Basketball players who make several baskets in succession are described as having a hot hand. Fans and players have long believed in the hot hand phenomenon, which refutes the assumption that each shot is independent of the next. However, a 1985 paper by Gilovich, Vallone, and Tversky collected evidence that contradicted this belief and showed that successive shots are independent events. This paper started a great controversy that continues to this day, as you can see by Googling hot hand basketball.
We do not expect to resolve this controversy today. However, in this lab we’ll apply one approach to answering questions like this. The goals for this lab are to (1) think about the effects of independent and dependent events, (2) learn how to simulate shooting streaks in R, and (3) to compare a simulation to actual data in order to determine if the hot hand phenomenon appears to be real.
Our investigation will focus on the performance of one player: Kobe Bryant of the Los Angeles Lakers. His performance against the Orlando Magic in the 2009 NBA Finals earned him the title Most Valuable Player and many spectators commented on how he appeared to show a hot hand. Let’s load some necessary files that we will need for this lab. Specifically, read in the dataset kobe_basket.dta into Stata.
This data frame contains 133 observations and 6 variables, where every row records a shot taken by Kobe Bryant. The shot variable in this dataset indicates whether the shot was a hit (H) or a miss (M).
Just looking at the string of hits and misses, it can be difficult to gauge whether or not it seems like Kobe was shooting with a hot hand. One way we can approach this is by considering the belief that hot hand shooters tend to go on shooting streaks. For this lab, we define the length of a shooting streak to be the number of consecutive baskets made until a miss occurs.
For example, in Game 1 Kobe had the following sequence of hits and misses from his nine shot attempts in the first quarter:
\[ \textrm{H M | M | H H M | M | M | M} \]
You can verify this by viewing the first 8 rows of the data in the Data Browser.
Within the nine shot attempts, there are six streaks, which are separated by a “|” above. Their lengths are one, zero, two, zero, zero, zero (in order of occurrence).
Counting streak lengths manually for all 133 shots would get tedious, so we’ll use another dataset, kobe_streak.dta to view all the streaks from the kobe_basket dataset.
use "kobe_streak.dta"We can then take a look at the distribution of these streak lengths.
qnorm lengthWe’ve shown that Kobe had some long shooting streaks, but are they long enough to support the belief that he had a hot hand? What can we compare them to?
To answer these questions, let’s return to the idea of independence. Two processes are independent if the outcome of one process doesn’t effect the outcome of the second. If each shot that a player takes is an independent process, having made or missed your first shot will not affect the probability that you will make or miss your second shot.
A shooter with a hot hand will have shots that are not independent of one another. Specifically, if the shooter makes his first shot, the hot hand model says he will have a higher probability of making his second shot.
Let’s suppose for a moment that the hot hand model is valid for Kobe. During his career, the percentage of time Kobe makes a basket (i.e. his shooting percentage) is about 45%, or in probability notation,
\[ P(\textrm{shot 1 = H}) = 0.45 \]
If he makes the first shot and has a hot hand (not independent shots), then the probability that he makes his second shot would go up to, let’s say, 60%,
\[ P(\textrm{shot 2 = H} \, | \, \textrm{shot 1 = H}) = 0.60 \]
As a result of these increased probabilites, you’d expect Kobe to have longer streaks. Compare this to the skeptical perspective where Kobe does not have a hot hand, where each shot is independent of the next. If he hit his first shot, the probability that he makes the second is still 0.45.
\[ P(\textrm{shot 2 = H} \, | \, \textrm{shot 1 = H}) = 0.45 \]
In other words, making the first shot did nothing to effect the probability that he’d make his second shot. If Kobe’s shots are independent, then he’d have the same probability of hitting every shot regardless of his past shots: 45%.
Now that we’ve phrased the situation in terms of independent shots, let’s return to the question: how do we tell if Kobe’s shooting streaks are long enough to indicate that he has a hot hand? We can compare his streak lengths to someone without a hot hand: an independent shooter.
While we don’t have any data from a shooter we know to have independent shots, that sort of data is very easy to simulate in Stata. In a simulation, you set the ground rules of a random process and then the computer uses random numbers to generate an outcome that adheres to those rules.
