they were not really as good or bad as we thought in the first place, based on any of their stats.  That goes for Sabathia, Halladay, Bonds, Chipper Jones, etc., etc.  Chipper Jones is not as good as his career stats tell us, even after you do all the appropriate adjustments.  Same for Halladay.  And Sabathia.  And everyone else who has been above average and we think has true talent X.  When I say “as we think” I mean as their stats suggest, not as we think based on a credible projection which already does the regression.  And of course, there is some chance that any given player is better than his prior stats - it is just that the chances of him being worse is greater than the chances of him being better.  That is ALWAYS the case, as long as we properly define the mean for that player.

That is the KEY to understanding regression toward the mean and is what most people don’t understand, even if they think they understand the concept.

Repoz Posted: July 27, 2009 at 12:19 PM | 244 comment(s) Login to Bookmark
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   201. Mike Emeigh Posted: July 28, 2009 at 01:21 PM (#3270149)
if you have ONE PA, then the result can only be 0 or 1. And what causes that? Random variation around a true mean + environment.


It's not random; it's just unexplained. If we had fine enough tools (which we don't have and probably never will have) we could explain more of it.

Some people like to think that everything is preordained so that if you can replicate everything (including the environment) and if Pujols did everything identical, then he was destined to get that out and would do so every single time out. If that is true, then at that moment in time, Pujols had no talent whatsoever for getting a hit.


It's a long leap from saying, as I do, that if you could capture everything about the environment you could explain why Pujols didn't get a hit in that PA, to saying that everything is "preordained" - just because we can explain "what" happened doesn't mean that we know "why" it happened (ir even that we can possibly know "why"). The specific set of factors that created a situation in which Pujols made an out may not even BE replicable, at least not by human beings.

I would think that the search for knowledge would be leading people to try to capture as much information about factors that affect hitter (and pitcher, and fielder) performance as we possibly can so that we could better determine whether a player has really taken a step forward when we see a performance improvement.

-- MWE
   202. BFFB Posted: July 28, 2009 at 01:43 PM (#3270165)
Some people like to think that everything is preordained so that if you can replicate everything (including the environment) and if Pujols did everything identical, then he was destined to get that out and would do so every single time out. If that is true, then at that moment in time, Pujols had no talent whatsoever for getting a hit.


The outcome is a result of the initial conditions. If you can keep replicating the initial conditions exactly you will keep getting the same result and this statement "Pujols had no talent whatsoever for getting a hit." would be true, for a given value of "true". But that situation would nothing more than thought experiment to illustrate an esoteric point about systems and the "unknowability" (is that a word?) of all the initial conditions and thusly not applicable at all to any real world situation. Which is why if you are talking about "true talent" it has to be a "mean true talent over a timeframe of x".
   203. Slivers of Maranville descends into chaos (SdeB) Posted: July 28, 2009 at 02:08 PM (#3270196)

Let's try another one. You believe Pujols IS the best hitter in baseball. He goes in an 0-fer slump. Do you still say that Pujols IS the best hitter in baseball?

If you answer yes, then you accept "true talent" as I'm trying to use it.


The error (as I see it) is that that IS is, in fact, prospective. If I said that Pujols is the best hitter in baseball, that means if we had a hypothetical FUTURE at bat, I would expect him to hit more often than another player. I agree that there can be error in that projection.

The issue about the Phillies and Devil Rays is, however, interesting. I would accept the proposition that an inferior team can beat a better team. So I have no problem with saying (for the sake of argument) say, the 2004 Cardinals were better than the Red Sox. So maybe you've got me. But I'm not so sure. For I could also believe that the 2004 Yankees were better than the 2004 Cardinals. And that the 2004 Red Sox were better than the 2004 Yankees. So where does that leave us? I'm not sure. Let me think about it.
   204. Juilin Sandar to Conkling Speedwell (Arjun) Posted: July 28, 2009 at 02:09 PM (#3270198)
And I want to thank Tango for not just responding to every question, but for actually answering them in a way that a dolt like me could understand.


