Many basketball players and fans alike believe in the "hot hand" phenomemon: the idea that making several shots in a row increases a player's chance of making the next shot. Is there really a hot hand in basketball? More generally, in success/failure trials is there a tendency for trials following streaks of successes to be more likely to result in success? This app can be used to perform a statistical test for "hot hand" type behavior in sequences of success/failure trials, such as the shot attempts of a basketball player.
The data consist of a fixed number of success/failure trials with the outcomes recorded in sequence. (The probability that a trial results in success is assumed to be the same for all trials.) The method to input the observed data is controlled by the radio buttions:
After entering the results sequence or the summary statistics (number of trials and number of successes), as well as the streak length (see below), click the button to Accept inputs and initialization simulation. The initialization process might take a minute; when the "please wait" message disappers, the app is ready to perform repetitions of the simulation (see Permutation test below). Note that the Accept inputs button should be pushed whenever these inputs change.
While there is no consensus definition of what constitutes the "hot hand", the term generally refers to a tendency for trials following streaks of successes to be more likely to result in success. Several statistics are used in practice; those included in the app are described below.
For statistics 1 through 4 below, the user must define the streak length: How many successes must be observed in a row in order to consider it a hot streak? The minimum possible streak length is 1; the maximum allowed streak length in the app is 7. Note: in applications concerning the hot hand in basketball, 3 is commonly used for the streak length.
In the app, the user can choose a streak statistic from the following.
The app can be used to perform a randomization-based hypothesis test of the hot hand phenomenon. The null hypothesis is that there is no hot hand; that is, that the trials are independent. Under the null hypothesis (together with the assumption of constant probability of success on all trials), given the number of successes any possible ordering of the successes and failures in the sequence is equally likely. Therefore, to simulate one hypothetical value of the streak statistic under the null hypothesis of no hot hand:
An approximate p-value can be computed by comparing the observed value of the streak statistic relative to its null distribution, by checking the box for Compute p-value. Since we are interested if there is evidence of presence of the hot hand, one-sided p-values are computed.
As illustrated by the example for streak statistic 3 above, for streak statistics 1 through 4 the value of the statistic cannot be computed for a permutation in which there are no streaks of the specified length. Thus the app distinguishes between the number of repetitions performed and the number of simulated values of the statistic. The latter count is the denominator of the simulated p-value.
Note: The clear plot button will simply clear the plot. To rerun the simulation the Accept inputs and initialize simulation button must be pressed first.
The hot hand refers to a tendency for successes to cluster together. The "cold" hand refers to a similar tendency for failures. The app can be used to analyze the cold hand by simply interchanging the roles of success and failure. For example, to assess evidence of the cold hand based on the observed sequence 0,0,0,1,0,0,1, enter the data as 1,1,1,0,1,1,0 and perform a "hot hand" analysis. KEVIN ROSS
The following paper accompanies this app and provides a discussion of the methods upon which the app is based, illustrations of hot hand analyses that can be performed using the app, and suggested activities.
This data set accompanies the paper. It contains the results from the NBA Three-Point Contest for 2013-2017. For each year, player, and round, the results of all the player's attempts in the round are recorded in sequence, with 1 indicating a made field goal (success) and 0 indicating a miss (failure). (In most cases there are 25 attempts in a round.) The results sequence for a particular player can be copied and pasted into the Enter observed results sequence box.
The paper accompanying the app also uses data from the following seminal study on the hot hand in basketball, which is referenced numerous times in the Miller and Sanjurjo papers and related literature.