Yongqi Gan
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We are examining two datasets: The NBA player statistics and their historical salaries for the last 4 years.
Sources: https://stats.nba.com/ for performance data and https://hoopshype.com/salaries/players/ for salary data.
Let us first load the performance data into our R environment using the read.csv function. The header = TRUE flag tells the console that the first row of the file contains column names.
players1819 <- read.csv(file = "players18-19.csv", header = TRUE)
players1718 <- read.csv(file = "players17-18.csv", header = TRUE)
players1617 <- read.csv(file = "players16-17.csv", header = TRUE)
players1516 <- read.csv(file = "players15-16.csv", header = TRUE)and the salary data...
salary1819 <- read.csv(file = "18-19_salary.csv", header = TRUE)
salary1718 <- read.csv(file = "17-18_salary.csv", header = TRUE)
salary1617 <- read.csv(file = "16-17_salary.csv", header = TRUE)
salary1516 <- read.csv(file = "15-16_salary.csv", header = TRUE)We will now merge the four dataframes for salary data and remove the individual datasets from our working environment. Note that we only include players that are accounted for in all four datasets to ensure consistency in our analysis.
#merge concactenates two dataframes together by values in a certain row or column.
df_salary <- merge(salary1516, salary1617, by.x = "Player", by.y = "Player")
df_salary <- merge(df_salary, salary1718, by.x = "Player", by.y = "Player")
df_salary <- merge(df_salary, salary1819, by.x = "Player", by.y = "Player")
#rm removes dataframes and vectors from our local working environment.
#rm(salary1516, salary1617, salary1718, salary1819)There are some issues with our data. If you click on the df_salary dataframe in the "environment" pane on the right, you will see that the year columns have weird names and that the dollar amounts are treated as factors. Factors are categorical variables best used to describe types of values (animals classified as "dog" "cat" etc.) This is undesirable for use in numerical data, and will impact any plotting or statistical analysis we want to do. Therefore, we will convert the factor vectors into numeric vectors and rename them using the code snippet below.
names(df_salary) <- c("Player", "y1516", "y1617", "y1718", "y1819")
df_salary$Player <- as.character(df_salary$Player)
df_salary$y1516 = as.numeric(gsub("[\\$,]", "", as.character(df_salary$y1516)))
df_salary$y1617 = as.numeric(gsub("[\\$,]", "", as.character(df_salary$y1617)))
df_salary$y1718 = as.numeric(gsub("[\\$,]", "", as.character(df_salary$y1718)))
df_salary$y1819 = as.numeric(gsub("[\\$,]", "", as.character(df_salary$y1819)))
#Trim unnecessary whitespace to prevent string matching errors
df_salary$Player = trimws(df_salary$Player)We want to get an overview of the data in order to get an idea of how to consider analyzing it. We can start by simply viewing the header of the salary data, shown below.
head(df_salary)## Player y1516 y1617 y1718 y1819
## 1 Aaron Gordon 4405071 4549389 5662481 21590909
## 2 Al Horford 12671359 27748190 28530811 28928710
## 3 Al Jefferson 14255279 10695850 10050366 4000000
## 4 Al-Farouq Aminu 8492868 8030598 7529204 6957105
## 5 Alan Williams 120677 914448 6172292 77250
## 6 Alec Burks 9728947 10355341 11156939 11536515
We can make a dotplot of 2018-2019 season salaries against points scored using the ggplot2 package, a popul. We will also show minutes played using a color gradient.
require(ggplot2)## Loading required package: ggplot2
require(scales)## Loading required package: scales
## Warning: package 'scales' was built under R version 3.4.4
data1819 <- merge(df_salary, players1819, by.x = "Player", by.y = "PLAYER")
p <- ggplot(data = data1819, aes(y = y1819, x = PTS, colour = MIN))
p + scale_y_continuous(labels = comma) + ylab("2018-2019 Salary") +
xlab("Points") + ggtitle("2018-2019 Season: Salary Plotted Against Points") +
scale_color_gradient(low="red", high="green") + geom_point()
Interesting. We see that a higher salary is generally correlated with more scored points and more minutes played on average. This is likely an obvious observation to most of you, but it is helpful to see it confirmed graphically.
The visual examination leads us to believe that there is a positive relationship between salary and scored points. Let's run a linear regression model to examine this numerically.
