RE: [APBR_analysis] Hot hand, a little more data
- -----Original Message-----
From: igorkupfer@... [mailto:igorkupfer@...]
Sent: Friday, March 14, 2003 3:29 PM
>From: "Michael Tamada" <tamada@...>I'm pretty much agnostic on the issue of hot hands with free throws. I can easily believe that they don't exist.
>Sent: Friday, March 14, 2003 2:07 AM
>> I bring this up, not as yet another new thread in the never-ending mega-thread
>> of the existence of hot hands, but rather as a follow-up to our discussion last
>> season. Many of the original studies used a narrow, non-intuitive definition of
>> a hot hand (such as dependence of the probability of making a free throw on
>> the make/miss of the preceding free throw).
>I did a study on the All Star 3 Point Shootout, which IMO provides an opportunity for
>the display of a more traditional "hot-hand."
>Result: no hot-hand.
Two problems however:
Runs of 25 FTs are small samples and are going to have what statisticians call "low power". In simple terms, it's hard to reject a null hypothesis with a small sample, and 25, though not tiny, is small.
The bigger problem though is that personally at least I don't think free throws have much to do with "hot hands". I never find myself thinking: gosh, Payton's on a hot streak, so he's going to sink these free throws.
To me, a hot hand means field goals. And it doesn't mean the probability of making any one field goal being higher, it means that the player was hot for several minutes or a quarter or even (especially in van Exel's case) an entire game. I.e. if Alan Iverson goes 3-7 in a quarter, that's no big deal, about average for him. If he goes 7-10, then we say he was "hot". That to me is what a "hot hand" means.
Now of course with those tiny sample sizes, there no way we can conclude from any single quarter that Iverson was unusually hot, vs. simple random chance leading to him going 7-10.
At this basic level, my expectation is that hot hands do exist, if for no other reason simply because some defenders and some teams are pretty weak, and an Iverson or van Exel can light up certain opponents. Or in van Exel's case, due to an anti-Sonics grudge as RickA suggests.
What if we were able to correct for that somehow, by calculating an expected FG% against a given team or player, and see if players have streaks of hotness or coolness that are more than what simple chance would procude? I think that'd be the ideal study, but hard to do.
We could also do runs tests, of the sort that you ran. But the weakness of those is that they only measure one type of hot hand, namely a streak of shots. If Kobe hits 9 3-pointers in a row, then yes, we'd say that was a hot hand and we could apply runs tests to see if Kobe's run lengths are within random chance range. But most of us would agree that going 8-9 or 7-9 also is a sign of a hot hand, even 5-9 is pretty darned warm if not hot. And a runs test ignores those 8-9 and 7-9 streaks.
A do-able test: look at game-by-game data. Maybe Kobe's 8-20 one night, 11-20 the next. Do his game-to-game fluctuations reflect random chance, or do they show greater variability than we'd expect from random chance?
This test has the opposite disadvantage of the runs tests and Tversky's "probability of hitting the next shot" tests: the game stats aggregate 48 minutes worth of activity, and most people would agree that many hot streaks last for less than a game, indeed less than a quarter.
But quarterly data are harder to come by. And most casual observers would agree that sometimes Kobe seems to have a hot game (record-setting 3-point eruption against Sonics) and sometimes not. So this proposed test, while not a complete test for the existence of hot hands, would at least be a test for the existence of hot *games*.
Some assumptions would have to be made: in particular, the analysis is a lot simpler if the number of FGA is simply taken as an exogenous variable, ie. a number which is fixed and determinate for that game (it will in general be different for the next game, but is still assumed to be a fixed number for each game). A more complete model of basketball would note that a player's FGA is itself a random variable which changes from game to game. But modelling that would probably double the complexity of the analysis without adding useful information to the question of hot hands.
My expectation is that many, perhaps most, players would show greater than random variablity in their hotness from game to game, as measured by FGM/FGA.
The problem is the one that I already mentioned: one major source of this variability would be the quality of the defender and team defense. Mike Bibby lighting up Steve Nash by going 10-18 is not the same as Bibby getting 10-18 against say Payton. So a true test for hot hands would have to correct for this somehow ... and once again that gets hard.