As a simple example, you can simulate flipping a fair coin. We first will clear out the existing data from Stata. We can always read the data back in the future.
clearThen, we set how many samples we want to take, for example 100.
set obs 100If want to sample “heads” or “tails”, we could set “heads” as 1 and “tails” as 0. This is sampling from a binomial distribution, so we use the rbinomial function. The arugments of the function correspond to having each observation represent a coin flip (1), and that the coin is fair (0.5).
generate sample = rbinomial(1, 0.5)The function rbinomial can be thought of as a hat with two slips of paper in it: one slip says heads and the other says tails. The function draws one slip from the hat and tells us if it was a head (1) or a tail (0).
Look at the variable sample in the Data Browser. Just like when flipping a coin, sometimes you’ll get a heads, sometimes you’ll get a tails, but in the long run, you’d expect to get roughly equal numbers of each.
To view the results of this simulation, use table to count up the number of heads and tails.
table sampleSay we’re trying to simulate an unfair coin that we know only lands heads 20% of the time. We can adjust for this by changing the second argument in rbinomial.
generate sample2 = rbinomial(100, 0.2)This indicates that for the two options (heads and tails), we want to select the first one, heads, with probability 0.2 and the second one, tails with probability 0.8. Another way of thinking about this is to think of the outcome space as a bag of 10 chips, where 2 chips are labeled “head” and 8 chips “tail”. Therefore at each draw, the probability of drawing a chip that says “head”" is 20%, and “tail” is 80%.
A note on setting a seed: Setting a seed will cause Stata to select the same sample each time you run your code. This will make sure your results don’t change each time you run your code, and it will also ensure reproducibility of your work (by setting the same seed it will be possible to reproduce your results). You can set a seed like this:
set seed 35797 # make sure to change the seedThe number above is completely arbitraty. If you need inspiration, you can use your ID, birthday, or just a random string of numbers. The important thing is that you use each seed only once. Remember to do this before you sample in the exercise above.
In a sense, we’ve shrunken the size of the slip of paper that says “heads”, making it less likely to be drawn and we’ve increased the size of the slip of paper saying “tails”, making it more likely to be drawn. When we simulated the fair coin, both slips of paper were the same size.
If you want to learn more about rbinomial or any other function, recall that you can always check out its help file.
help rbinomialSimulating a basketball player who has independent shots uses the same mechanism that we use to simulate a coin flip. It is still choosing between two options: (“Miss”, “Hit”).
To make a valid comparison between Kobe and our simulated independent shooter, we need to align both their shooting percentage and the number of attempted shots. We first need to read in our dataset kobe_basket.dta again. Notice we can apply the option clear to clear our simulated data.
use "kobe_basket.dta", clearNow we need to generate a new variable that is our simulated shots, assuming no hot hand.
rbinomial function so that it reflects a shooting percentage of 45%? Make this adjustment, then run a simulation to sample 133 shots. Note that because we already have a dataset loaded, we do not need to set the observations using set obs. Assign the output of this simulation to a new variable called sim_basket.With the results of the simulation saved as sim_basket, we have the data necessary to compare Kobe to our independent shooter.
Both variables, shot and sim_basket represent the results of 133 shot attempts, each with the same shooting percentage of 45%. We know that our simulated data is from a shooter that has independent shots. That is, we know the simulated shooter does not have a hot hand.
Using the dataset ind_streak.dta, describe the distribution of streak lengths. Make sure to include a plot in your answer.
What is the typical streak length for this simulated independent shooter with a 45% shooting percentage? How long is the player’s longest streak of baskets in 133 shots?
If you were to run the simulation of the independent shooter a second time, how would you expect its streak distribution to compare to the distribution from the question above? Exactly the same? Somewhat similar? Totally different? Explain your reasoning.
How does Kobe Bryant’s distribution of streak lengths compare to the distribution of streak lengths for the simulated shooter? Using this comparison, do you have evidence that the hot hand model fits Kobe’s shooting patterns? Explain.
This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted for Stata by Jenna R. Krall and John Muschelli and adapted for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel from a lab written by Mark Hansen of UCLA Statistics.