I'll second this - I actually understand the point being made (now. It's kinda surprising, actually), which is pretty nice. I can follow what's being said, too.
   205. BFFB Posted: July 28, 2009 at 02:21 PM (#3270209)
When making pronouncements on the team level you run into exactly the same definitional problems as you would on a player level. All you are doing is changing the "thing" you are making a true-talent pronouncement for. In the second instance you are doing it for a player in the former you are doing it for a team. With the true-talent of the team at a particular instance being a composite of the true-talent of the individual players on that team in that instance.

I'll second this


Thirded.
   206. Jeff K. Posted: July 28, 2009 at 02:21 PM (#3270210)
I am very disappointed in the metaphor of a weighted coin for random variation.

Yeah? Well, I'm disappointed. In...your face.
   207. Tango Posted: July 28, 2009 at 02:48 PM (#3270239)
My philosophical position is that to not believe in random variation is to believe in fate.

I preach at the alter of the binomial distribution, and therefore random variation is in play. If you can't accept that as the basis, in no way can we have a fruitful discussion on the matters of regression toward the mean or the true talent level.
   208. BFFB Posted: July 28, 2009 at 03:02 PM (#3270258)
I have no problems with random variation as long as it's applied solely to models of the physical world rather than the physical world itself where random variation is just short hand for "the ability to predict an outcome is limited by knowledge of and ability to measure factors which effect said outcome".
   209. Steve Treder Posted: July 28, 2009 at 03:24 PM (#3270287)
The error (as I see it) is that that IS is, in fact, prospective.

So, what you're saying is, it depends on what the definition of "is" is.
   210. Tango Posted: July 28, 2009 at 03:47 PM (#3270326)
"Is" is IS.

Pujols IS a better hitter than anyone else in the world. It's present tense. My certainty of that is not 100%, because Pujols is human, and therefore, at this moment in time, he may have some injury, or other thing that prevents him from being "all he can be". Or, Mauer or Zobrist or someone else may actually have figured it all out.

Whether he PERFORMED as the best hitter in the world is subject to random variation. Felix Hernandez, for one moment in time, may as well have performed as the best hitter in the world (or that Johan Santama may have performed as the worst pitcher in the world).
   211. phredbird Posted: July 28, 2009 at 05:09 PM (#3270451)
but ... but ... how did you arrive at the conclusion that pujols is the best unless you measure a performance? you had to have some data set at some time. we don't agree he's the best hitter in the game because we had some epiphany.

are you saying there's still this fuzzy 'true talent' version of him we can't know because we don't have the tools? i'd say we never will based on my understanding of the discussion so far.

nevermind. i'm getting this ... maybe. he's established that status in our minds at some point, but even so we have to say its provisional because it's actually possible he could go 0-fer-2010 and we'd have to adjust our pronouncements.
   212. Russ Posted: July 28, 2009 at 05:18 PM (#3270466)
But when their careers are finished, the question becomes moot.


However, the counterfactual careers are very important when you consider trying predict the performances of future players. If Bonds and Perez had exactly the same talent, but Bonds randomly had such a better career, then that would say something about the underlying population of player careers. Obviously if you condition on their whole career, the point becomes moot FOR THOSE PLAYERS, but it still doesn't change how their careers affect how you view the careers of others.

One of my philosophical ruminations is how to adjust for the fact that players who play in the major league early may have better careers in part because they are better players but also because they had the opportunity to play in the major leagues sooner. There is a selection bias for major league players that is never really accounted for. That is, if Player A has an MLE of 0.280 in major league play and Player B has an MLE of 0.280 in AAA, then if they are of the same age, position, etc., most statistical models would model them as having the same projection going forward. However, I think that Player A has had the advantage of being able to experience major league pitching, so he should project better going forward due to his opportunities. And if he does do better going forward, than that says something about how we evaluate players who in the minors for a long time (the Ken Phelps-ers) and what affect staying in the minors will have on the careers.
   213. Jeff K. Posted: July 28, 2009 at 05:20 PM (#3270470)
but ... but ... how did you arrive at the conclusion that pujols is the best unless you measure a performance? you had to have some data set at some time.