#lm stands for "linear model"
points_salary <- lm(y1819 ~ PTS, data = data1819)
summary(points_salary)##
## Call:
## lm(formula = y1819 ~ PTS, data = data1819)
##
## Residuals:
## Min 1Q Median 3Q Max
## -21203611 -4704150 -789194 4028796 20410886
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2174352 830607 2.618 0.00938 **
## PTS 843156 65573 12.858 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6814000 on 254 degrees of freedom
## Multiple R-squared: 0.3943, Adjusted R-squared: 0.3919
## F-statistic: 165.3 on 1 and 254 DF, p-value: < 2.2e-16
Let's interpret this regression output. The line of best fit is given by data1819 = 2,174,352 + 843,156 * PTS. We can interpret this to mean that for every additional point scored, on average the salary of that player will increase by 843,156 dollars. We note the small p-value (less than 2−16), implying our results are statistically significant at the 99.99% confidence level.
The original plot with our newly found regression line is displayed below.
p + scale_y_continuous(labels = comma) + ylab("2018-2019 Salary") +
xlab("Points") + ggtitle("2018-2019 Season: Salary Plotted Against Points") +
scale_color_gradient(low="red", high="green") + geom_point() + geom_abline(intercept = 2174352, slope = 843156)
To demonstrate the power of R packages, let's consider a hypothetical scenario: You are the manager of an NBA team. Your goal is to score at least 100 points every game on average, while spending as little as possible on the players' salaries.
This is an optimization problem. We are trying to minimize a cost function (total salary cost), given a constraint (average at least 100 points). We can solve this optimization problem using a technique known as linear programming. For technical details you are welcome to review https://www.math.ucla.edu/~tom/LP.pdf; however, a technical understanding of the algorithm is not necessary for this example.
We will use a R package built by Michel Berkelaar, lpSolve, to solve this problem. R packages are custom-written libraries developed by statistians, companies, and other users to extend or the functionality of R or to simplify certain procedures.
require(lpSolve)## Loading required package: lpSolve
# obj represents the objective function. This is the function that calculates the total cost of the team and is what we are trying to minimize.
obj <- data1819$y1819
#constr represents the constraint functions. These are the conditions that a valid solution must satisfy. The conditions we impose are that the "team" must have a minimum sum of 100 points per game, the team must have at least 12 players, and each player can only be chosen at most once.
constr <- matrix(append(append(data1819$PTS, rep(1, times = 256)), as.vector(diag(nrow = 256))), nrow = 258, byrow = TRUE)
#right represents the right hand side of the constraint functions.
right <- c(100, 12, rep(1, times = 256))
#constraints_direction represents the sign of the constraint functions.
constranints_direction <- c(">=", ">=", rep("<=", times = 256))
optimum <- lp(direction="min",
objective.in = obj,
const.mat = constr,
const.dir = constranints_direction,
const.rhs = right,
all.int = T)
best_sol <- optimum$solution
names(best_sol) <- data1819$Player
print(best_sol)## Aaron Gordon Al Horford Al-Farouq Aminu
