The Myth of the Hot Hand - Mathematically Vindicated?

[BD80]

How about in tennis?

There are some players who get on a run where their serve is absolutely unhittable, and they could hit a dime placed on the back line of the service box. The same player can go through a stretch where they can't get a serve in even if they back it down by 20%. There is no way you can say the odds of such a player serving an ace are the same in middle of either stretch.

[cspan37421]

There is no way you can say the odds of such a player serving an ace are the same in middle of either stretch.

By definition, I'd say you're right.

[cspan37421]

I agree with your take, but yes, those who deny the concept of the hot streak also deny the existence of the cold streak.

No one denies the existence of streaks. Streaks happen even in random coin flipping, as others have noted above (4 heads in a row, 6.25% of the time, on average). As noted, random streaks occur more often than most people think. The denial of hot hand does mean streaks don't happen. It means that they can't be reliably identified in advance at a rate greater than chance. It also means that, upon the ex-post identification of a streak, the odds of the next shot going in are not really changed from the player's average skill (say, corrected for quality of defense).

It seems to me that if the hot hand were real, there would be no reversion to the mean. Under hot hand, a player who started hitting a few in a row would logically hit the next few at a higher % chance. This keeps going, hitting at a higher and higher rate until they are deadeye perfect! It doesn't happen. If the hot hand theory were real, you need a theory to explain how and why it stops. If you say, "cold hand", well, good luck identifying in advance when that switch from hot to cold takes place. At some point, the notion starts sounding like epicycles or something.

Those who think they can predict hot hand should take their talents to Las Vegas. Identifying "hot hands" ex post isn't really what the notion of "hot hand" is all about. It is: should you pass to the guy (or gal) with the hot hand, even if their shooting % is lower than someone else that is available to shoot it?

Maybe, if there's a defensive mismatch that the other team can't adjust to. But that's not hot hand, that's a mismatch, which causes the baseline % being compared to be truly different than their career (or season) average. But most of the time it's not wise to assume a worse player on a hot streak will continue that indefinitely, nor to assume a better player on a cold streak will continue to fail. Again, unless there are sound reasons that effectively change the baseline % under consideration (defensive quality, illness/injury, etc).

[cspan37421]

This discussion reminds me of a quote I recently saw from someone named Jon Ronson:

"Ever since I first learned about confirmation bias, I've been seeing it everywhere."

Obligatory XKCD:

correlation.png

[uh_no]

This argument is so stupid and comes up all the time. Hot Hands are real.

http://www.amazon.com/HotHands-Hand-.../dp/B00PX20LO0

If I have to have some delivered to your houses to make you believe it, I will!

[BD80]

This argument is so stupid and comes up all the time. Hot Hands are real.

http://www.amazon.com/HotHands-Hand-.../dp/B00PX20LO0

If I have to have some delivered to your houses to make you believe it, I will!

But can they cure cold feet?

[MChambers]

It seems to me that if the hot hand were real, there would be no reversion to the mean. Under hot hand, a player who started hitting a few in a row would logically hit the next few at a higher % chance. This keeps going, hitting at a higher and higher rate until they are deadeye perfect! It doesn't happen. If the hot hand theory were real, you need a theory to explain how and why it stops. If you say, "cold hand", well, good luck identifying in advance when that switch from hot to cold takes place. At some point, the notion starts sounding like epicycles or something.

This oddity has occurred to me, too. I suppose it depends on how you define and/or measure a hot hand. Is it simply a higher probability of hitting the next shot after making one shot, or perhaps several shots in a row? And how much higher of a probability? Maybe it's not all that much of a delta.

[swood1000]

Obligatory XKCD:

correlation.png

See also spurious-correlations.

[darthur]

Yeah. Folks who do serious stats should bring their stats to a defense of the "hot hand".

-jk

So here's the original paper: http://psiexp.ss.uci.edu/research/te...ersky_1985.pdf

I was curious and tried to measure how just big the effect of their error is here. So first, let's be clear: their math is wrong. They looked at professional basketball players and counted P[made shot | three previous makes] and P[made shot | three previous misses]. They then averaged these across players: Average P[made shot | three previous makes] = 0.46 and Average P[made shot | three previous misses] = 0.56, saw the first one is smaller, and concluded there is no hot hand.