As I said last night, like coins, actual (true, whatever) talent obviously determines performance, but unlike coins the performance is the only measure of talent that we really have for players. That doesn't mean that in all samples performance = true talent, but to me for large, varied, and consistent enough samples it does (again, where mgl and I disagree.)

are you saying there's still this fuzzy 'true talent' version of him we can't know because we don't have the tools? i'd say we never will based on my understanding of the discussion so far.

Tango would agree with you, that's the whole 99.9999999999% discussion from last page.

he's established that status in our minds at some point, but even so we have to say its provisional because it's actually possible he could go 0-fer-2010 and we'd have to adjust our pronouncements.

Or there is however minute a chance that he's actually considerably worse and has been extraordinarily lucky over and over again and manifested in ways that aren't similar to others (bloop hits and the like.)
   214. Jeff K. Posted: July 28, 2009 at 05:22 PM (#3270478)
That is, if Player A has an MLE of 0.280 in major league play and Player B has an MLE of 0.280 in AAA, then if they are of the same age, position, etc., most statistical models would model them as having the same projection going forward.

I don't know if they would, but I know they damn well shouldn't. To do that would have to presuppose and factor into subsequent calculations a zero error margin on whatever MLE they're using for B, and that is not the case.
   215. phredbird Posted: July 28, 2009 at 05:24 PM (#3270481)
Or there is however minute a chance that he's actually considerably worse and has been extraordinarily lucky over and over again


then we'd say that 0-fer-2010 is him regressing toward his true mean and bob's your uncle.
   216. Edmundo got dem ol' Kozma blues again mama Posted: July 28, 2009 at 05:31 PM (#3270492)
Tango would agree with you, that's the whole 99.9999999999% discussion from last page.

he's established that status in our minds at some point, but even so we have to say its provisional because it's actually possible he could go 0-fer-2010 and we'd have to adjust our pronouncements.

Or there is however minute a chance that he's actually considerably worse and has been extraordinarily lucky over and over again and manifested in ways that aren't similar to others (bloop hits and the like.)

Am I thinking right here? If you had a google universes, each with a parallel Neifi and a parallel Bonds, in at least 1 of those universes, its Neifi would have a better career than its Bonds. Or vice versa, which happened to have happened here.
   217. Shock Posted: July 28, 2009 at 05:33 PM (#3270498)
Sigh. I wrote a really long post late last night but I did nit submit it and now it's sitting in the reply box at home.

But basically, with the coin example, the only reason there is randomness with the coin is because we do not measure everything about the coin's environment. The coin does not land heads 75% of the time because of "randomness," not in an absolute sense that is. It lands that way because 75 times out of a hundred the flipper exerted the amount of force necessary so that it would land heads due to its environment blah blah.

If we could gather all data that is necessary for determining how the coin will land (force, air resistance, etc.) we could use this flawless model to predict how the coin will land 100% of the time. There will be no randomness in this universe, the coin will not have a 50% chance of landing heads on any given flip, but will always land 100% how we expected it to land given our flawless model.

Randomness in the coin flip exists because we do not have a flawless model, and we do not know how the coin will land because we don't have the tools necessary to take everything into account.
   218. Tango Posted: July 28, 2009 at 05:37 PM (#3270509)
nevermind. i'm getting this ... maybe. he's established that status in our minds at some point, but even so we have to say its provisional because it's actually possible he could go 0-fer-2010 and we'd have to adjust our pronouncements.


You got it.

"True Talent" is unknowable. We can only estimate it. We estimate it based on past performance, what we know about the player, and how other players of the same population ended up performing.
   219. phredbird Posted: July 28, 2009 at 05:59 PM (#3270542)
well, that sounds pretty flux-y and would offend my platonic sense of order if i had one.

it will have to do!
   220. Slivers of Maranville descends into chaos (SdeB) Posted: July 28, 2009 at 06:03 PM (#3270549)

"True Talent" is unknowable. We can only estimate it. We estimate it based on past performance, what we know about the player, and how other players of the same population ended up performing.