## 0 0 0
## Alan Williams Alec Burks Alex Len
## 1 0 0
## Allen Crabbe Amir Johnson Andre Drummond
## 0 0 0
## Andre Iguodala Andrew Bogut Andrew Wiggins
## 0 1 0
## Anthony Davis Anthony Tolliver Aron Baynes
## 0 0 0
## Austin Rivers Avery Bradley Ben McLemore
## 0 0 0
## Bismack Biyombo Blake Griffin Boban Marjanovic
## 0 0 0
## Bobby Portis Bojan Bogdanovic Bradley Beal
## 0 0 0
## Brandon Knight Brook Lopez Bruno Caboclo
## 0 0 1
## Cameron Payne Carmelo Anthony Chandler Parsons
## 0 0 0
## Channing Frye Chris Paul CJ McCollum
## 0 0 0
## CJ Miles Clint Capela Cody Zeller
## 0 0 0
## Corey Brewer Cory Joseph Courtney Lee
## 1 0 0
## Cristiano Felicio D'Angelo Russell Damian Lillard
## 0 0 0
## Danilo Gallinari Danny Green Dante Cunningham
## 0 0 0
## Dante Exum Darren Collison DeAndre Jordan
## 0 0 0
## Delon Wright DeMar DeRozan DeMarcus Cousins
## 0 0 0
## DeMarre Carroll Derrick Favors Derrick Rose
## 0 0 1
## Devin Booker Devin Harris Dewayne Dedmon
## 1 0 0
## Dion Waiters Dirk Nowitzki Doug McDermott
## 0 0 0
## Draymond Green Dwight Howard Dwight Powell
## 0 0 0
## Dwyane Wade E'Twaun Moore Ed Davis
## 0 0 0
## Elfrid Payton Emmanuel Mudiay Enes Kanter
## 0 0 0
## Eric Bledsoe Eric Gordon Ersan Ilyasova
## 0 0 0
## Evan Fournier Evan Turner Frank Kaminsky
## 0 0 0
## Garrett Temple Gary Harris George Hill
## 0 0 0
## Gerald Green Giannis Antetokounmpo Glenn Robinson III
## 1 0 0
## Goran Dragic Gordon Hayward Gorgui Dieng
## 0 0 0
## Greg Monroe Harrison Barnes Hassan Whiteside
## 0 0 0
## Ian Clark Ian Mahinmi Iman Shumpert
## 0 0 0
## Isaiah Canaan Isaiah Thomas Jabari Parker
## 1 0 0
## Jae Crowder Jahlil Okafor Jamal Crawford
## 0 0 0
## James Harden James Johnson JaMychal Green
## 0 0 0
## Jared Dudley Jarell Martin Jason Smith
## 0 0 0
## JaVale McGee Jeff Green Jeff Teague
## 0 0 0
## Jerami Grant Jeremy Lamb Jeremy Lin
## 0 0 0
## Jerian Grant Jerryd Bayless Jimmy Butler
## 0 0 0
## JJ Redick Joakim Noah Jodie Meeks
## 0 0 0
## Joe Harris Joe Ingles Joel Embiid
## 0 0 0
## John Henson John Wall Jon Leuer
## 0 0 0
## Jonas Jerebko Jonas Valanciunas Jonathon Simmons
## 0 0 0
## Jordan Clarkson Jose Calderon Josh Richardson
## 0 0 0
## JR Smith Jrue Holiday Julius Randle
## 0 0 0
## Justin Anderson Justin Holiday Justise Winslow
## 0 0 0
## Jusuf Nurkic Karl-Anthony Towns Kawhi Leonard
## 0 0 0
## Kelly Olynyk Kemba Walker Kenneth Faried
## 0 0 0
## Kent Bazemore Kentavious Caldwell-Pope Kevin Durant
## 0 0 0
## Kevin Love Kevon Looney Khris Middleton
## 0 0 0
## Klay Thompson Kosta Koufos Kyle Anderson
## 0 0 0
## Kyle Korver Kyle Lowry Kyle O'Quinn
## 0 0 0
## Kyrie Irving LaMarcus Aldridge Lance Stephenson
## 0 0 0
## Lance Thomas Langston Galloway LeBron James
## 0 0 0
## Luc Mbah a Moute Luol Deng Marc Gasol
## 0 0 0
## Marcin Gortat Marco Belinelli Marcus Morris
## 0 0 0
## Marcus Smart Mario Hezonja Markieff Morris
## 0 0 0
## Marvin Williams Mason Plumlee Matthew Dellavedova
## 0 0 0
## Meyers Leonard Michael Beasley Michael Carter-Williams
## 0 0 0
## Michael Kidd-Gilchrist Mike Conley Mike Muscala
## 0 0 0
## Mike Scott Miles Plumlee Montrezl Harrell
## 0 0 0
## Myles Turner Nemanja Bjelica Nerlens Noel
## 0 0 0
## Nick Young Nicolas Batum Nik Stauskas
## 1 0 0
## Nikola Jokic Nikola Mirotic Nikola Vucevic
## 0 0 0
## Noah Vonleh Norman Powell Omri Casspi
## 0 0 0
## Pat Connaughton Patrick Beverley Patrick Patterson
## 0 0 0
## Pau Gasol Paul George Paul Millsap
## 0 0 0
## PJ Tucker Quincy Acy Quincy Pondexter
## 0 1 0
## Rajon Rondo Raul Neto Raymond Felton
## 0 0 0
## Reggie Bullock Reggie Jackson Richaun Holmes
## 0 0 0
## Ricky Rubio Robert Covington Robin Lopez
## 0 0 0
## Rodney Hood Rondae Hollis-Jefferson Rudy Gay
## 0 0 0
## Rudy Gobert Russell Westbrook Ryan Anderson
## 0 0 0
## Salah Mejri Sam Dekker Serge Ibaka
## 0 0 0
## Seth Curry Shabazz Napier Shaun Livingston
## 0 0 0
## Shelvin