If we assume that a player shoots 50%, they are basically relying on the false fact that if I flip a coin N times, Average (Occurrences of HHHH) / (Occurrences of HHHH + HHHT) is 0.5. This statement is not true. It's true that the average occurrences of HHHH will be almost exactly half the average occurrences of HHHT, but it's not true that the average of the ratios will be 0.5. Anyway, I wanted to see how not true it is, so I tried a million simulations and recorded what the average of the ratio actually is for each N:

50: 0.424687
100: 0.464879
150: 0.478439
200: 0.484524
250: 0.488079
300: 0.490185
350: 0.491675
400: 0.492764
450: 0.493608
500: 0.494284
550: 0.494830
600: 0.495293
650: 0.495690
700: 0.496017
750: 0.496295
800: 0.496537
850: 0.496735
900: 0.496931
950: 0.497096
1000: 0.497253

They have 9 players with N values ranging from 248 to 884, so we are only talking about a 1-2% difference here. I didn't try to account for different people having different shooting percentages but it does seem to me like this mistake doesn't really change anything. But it's still very interesting.

(More subjectively, I have some larger doubts about the paper's methodology and sample size though. I also think the argument being thrown around here that a "hot hand" implies spiraling off to 100% or 0% to be a pretty ridiculous straw man.)

[killerleft]

I'll bet Coach K had the "hot hand" on occasion when he played, and he's obviously seen players do some spectacular things over the years. I wonder how he feels about it now, after all these years as a coach?

[cspan37421]

Darthur,

Though I have a degree in math from our favorite revenue sport school, I've been out of the proof business for quite some time. And was more of an applied guy anyway. Nevertheless, I would like to see something more closed-form to prove your claim, instead of the fact that you did a million simulations and this is what you came up with. Unless there's been a revolution in random number generation, I don't think we can rely on Monte Carlo type methods to prove our point (well, those are kind of separate anyway). It might call our attention to something potentially interesting (much in the way that Julia Galef describes correlation as a phenomenon that whispers to us "Pssst! Look over here!" , but it does not form a conclusive case. I have done many simulations before where the closed-form solution was known with certainty (say 50%), yet the average result was at least as far off as what you're getting here. Might it not be the case that whatever epsilon you choose, you just need a sufficiently high N in order to get in the neighborhood?

I may take pen to paper with this out of curiosity, and see if I get anywhere. [edit: I'm not sure you defined the problem unambiguously enough for me to get very far, unfortunately]

You may be right that the hot hand claim does not imply that shooting percentages will approach 100%, but does the notion not say that a hit is more likely after a series of hits? Then what will bring about mean reversion? Random, "unlucky" misses? So to keep it simple, if you are a 50% FG shooter and experience +++, the hot hand suggests the prob of the next shot is higher. Let's say 60% for sake of argument. If you score, the next one is higher yet. If you miss, are you saying that the probability of the 5th shot would then be less than 60%? Less than 50%? In between? It's like a moving average with some perturbation based on recent success?

My understanding is that there is no convincing empirical evidence that hot hand exists, though there may be some disagreement over what the formal definition is, or what hypothesis should be tested.

[LastRowFan]

From the WSJ http://www.wsj.com/articles/the-hot-...ead-1443465711

-----------------------
Los Angeles Clippers guard J.J. Redick once studied the hot hand with a psychologist’s dream lab rat: himself.

For a college statistics course, Redick experimented in Duke’s gym by attempting 100 shots per day for a week, recording not only the result of every attempt but also how he felt at the time. What happened when he thought he was hot? “Every day was the same thing,” he said. “No matter what I thought I was feeling, makes or misses, it was plus or minus the statistic I was looking for.”

Redick says that helps him keep poor shooting nights in perspective. Last season, through five games, Redick was shooting 23% on 3-pointers, which would’ve been his worst shooting season ever. “The media kept asking me about it,” he said. “But at some point I’m going to start making shots and my percentage will be in the 40s. Then it’ll be like: Oh, he’s on fire!” Redick was right. He ended up shooting 43.7% on threes—slightly above his lifetime average of 40%.

One person who was happy to hear Redick recalled his study was his professor. Duke statistician David Banks wasn’t aware that his old student is now in the NBA. He also says he has never seen a full basketball game. “I have to confess,” he said, “it just strikes me as incredibly dull.” When told of Redick’s whereabouts, though, Banks said: “I’m delighted. It’s always gratifying when my students aren’t living in refrigerator boxes behind bus stations.”