It is this kind of probablistic definition that I have difficulties with. Again, it's fine as a model. But I am not sure it is fine as a description of reality.

What you are trying to do is to isolate a person's 'talent' divorced from all context -- what ballpark he plays in, what pitching he is facing, the liveliness of the ball, whether he is hungover, whether there was dust in his eyes in a particular at-bat, etc.

Which is fine up to a point. But human beings don't really exist independent of their context. The player is in part a product of his environment, it's not all just extraneous noise. Correcting Hank Aaron's stats for Fulton County Stadium has all sorts of uses. But I don't know that it gets us closer to the 'true' Aaron, since that's where the real flesh-and-blood player played.
   221. Jeff K. Posted: July 28, 2009 at 06:05 PM (#3270555)
with the coin example, the only reason there is randomness with the coin is because we do not measure everything about the coin's environment.

I didn't say there was randomness, I said that the individual results that make up the performance are discrete. Outside of a 50/50 coin, the discreteness impacts. For instance, if an object were to fall side A down 70% of the time and side B 30%. If I flip it 10 times and it comes up 7/3, that may come AAABAABAAB or ABAABAABAA or AAAAAAABBB. It's the same coin, whatever you want to call the building up of non-integer probability percentages from discrete and whole outcomes (variation, chance...), the points is that it's there.

Measuring everything about the environment, or "the perfect model" you proscribe, frankly I don't see the point. Yes, if you could factor in (or automate out, say using a robot arm) every single thing to get a perfect model, you'd know what's going to happen. That's definitional, and so what? That's like saying if we have a time machine then there's no randomness because we can just go see what will happen and then come back. While true, we don't have a time machine or perfect model, and in their absence their ability to deny the concept of randomness impacts the discussion zero.
   222. Jeff K. Posted: July 28, 2009 at 06:08 PM (#3270561)
I somewhat agree with 222 that the definition of true talent as something "unknowable" is getting a little too philosophical for my taste. I know we can't get exactly to it now and probably never would, but it's starting to sound a little midichlorian-y to me.
   223. Best Regards, President of Comfort Posted: July 28, 2009 at 06:22 PM (#3270585)
Search your feelings, Jeff.
   224. Rickey! In a van on 95 south... Posted: July 28, 2009 at 06:28 PM (#3270597)
True Talent" is unknowable. We can only estimate it.

Oh please, Immanuel. Are you seriously arguing that we are separated from the talent-in-itself by an unbridgable gap of understanding? Perhaps you can tell us how statistics is an arrow of longing always already in flight across a unfathomless abyss?

If "true talent" is unknowable it is also meaningless.
   225. Shock Posted: July 28, 2009 at 06:32 PM (#3270605)
That's definitional, and so what?


So, relative to its environment, the coin does not have an ability to perform above or below its true talent level. It performs exactly as expected based on its true talent + the environment. If you can control for the environment, then the true talent = observed performance. There is no third factor.
   226. Jeff K. Posted: July 28, 2009 at 06:34 PM (#3270608)
Oh please, Immanuel. Are you seriously arguing that we are separated from the talent-in-itself by an unbridgable gap of understanding?

I think he is indeed saying we Kant understand it. Perhaps a talented Heidegger can unLocke the secret, but it's not a run of the Mill question, it's pretty Nietschze.
   227. Jeff K. Posted: July 28, 2009 at 06:36 PM (#3270610)
If you can control for the environment, then the true talent = observed performance. There is no third factor.

The same could also be equally ridiculously true for players, or anything. If one controls for all outside factors, then the performance resulting is the true talent. Again, it's defintional, and I again ask where that impacts the discussion.
   228. Shock Posted: July 28, 2009 at 06:44 PM (#3270619)
If one controls for all outside factors, then the performance resulting is the true talent.


If that IS the case, then the idea that one "out-performs" or "under-performs" their true talent is incorrect. They always perform at their true talent, but unmeasured outside factors cause our measurements OF that performance to be all over the place.
   229. Edmundo got dem ol' Kozma blues again mama Posted: July 28, 2009 at 06:44 PM (#3270620)
Perhaps you can tell us how statistics is an arrow of longing always already in flight across a unfathomless abyss?

