A little while back, Jonah Lehrer did a nice blog post about reasoning that used the famous study by Gilovich, Vallone and Tversky, The Hot Hand in Basketball (PDF link) as an example of a case where people don’t want to believe scientific results. The researchers found absolutely no statistical evidence of “hot” shooting– a player who had made his previous couple of shots was, if anything, slightly less likely to make the next one. Lehrer writes:
Why, then, do we believe in the hot hand? Confirmation bias is to blame. Once a player makes two shots in a row – an utterly unremarkable event – we start thinking about the possibility of a streak. Maybe he’s hot? Why isn’t he getting the ball? It’s at this point that our faulty reasoning mechanisms kick in, as we start ignoring the misses and focusing on the makes. In other words, we seek out evidence that confirms our suspicions of streakiness. The end result is that a mental fiction dominates our perception of the game.
Here’s where things get meta: Even though I know all about Tversky and Gilovich’s research – and fully believe the data – I still perceive the hot hand. I can’t help but watch the NBA playoffs and marvel at the streakiness of shooters, from Kobe to Rose. (Personally, I’d love to see an analysis of Ray Allen. If that man doesn’t show the hot hand, then it really doesn’t exist.) And I’m not alone in my stubborn skepticism. Red Auerbach, the legendary coach of the Celtics, reportedly responded to Tversky’s statistical analysis with a blunt dismissal. “So he makes a study,” Auerbach said. “I couldn’t care less.”
Elsewhere in blogdom, Paul Waldman looks for ways the study could be wrong, and Kevin Drum pokes fun at him for doing so, noting that “it’s interesting how unwilling most athletes are to accept the results of this study.”
It’s not hard to see why people who play basketball find this result surprising– shooting a basketball in a game situation is a complicated process involving lots of factors– balance, timing, sight lines, defense (though, to be fair, the study uses data from the NBA in the early 1980’s, so it’s probably safe to exclude the effects of defense…)– and it seems difficult to believe that all of those combine to give a single unchanging chance of success. And if you’ve played a lot of basketball, you know that there are days when one or more of those things just doesn’t feel right– where you’re rushing your shots for some reason, or the ball keeps slipping, or something like that.
The study itself looks pretty solid, though. There’s only one real weak point to it, that I see, which is that not all shots are equal.
By that I mean that they appear to have used a single number– the overall field goal shooting percentage– to characterize each player’s shooting, and looked for deviations from that. Anybody who has played the game, though, knows that different sorts of shots have different probabilities of success. Even great players tend to shoot a fairly low percentage from 3-point range– taking a look at this fan-generated list of great shooters, a lot of them are under 40% from three. Larry Bird shot 37.6% from three-point range for his career, and Reggie Miller 39.5%, for example, and Andrew Toney, the player in the study called out as an example of a “streak shooter” who wasn’t really shot just 34.2% from three.
All of those guys, though, have overall shooting percentages that are significantly higher– 49.6%, 47.1%, and 50.0%, respectively. That’s because they shot a considerably higher percentage from closer in– 50.9%, 51.6%, and 51.2%, respectively. So it can’t really be true that all shooting is described by a single percentage– there’s quite clearly a difference between different categories of shots, at least between very long shots and not-so-long shots. This is not factored into the statistical analysis of the “hot hand” paper, or if it is, they didn’t bother to tell anybody.
That leaves a tiny bit of a loophole for believers in the “hot hand,” especially when you take into account fan psychology. That is, when people talk about a player being “hot,” they don’t generally mean that he’s hitting lots of lay-ups. “Hot” in basketball circles generally refers to hitting a lot of shots for which the expected percentage is low. This is why if you go down lists of great shooters, you don’t find any dominant centers, despite the fact that guys like Wilt Chamberlain (career 54.0% shooting) or Kareem Abdul-Jabbar (career 55.9% shooting) blow away the shooting percentages of anybody on the list of great NBA shooters, percentage-wise. Those guys are shooting most of their shots from very close to the basket, often with a large size advantage over their opponents, and so they don’t tend to be deemed “hot” or “cold”– to the average fan, those are shots you’re expected to make.
A player is generally said to be “hot” after hitting either a couple of long jump shots in a row, or hitting some improbable-looking shots in the lane. Very few fans or commentators would describe a guy as having a hot night if he hit six open lay-ups in a row– instead, they’d likely talk about how bad the defense was. You’re judged to be “hot” when you hit a couple of three-pointers, or rattle in a couple of off-balance floaters in the lane.
(Similarly, free-throw shooting doesn’t really affect fan perception of “hot” or not– if an otherwise poor shooter hits several in a row, it will be remarked upon, but nobody has ever gone into work the day after a game and said “Man, Larry Bird was hot last night– did you see the way he hit all those free throws?”)
It’s possible, then, that the difference in types of shots could mask a real deviation from chance. That is, the tests that they did on NBA team statistics looked at total shooting percentage, and failed to see any evidence of “hot” shooting, but it could be that there really were cases where guys shot a higher percentage than would be expected for the subset of shots that “count” toward being “hot”, but the difference was masked by shot selection effects. A career 33% 3-point shooter going 6-for-9 might qualify as “hot,” but it could be masked statistically if he also went 5-for-10 from two-point range, putting him only a little above average.
They attempt to control for this by an additional experiment, in which players from the Cornell men’s and women’s varsity teams were asked to shoot from a distance where they shot about 50%, and they looked for correlations in their shooting. And, interestingly, this was the one area where they did find a correlation– player 9 on the men’s team showed a significantly higher probability of hitting the next shot after making a couple in a row (83% for a shot after making three in a row, compared to 54% overall).
Do I think this is the real explanation of the difference? No– I suspect that “people are bad at statistics” is the real explanation for the perception of “streak” shooting. If you were to repeat their analysis tracking 3-point shooting percentages separately, I doubt the results would be any different (in large part because it would be difficult to get the same level of statistical certainty– of the three players cited above, Reggie Miller was the only one to take more than 10% of his shots from long range. And characterizing all two-point shots by “degree of difficulty” would be way too difficult to undertake, unless you found yourself with a really obsessive graduate student who was also a basketball fan.
But the recent discussion (plus the fact that I’m taking a little time off from hoops, and find myself missing it) got me to read the paper, and that jumped out at me as the possible loophole, so I thought I might as well get a blog post out of the deal…
GILOVICH, T. (1985). The hot hand in basketball: On the misperception of random sequences*1 Cognitive Psychology, 17 (3), 295-314 DOI: 10.1016/0010-0285(85)90010-6