[gus]

From the WSJ http://www.wsj.com/articles/the-hot-...ead-1443465711

-----------------------
Los Angeles Clippers guard J.J. Redick once studied the hot hand with a psychologist’s dream lab rat: himself.

For a college statistics course, Redick experimented in Duke’s gym by attempting 100 shots per day for a week, recording not only the result of every attempt but also how he felt at the time. What happened when he thought he was hot? “Every day was the same thing,” he said. “No matter what I thought I was feeling, makes or misses, it was plus or minus the statistic I was looking for.”

Redick says that helps him keep poor shooting nights in perspective. Last season, through five games, Redick was shooting 23% on 3-pointers, which would’ve been his worst shooting season ever. “The media kept asking me about it,” he said. “But at some point I’m going to start making shots and my percentage will be in the 40s. Then it’ll be like: Oh, he’s on fire!” Redick was right. He ended up shooting 43.7% on threes—slightly above his lifetime average of 40%.

One person who was happy to hear Redick recalled his study was his professor. Duke statistician David Banks wasn’t aware that his old student is now in the NBA. He also says he has never seen a full basketball game. “I have to confess,” he said, “it just strikes me as incredibly dull.” When told of Redick’s whereabouts, though, Banks said: “I’m delighted. It’s always gratifying when my students aren’t living in refrigerator boxes behind bus stations.”

JJ clearly doesn't know what he's talking about. Anyone who's played basketball at any level knows that the hot hand exists.

I can't read the article (not a subscriber), but I have to think Banks has to be pulling the interviewer's leg.

[darthur]

Darthur,

Though I have a degree in math from our favorite revenue sport school, I've been out of the proof business for quite some time. And was more of an applied guy anyway. Nevertheless, I would like to see something more closed-form to prove your claim, instead of the fact that you did a million simulations and this is what you came up with.

Well I do too, but alas I could not come up with a closed form expression for a general N. However, it's easy to analyze a specific N by hand. Let's look at N=5. So I'm doing 5 coin flips. The article takes as granted that E[occurrences of HHHH / occurrences of HHHT + occurrences of HHHH] = E[occurrences of TTTH / occurrences of TTTH + occurrences of TTTT]. (I'm going to assume you believe me on this and focus on the mathematical question.) Anyway, this statement is not true.

First of all, it's not even well defined because there could be 0 occurrences of HHHH or HHHT, so this is an expected value of a random variable that is sometimes 0/0. To try to give the article the benefit of the doubt, let's condition the expectation on the fact that HHHT or HHHT occurs at least once, which I think best matches what they did. (In my simulation, I cheated a different way by saying 0/0 = 0.5). Anyway, by symmetry, the claim is equivalent to saying E[occurrences of HHHH / occurrences of HHHT + occurrences of HHHH] = 0.5.

So for N=5, here are the sets of coin flips with at least one occurrence of HHHT or HHHH, and the value of #HHHH/#HHHT+#HHHH. Each of these is equally likely:

HHHHH: 2/2
HHHHT: 1/2
HHHTT: 0/1
HHHTH: 0/1
THHHH: 1/1
THHHT: 0/1

The average is (1 + 0.5 + 0 + 0 + 1 + 0) / 6 < 0.5. But notice that the expected number of occurrences of HHHT and HHHH are the same. It's just the ratio doesn't behave in a way you'd expect.

Unless there's been a revolution in random number generation, I don't think we can rely on Monte Carlo type methods to prove our point (well, those are kind of separate anyway). It might call our attention to something potentially interesting (much in the way that Julia Galef describes correlation as a phenomenon that whispers to us "Pssst! Look over here!" , but it does not form a conclusive case.

As it turns out, I did a PhD in algorithm theory with a focus on randomization after Duke, so this is something I can speak to with confidence . It is very possible to prove things rigorously about the output of a randomized algorithm if you know what you're doing. Suppose the real expected value of this distribution is 0.5. By the Chernoff bounds (https://en.wikipedia.org/wiki/Chernoff_bound - see the last special case), I calculate there is ~99% chance that the average of my 1000000 trials should be at least 0.4985. In other words, there's no bloody way I could get results so consistently low (unless I made a mistaken in my calculation or implementation of course!).