You, sir, are a 99.999999% true poet.
   230. Edmundo got dem ol' Kozma blues again mama Posted: July 28, 2009 at 06:50 PM (#3270625)
then the idea that one "out-performs" or "under-performs" their true talent is incorrect.
Except for the Black Sox and Operation Shutdown. And Gary Sheffield.
And Chris Truby, who obviously made a deal to outperform his true talent.
   231. Shock Posted: July 28, 2009 at 06:55 PM (#3270630)

And Chris Truby, who obviously made a deal to outperform his true talent.


By sacrificing Albert Belle?
   232. Tango Posted: July 28, 2009 at 06:55 PM (#3270632)
If one controls for all outside factors


There are two parts of the environment:
1. the properties of the environment
2. the timing of the events

Let's say you put Pujols in a game against Santana, and at the time Pujols has to make a decision, he chooses to not swing.

Let's say then you can reset the clock, make it so that this event doesn't happen, and let Santana throw exactly the same pitch in the same way in the same park at the same time. How Pujols responds to this pitch will not necessarily be identical to how he responded it the first time.

If you could control both the environment, and the timing of the events, that'd be fate. There'd be no random variation to consider.

Random variation is about the timing of events given the conditions, known and unknown.
   233. Jeff K. Posted: July 28, 2009 at 06:56 PM (#3270633)
If that IS the case, then the idea that one "out-performs" or "under-performs" their true talent is incorrect.

Of course, within that set of results/sample. In the ridiculous world where we can control for all externalities, a player's performance equals his true talent, and because it's definitional it is impossible for performance to differ from true talent *within that data set*. However, a player's true talent over a larger set inclusive of the smaller could be different, and that's what people are generally talking about. I'm still not seeing a big thing here.
   234. Jeff K. Posted: July 28, 2009 at 06:58 PM (#3270635)
Thanks, tango. I was trying to say that as part of my argument, but you put it better than I could come up with.

Shock, what I'm saying is this: Within a dataset, there will be smaller datasets. At some point it's not instructive to say "Oh Pujols made an out that PA in our 500 PA dataset, so while his 500 PA true talent was 50 bajillion, his true talent for that PA was 0." Instead, it's more intuitive, easier, and says more if we think of the timing issue. We know he won't get a hit every time, so overanalyzing and exaggerating the importance of a PA where he didn't has no purpose. He 'underperformed' during that PA, even though he again must make outs some times, and the only real factor of note is the timing of discrete events.
   235. Tango Posted: July 28, 2009 at 07:04 PM (#3270646)
Sam, I defined it this way:

I'm defining true talent as "the innate physical, emotional and mental skill that surfaces in some combination in the face of competition at a particular moment in time and space".


Then obviously it is unknowable. That doesn't make it meaningless, because we can estimate it.

It is unknowable to the braintrust at Harvard as to who are the brightest and most motivated kids who apply. That doesn't make the processes to try to identify those kids meaningless.
   236. Shock Posted: July 28, 2009 at 07:05 PM (#3270650)
How Pujols responds to this pitch will not necessarily be identical to how he responded it the first time.


Well, why is that? Are human brains capable of behaving completely randomly, or is there another factor that we did not discuss that lead to Pujols deciding not to swing? :-)

Jeff, I'm sorry if I seem too tangential. I'm trying to get my brain around the notion that Juan Pierre could (1*10^-1000) be better than Albert Pujols, but performing worse. I keep coming back to the thought that if Juan Pierre is actually better than Pujols, the reason he has put up inferior statistics is because he has consistently played within some wacky environment that has harmed his performance in such a way that our measurements of his performance don't take into account.

