Hot Hand in Basketball - Extra Research

While creating the Hot Hand in Basketball issue, I researched multiple papers and publication. I am surfacing here a short summary of the most interesting findings.

Apr 13, 2024

For the ones that really are interested in a more thorough analysis of the research, here are some more insights from the publications discussed previously in the issue about Hot Hand in Basketball.

GVT Deeper Dive

Aside from what was discussed before, Gilovich (Cornell University), Vallone (Stanford), Tversky (Stanford), for reference GVT, as part of their paper The Hot Hand in Basketball: On the Misperception of Random Sequences”, they also performed more analyses on their data and performed a controlled experiment.1

Introducing Partitions

They partitioned the records of shots of players into non-overlapping sets of four consecutive shots. They created partitions starting from the 1st, 2nd, 3rd, 4th shot of each player.

They grouped the partitions to 3 groups:

high performance (4/4 or 3/4 made shots)
moderate (2/4 shots)
low performance (1/4 or 0/4 made shots)

They compared the number of high, moderate, low sets for each of the nine players of 76ers to the values expected by chance, assuming independent shots with the average shot-making field goal percentage as probability(making a shot | shooting a shot).

They found no evidence to support that some players have more high performance shots than expected.

Hot VS cold nights

They also compared the variability of shooting percentages of players across games. They used the Lexis ratio of the standard errors (SE) of their shooting percentage:

SE observed in a game / SE expected using the average observed shooting percentage.

They found no player that the Lexis ratio was “significantly” greater than 1, concluding that the shooting percentages across games do not deviate from their overall shooting percentage more than expected.

Controlled shooting experiment

They also designed an experiment in a controlled setting. For each player of the 26 players of the varsity teams of Cornell, 14 males and 12 females, they found the distance that each player was shooting 50%.

They drew two 4.5 meter in length arcs on the floor from which each player would take their shots with an angle of 60 degree from the left and right sides of the basket. The players shot 100 shots, 50 moving on each arc designed individually for them.

They analyzed also there the runs (consecutive makes or misses) similar to the 76ers and again found only one player with less runs (more clustering of shots) than expected.

Miller & Sanjuro Deeper Dive

Miller and Sanjuro challenged the findings by GVT in their paper “Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers”.2 Following their general bias correction, in their paper they also had two more interesting sections.

Player Analysis

They applied the bias correction and calculated for each player of the Cornell team of the GVT study the effect of each player getting on-fire as following:

Bias-Corrected Difference = P(shot made | 3 made shots) - P(shot made | 3 missed shots)

Then they displayed the results for each player in y-axis of Figure 1 sorted in increasing difference value.

Pasted Graphic.png — Figure 1 - Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers, Joshua B. Miller and Adam Sanjurjo, Published: Econometrica, Vol. 86, No. 6 (November 2018), 2019-2047, https://arxiv.org/pdf/1902.01265.pdf

You can see that for 5 out of the 25 players the lower bound of the 95% Confidence interval is above 0, which indicates that these players could be getting the “hot hand”.

Streaky shooting

Every calculation of Miller and Sanjuro assumed that the player has a fixed probability of making a shot which does not take into account the actual “hot hand” / streaky shooting.

In the Appendix B of their paper they estimated the size of the bias given that the process of shooting is not characterized by a fixed probability of success. They hypothesized that being “hot” is characterized by an increase in your probability of making a shot. They used a hidden markov chain over the player’s ability to make a shot.

In summary, they displayed that if being “hot” increases your probability of making a shot by 10%, then the bias is around 2x than the baseline bias.

Essentially, the results they computed were an underestimate of the actual difference, thus the effect of being hot is even larger in the shooting of players.