More "hot hand" thoughts

There are two ways to define (or measure) a "hot hand": 

(A) you can check if a hit is more likely than normal to be followed by a hit, and if a miss  is more likely that normal to be followed by a miss.

(B) you can check if a *high-probability opportunity* is more likely than normal to be followed by another high-probability opportunity, and if a low-probability opportunity is more likely to be followed by a low-probability opportunity.

The measure in Type A is usually what fans mean.  After a team wins, say, six consecutive games, are they more likely than average to win the next one?

The measure in Type B is a bit more obscure.  Type B is saying something like, "are there streaks where a .500 team is really a .550 team, regardless of whether they win or lose during that streak?"

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Now, you can always convert Type B to Type A.  That is, suppose your model says you get 100 consecutive chances at 40%, and then 100 more at 60%.  That's a hot hand in terms of probabilities, in terms of Type B.  Indeed, it's a *very* hot hand: with only one exception, a 40 is always followed by a 40, and a 60 is always followed by a 60.

But, in terms of Type A, a hit being followed by a hit, not so much.  If you work it out, you get

-- After a hit, the chance of a hit is 52%.
-- After a miss, the chance of a hit is 48%.

That is: if we assume a "hot hand" of type Type B that's absolutely huge -- 20 percentage points -- the corresponding Type A effect is much smaller, at only 4 percentage points.

With lesser effects, it's even worse.  Suppose that the Toronto Blue Jays show type Type B streakiness in probabilities.  If the weather is warm, they play .510 ball.  If the weather is cold, they play .490 ball.  That's a streaky pattern because temperature is streaky -- warmer in the summer and colder in spring and fall.

If you didn't know about this weather thing, would you be able to observe any streakiness?  No way.  After a win, the Jays are a .5002 team.  After a loss, they're a .4998 team.  You wouldn't notice that even after 1,000,000 games.  Seriously.  We're talking two games every 10,000.

Even if Type B is reasonably large, Type A is likely still small.

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That's what we saw in the previous post.  In his paper, Dan Stone showed a model where the "Type B" effect was 10 percent of the difference from .500.  Then, he added a bunch of randomness.  And, what happened?  The equivalent "Type A" effect was almost zero:

-- after a hit, the chance of a hit is 75.049 pct.
-- after a miss, the chance of a hit is 74.875 pct.

If the academic standard for this kind of research is a "Type B" model, I wish authors would also show us the "Type A" effect, so we can really see what we're dealing with, as fans.  In this case, we're dealing with one extra hit every 575 attempts, which is barely a "hot hand" effect at all.

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Of course, all this partially depends on what you're looking for.  If you just want to discover an effect -- any effect -- then it doesn't matter how big it is.  But, in a way, proving "existence" is unnecessary.  We *already know* there's a "hot hand" effect -- because of home field advantage.  

Players do better at home than on the road.  Therefore, a hit is more likely to be a home hit than a road hit, which means a hit is more likely after a hit (probably home) than after a miss (probably road).

Now, that effect is very, very small.  If you assume a .750 foul shooter is .751 at home, but .749 on the road, then the probability of a hit, given that the prevous attempt was successful, is .750001333.  Barely there at all -- but it's there, and we know it's there!

So, if you're looking just to prove the *existence* of an effect, there's no need.  I just did it.

And, of course, there certainly must other effects, of some magnitude.  I'm sure if a player's wife is mean to him one morning, he'll play differently that day.  Maybe better, probably worse.  It would be silly to insist that doesn't happen, that a .750 shooter is still a .7500000000 shooter no matter what's preoccupying him that day. 
  
But ... the effect might be very, very small.  If you want to prove it's there, good luck.  As I said, if he's .751 on good days and .749 on bad days, the Type A effect is one shot in 750,000.

Even if he's .760 on good days, but .740 on bad days -- which would be pretty significant, in my book -- he'd be .750133 after a success.  That's one shot in 7,519.

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So it's not about existence -- it's about size.  We have evidence that *some* streakiness exists, albeit in very, very small amounts.  What the skeptics have to show is that it exists in some *significant* amount, in the real-world, "type Type A" sense. 

And, even that's probably not really enough.  Because, "hot hand" implies more than just correlation.  Suppose a baseball team trades for a bunch of superstars in August.  That  team will definitely show a "hot hand" effect.  If you check when it won three straight, it was most likely after the trade than before.  And, therefore, the chance of it winning the fourth game is higher than its average for the season.

But is that really a "hot hand," in the sense that people mean?  I doubt it.  I think they're talking about psychology, about how the three consecutive wins convinced the team it could compete, and that confidence and momentum extends into the fourth game and lifts them higher.  I think fans won't call it a "hot hand" if it's just caused by extra superstars, something they know is already there.