Nylon Calculus: Tossing a wet blanket on the hot hand

It started with a pair of 3-pointers and a dunk on the break. He shot the next 3-pointer off the dribble, then fading away, then falling out of bounds. He pointed to the floor and pulled up from 28 feet. The opponents were double-teaming him, fouling him, and calling in reinforcements from the bench. It didn’t matter. He hit every shot. In the end, Klay Thompson finished with 37 points on 13-for-13 shooting during that impossibly perfect third quarter against the Kings in 2015. It was the type of performance that would make a true believer out of even the most cynical hot-hand skeptic. But do moments like these really prove that the hot hand exists?

The controversy was first sparked in 1985 by a paper in Cognitive Psychology which claimed that the hot hand was just a “misperception of random sequences.” The scientific community was convinced by this so-called “hot-hand fallacy”, with no less than Nobel laureates weighing in to explain the “massive and widespread cognitive illusion.” But a recent reanalysis of the 1985 data has rekindled the argument, suggesting that — due to the principle of restricted choice — the earlier tests were biased and the original conclusions wrong. The debate continues to rage on within community forums for statisticians all across the internet.

Last week, a Reddit post poured some more fuel on the fire of the hot-hand dispute by examining Thompson’s torrid 3-point shooting percentages after consecutive made 3-point attempts. It was a really compelling chart, but it left me with some questions. Primarily, what was the sample size for those bright-red bars on the right; the ones representing the 3-point attempts that followed streaks of 5, 6, and 7 buckets? And would this hot-hand trend track over Thompson’s entire career?

It turns out that, even when considering Thompson’s entire career – 8 seasons, 600 games, 4183 3-pointers…and counting – the number of the 3-point attempts that followed long streaks of makes or misses was too small to provide meaningful information. So, to get a robust estimate of 3-point percentage we were forced to lump together the bins at each end, e.g., 3-point percentage after 3 or more made 3-pointers. Interestingly, the gap between Thompson’s “hot” 3-point shooting and his “cold” 3-point shooting has persisted throughout his career; he has been 5 percentage points better from deep after hitting three in a row than he has been after missing three in a row.

So does Thompson really feel the hot hand or has it all been just random sequences we misperceived? Does Thompson get in-the-zone or is he just suffering from cognitive illusions? It’s difficult to know for sure, because there are so many factors, like increasing shot difficulty, for which we cannot adequately account. But in a simulation, it’s easy to control the playing field. So, let’s ask the computer!

We set up a program to simulate how Thompson’s career would have played out in an alternate universe where the hot hand does NOT exist. Using rbinom in R we generated random 1s and 0s from the binomial distribution to represent his 3-point makes and misses, replicating his career 42 percent shooting from deep. We plugged in Thompson’s actual log of 3-point attempts by game (a range from 0 to 24 3-point attempts) and then simulated the first 600 games of his career 10,000 times. This simulation process gives us a sense for the range of possible outcomes in a world without the hot hand.

As an aside: In my simulations, I did NOT see a lot of evidence of the bias predicted by the principle of restricted choice. That is, the typical 3-point percentage in the 10,000 simulated Thompson careers was 42 percent, regardless of the preceding sequence — first 3-point attempts of the game, after 3 made 3-point attempts, after 3 missed 3-point attempts – the median 3-point percentage was right around his career average in each instance. I’m not sure what to make of this (shrug emoji).

Remember, in real life, Thompson has made 45 percent of the 3-pointers which he has attempted after having hit three or more 3-point attempts. In the simulated universe where we KNOW the hot hand is NOT part of the matrix, Thompson was mostly unable to match that career achievement but occasionally he did. To be exact, he surpassed his real level of post-hot-streak 3-point shooting in 20 percent of his fake careers. In other words, Thompson’s 3-point shooting history would be unlikely but NOT impossible in a world where the hot hand was programmed NOT to exist.

Disappointingly, then, we can’t confirm the existence of the hot hand based on Thompson’s career alone. But what if we expand our search for hot hands to include more of the league’s most prolific 3-point shooters?

Among this group who shot 2,000+ total 3-pointers and averaged 5+ 3-point attempts per game from 2000-01 to 2018-19 (the period of time covered by the Event Finder tool on Basketball-Reference), we see that Thompson has indeed experienced some of the hottest hands. Overall though, there is no consistent evidence for a hot-hand effect across the group. In fact, most of these 3-point specialists shot WORSE after making three in a row than after missing three in a row.

OK, so my math is inconclusive. I can’t confirm that Thompson is definitive PROOF of the hot hand; he might just be a really fantastic shooter who sometimes experiences random sequences of especially fantastic shooting. I’ll leave it to you to draw your own conclusions. And if you are a strict adherent to p-values and model output, definitely swallow the blue pill. But personally, I want to believe the hot hand is real! So, I am eating the red pill.