You may be right that the hot hand claim does not imply that shooting percentages will approach 100%, but does the notion not say that a hit is more likely after a series of hits? Then what will bring about mean reversion? Random, "unlucky" misses? So to keep it simple, if you are a 50% FG shooter and experience +++, the hot hand suggests the prob of the next shot is higher. Let's say 60% for sake of argument. If you score, the next one is higher yet. If you miss, are you saying that the probability of the 5th shot would then be less than 60%? Less than 50%? In between? It's like a moving average with some perturbation based on recent success?

My understanding is that there is no convincing empirical evidence that hot hand exists, though there may be some disagreement over what the formal definition is, or what hypothesis should be tested.

I agree there's a lot of uncertainty. I just claim you could imagine something like a "hot hand" (whatever that means exactly) that could be a sane model. Whether that would be the right model is a big question, and it does seem as you say that there's not much empirical evidence for it.

[swood1000]

No one denies the existence of streaks. Streaks happen even in random coin flipping, as others have noted above (4 heads in a row, 6.25% of the time, on average). As noted, random streaks occur more often than most people think. The denial of hot hand does mean streaks don't happen. It means that they can't be reliably identified in advance at a rate greater than chance. It also means that, upon the ex-post identification of a streak, the odds of the next shot going in are not really changed from the player's average skill (say, corrected for quality of defense).

When people talk about a hot hand they are referring to an enhanced psychological and/or physical state that makes success more likely while that state persists. You imply that if the hot hand phenomenon were real it would be possible to reliably identify streaks in advance but I don't follow that. To say that a person has had a hot hand for his last five shots does not require us to say that this state will persist, although that might be likely.

It seems to me that if the hot hand were real, there would be no reversion to the mean. Under hot hand, a player who started hitting a few in a row would logically hit the next few at a higher % chance. This keeps going, hitting at a higher and higher rate until they are deadeye perfect! It doesn't happen. If the hot hand theory were real, you need a theory to explain how and why it stops. If you say, "cold hand", well, good luck identifying in advance when that switch from hot to cold takes place. At some point, the notion starts sounding like epicycles or something.

Don't follow this either. If we don't know what causes hot hands then what reason do we have for predicting that the player will continue being hot instead of reverting to his normal performance level? If we think of a hot hand as being 125% of the player's normal effectiveness then it stays at that level. Why would it inch higher and higher?

Those who think they can predict hot hand should take their talents to Las Vegas. Identifying "hot hands" ex post isn't really what the notion of "hot hand" is all about. It is: should you pass to the guy (or gal) with the hot hand, even if their shooting % is lower than someone else that is available to shoot it?

The question is: is the player in an enhanced state in which he or she has a temporary shooting % that is higher than that of another player who is available to shoot it. When I look at the recent championship MBB game it appears to me that during the latter part of that game Coach K was directing the ball into the hands of Grayson Allen at a higher than normal rate because he believed that Grayson was hot. Was Coach K wrong to do that?

[freshmanjs]

The World Series presents an opportunity to test how you feel about the idea of a player on a "hot streak".

- Throughout his career, Daniel Murphy has averaged 1 HR every 14.6 games (or 58.4 PA).
- In the 2015 playoffs, Murphy has averaged 1 HR every 1.3 games (or 5.6 PA).

That's about as "hot" as you will ever see a player.

Now consider the following hypothetical scenario. What it someone offered to give you free courtside final four tickets for life if you answered the following question correctly.

"Will Daniel Murphy hit at least one home run in the 2015 World Series?"

How would you answer?

Given that Murphy will likely get 20-30 PA in the World Series, mostly against above average pitching, your personal answer says a lot about how you feel about the idea of a player on a hot streak. If you think perceived hot streaks are mostly a factor of randomness, you would probably answer "no" to the question above. If you believe hot streaks are a significant factor in athlete performance, you would probably answer "yes".

Now, whether or not Murphy actually hits a home run won’t say much about whether hot streaks really exist (too small a sample), but your answer to the question above may reveal something about how you feel about the topic.

no would have been the winner.

[MChambers]

Found an article today on this topic. Haven't read it yet, since I'm work and actually working, for the most part, but I figured if I posted the link here someone would read it and summarize it here:

http://nymag.com/scienceofus/2016/08...-hot-hand.html

Matt

[cato]

Found an article today on this topic. Haven't read it yet, since I'm work and actually working, for the most part, but I figured if I posted the link here someone would read it and summarize it here:

http://nymag.com/scienceofus/2016/08...-hot-hand.html

Matt

Spoiler alert: Rosencrantz was right.