Mack Solomon Hill Spencer Dinwiddie
## 0 0 1
## Stanley Johnson Stephen Curry Steven Adams
## 0 0 0
## Taj Gibson Terrence Ross Terry Rozier
## 0 0 0
## Thabo Sefolosha Thaddeus Young Tim Frazier
## 0 0 0
## Tobias Harris Tony Parker Tony Snell
## 0 0 0
## Treveon Graham Trevor Ariza Trey Burke
## 0 0 0
## Trey Lyles Tristan Thompson Troy Daniels
## 0 0 0
## Tyler Johnson Tyler Zeller Tyreke Evans
## 0 1 0
## Tyson Chandler Tyus Jones Udonis Haslem
## 0 0 0
## Victor Oladipo Vince Carter Wayne Ellington
## 0 0 0
## Wesley Johnson Wesley Matthews Will Barton
## 0 0 0
## Willie Cauley-Stein Wilson Chandler Zach LaVine
## 0 0 0
## Zaza Pachulia
## 0
The players with a "1" under their name have been chosen for this "optimal" team.
print(paste("Total cost: ", optimum$objval, sep=""))## [1] "Total cost: 12387465"
We find that the total cost of this "optimal" team is 12.387 million dollars.
It is widely understood in the professional basketball world that switching teams generally results in an increase in salary. We will now quantitatively test this hypothesis and determine the size of this effect.
First, we need to extract only the players who were active in the last 4 seasons:
player <- merge(merge(merge(players1516[ , c("PLAYER", "TEAM")], players1617[ , c("PLAYER", "TEAM")], by = "PLAYER"), players1718[ , c("PLAYER", "TEAM")], by = "PLAYER"), players1819[ , c("PLAYER", "TEAM")], by = "PLAYER")## Warning in merge.data.frame(merge(merge(players1516[, c("PLAYER",
## "TEAM")], : column names 'TEAM.x', 'TEAM.y' are duplicated in the result
names(player) <- c("PLAYER", "TEAM1516", "TEAM1617", "TEAM1718", "TEAM1819")Now we can count the number of times each player switched teams:
player$switches <- 0
player$switches <- ifelse(player$TEAM1516 == player$TEAM1617, player$switches, player$switches + 1)
player$switches <- ifelse(player$TEAM1617 == player$TEAM1718, player$switches, player$switches + 1)
player$switches <- ifelse(player$TEAM1718 == player$TEAM1819, player$switches, player$switches + 1)Let's add the salary data to this dataset:
player <- merge(player, df_salary, by.x = "PLAYER", by.y = "Player")
player2 <- player
player2$change <- player2$y1819 - player2$y1516
require(reshape)## Loading required package: reshape
## Warning: package 'reshape' was built under R version 3.4.4
player <- melt(player, id = c("PLAYER", "TEAM1516", "TEAM1617", "TEAM1718", "TEAM1819", "switches"))We can create a plot of salaries based on the number of team switches:
p <- ggplot(data = player, aes(y = value, x = variable, colour = switches))
p + scale_y_continuous(labels = comma) + ylab("Salary") +
xlab("Season") + ggtitle("Salary Plotted Against Number of Team Switches") +
scale_color_gradient(low="red", high="green") + geom_point()
The graph doesn't seem to be too promising on first glance. Let's run a regression analysis on the change in salary across these four seasons given the change in teams:
switch_lm <- lm(change ~ switches, data = player2)
summary(switch_lm)##
## Call:
## lm(formula = change ~ switches, data = player2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -20935449 -4550918 -402481 4353721 18122651
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8427039 708532 11.894 < 2e-16 ***
## switches -3266267 477605 -6.839 6.23e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6925000 on 247 degrees of freedom
## Multiple R-squared: 0.1592, Adjusted R-squared: 0.1558
## F-statistic: 46.77 on 1 and 247 DF, p-value: 6.232e-11
We find that this is in fact exactly the case. A team switch on average will decrease a player's earnings by $3.26 million.