If we're just going to say that that is, as Backlasher said, 100% wrong, and that even if Pujols and Pierre have had the same environment in every way, Pierre could still perform worse despite having a higher true talent level simply because of random timing of the events, then okay I guess. The concept of random in this sense (ie the idea that Pujols chooses not to swing could be completely random,) is a tough nut for me to crack.
   237. Frisco Cali Posted: July 28, 2009 at 07:58 PM (#3270758)
Where's my dolly?
   238. Slivers of Maranville descends into chaos (SdeB) Posted: July 28, 2009 at 08:54 PM (#3270859)
The assumption that each at-bat is an independent, infinitely replicable event with an outcome randomly distributed around the true talent level of the batter modulo that of the pitcher is a useful abstraction, but it breaks down at least at the margins (and I would argue in the center as well, although that's harder to demonstrate).

A good player, like Pujols, can strike out in an AB. He can strike out in 2 consecutive AB, although that is less common ("less likely" in statistical parlance) or in 3 consecutive AB.

Could Pujols strike out in 100 consecutive AB? That's an interesting question. For one, I would argue that after striking out in, say 10 consecutive AB he would alter his approach at the plate, take additional batting practice, what have you. The 11th AB then is not really independent of the preceding 10.
   239. BDC Posted: July 28, 2009 at 09:09 PM (#3270891)
after striking out in, say 10 consecutive AB he would alter his approach at the plate, take additional batting practice, what have you. The 11th AB then is not really independent of the preceding 10

Except insofar as an ability to make adjustments is an important part of baseball talent. This was something widely publicized about Tony Gwynn, Rod Carew ... Willie Keeler, for that matter. They would alter their approach for the next PA, if they got fooled in the last one. Richie Sexson, not so much.
   240. Jeff K. Posted: July 28, 2009 at 11:23 PM (#3271111)
I keep coming back to the thought that if Juan Pierre is actually better than Pujols, the reason he has put up inferior statistics is because he has consistently played within some wacky environment that has harmed his performance in such a way that our measurements of his performance don't take into account.

Certainly a good possibility, within the unlikely scenario of Pierre having a true talent better than Pujols. I do see what you're getting at, now, and how it applies. It's not a bad thing at all to say that if we somehow later find out that Pierre was more talented, our first reaction should not be to say "It's chance" and instead should be "Was the problem in the metrics or was there some frighteningly unique thing that caused Pierre's performance such negative bias?" However, in general I'm okay with saying "There is a very small margin between our confidence level that Pujols is better than Pierre and 100%, and that represents the likelihood that the metrics are off, there is some externality uncaptured, or by dint of extraordinary bad luck on Pierre's part, Pierre is more talented and the numbers don't show that."

even if Pujols and Pierre have had the same environment in every way, Pierre could still perform worse despite having a higher true talent level simply because of random timing of the events, then okay I guess. The concept of random in this sense (ie the idea that Pujols chooses not to swing could be completely random,) is a tough nut for me to crack.

Ignoring the choosing to swing example for a moment, think of it this way: Every career is a function of the PAs that make it up. In general, there's no difference in the end between 10,000 PAs where every fourth PA was a hit (Out-Out-Out-Hit-Out-Out-Out-Hit) and 7500 consecutive outs followed by 2500 consecutive hits. Additionally, we know that the discrete nature of PAs can make distribution seem random while also masking some randomness. Given that, it is possible, *incredibly unlikely* but possible, that Pierre has all along been a true talent 200 OPS+ guy, that he'll end his career at 200 OPS+, and the mirage of suck is caused by timing. A hyper-exaggerated case where all of his "pre-ordained" outs for his career just bunched up at the front. There's a million more likely things, but it's there.
   241. AuntBea Posted: July 28, 2009 at 11:58 PM (#3271205)
If we were to strike a snooker ball as accurately as the quantum uncertainty of Nature permits, then it would take merely a dozen collisions with the sides of the table and other balls for this uncertainty to have amplified to encompass the entire extent of the snooker table. Laws of motion would henceforth tell us nothing about the individual trajectory of the ball.


--John D. Barrow, New Theories of Everything
   242. The District Attorney Posted: July 29, 2009 at 12:57 AM (#3271384)
When philosophy paints its gray on gray, then has a shape of life grown old. By philosophy’s gray on gray it cannot be rejuvenated, but only understood. The owl of Minerva spreads its wings only with the falling of the dusk.
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