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.
In this lab, we will explore and visualize the data using the
tidyverse
suite of packages. The data can be found in the
companion package for OpenIntro labs, openintro.
Let’s load the packages.
library(tidyverse)
library(openintro)
Your lab report template file is already included for you in the Files tab. Go ahead and open that up first, adding your name and lab number, as well as the names of your lab group, to the report.
Your investigation will focus on the performance of one former
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. The data file we’ll use
is called kobe_basket
.
glimpse(kobe_basket)
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 9 rows of the data in the data viewer.
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 the custom function calc_streak
to calculate
them, and store the results in a data frame called
kobe_streak
as the length
variable.
<- calc_streak(kobe_basket$shot) kobe_streak
We can then take a look at the distribution of these streak lengths.
ggplot(data = kobe_streak, aes(x = length)) +
geom_bar()
We’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 R. 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 with the following.
<- c("heads", "tails")
coin_outcomes sample(coin_outcomes, size = 1, replace = TRUE)
The vector coin_outcomes
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 sample
draws one slip
from the hat and tells us if it was a head or a tail.
Run the second command listed above several times. 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.
If you wanted to simulate flipping a fair coin 100 times, you could
either run the function 100 times or, more simply, adjust the
size
argument, which governs how many samples to draw (the
replace = TRUE
argument indicates we put the slip of paper
back in the hat before drawing again). Save the resulting vector of
heads and tails in a new object called sim_fair_coin
.
<- sample(coin_outcomes, size = 100, replace = TRUE) sim_fair_coin
To view the results of this simulation, type the name of the object
and then use table
to count up the number of heads and
tails.
sim_fair_cointable(sim_fair_coin)
Since there are only two elements in coin_outcomes
, the
probability that we “flip” a coin and it lands heads is 0.5. Say we’re
trying to simulate an unfair coin that we know only lands heads 20% of
the time. We can adjust for this by adding an argument called
prob
, which provides a vector of two probability
weights.
<- sample(coin_outcomes, size = 100, replace = TRUE,
sim_unfair_coin prob = c(0.2, 0.8))
prob=c(0.2, 0.8)
indicates that for the two elements in
the outcomes
vector, 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 R to select the same sample each time you knit your document. This will make sure your results don’t change each time you knit, 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 seed
The 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 set a seed in each code chunk where you
randomly sample, and make sure the set.seed
command comes
before the sampling. If you want to understand this better, try
repeatedly running the unfair coin simulation with and without setting a
seed.
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 you simulated the fair coin, both slips of paper were the same
size. This happens by default if you don’t provide a prob
argument; all elements in the outcomes
vector have an equal
probability of being drawn.
If you want to learn more about sample
or any other
function, recall that you can always check out its help file.
?sample
Simulating a basketball player who has independent shots uses the same mechanism that you used to simulate a coin flip. To simulate a single shot from an independent shooter with a shooting percentage of 50% you can type
<- c("H", "M")
shot_outcomes <- sample(shot_outcomes, size = 1, replace = TRUE) sim_basket
To make a valid comparison between Kobe and your simulated independent shooter, you need to align both their shooting percentage and the number of attempted shots.
sample
function so
that it reflects a shooting percentage of 45%? Make this adjustment,
then run a simulation to sample 133 shots. Assign the output of this
simulation to a new object called sim_basket
.Note that we’ve named the new vector sim_basket
, the
same name that we gave to the previous vector reflecting a shooting
percentage of 50%. In this situation, R overwrites the old object with
the new one, so always make sure that you don’t need the information in
an old vector before reassigning its name.
With the results of the simulation saved as sim_basket
,
you have the data necessary to compare Kobe to our independent
shooter.
Both data sets 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 calc_streak
, compute the streak lengths of
sim_basket
, and save the results in a data frame called
sim_streak
.
Describe the distribution of streak lengths. 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? Make sure to include a plot in your answer.
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.
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work is licensed under a
Creative
Commons Attribution-ShareAlike 4.0 International License.