Predicting NBA salaries with multiple linear regression

Modeling Inference
Data Science with R

Getting Started

Programming exercises are designed to provide an opportunity for you to put what you learn in the videos and readings. These exercises feature interactive code cells which allow you to write, edit, and run R code without leaving your browser.

When the ▶️ Run Code button turns to a solid color (with no flashing bubble indicating that the document is still loading), you can interact with the code cells!

Packages

We’ll use the tidyverse and tidymodels for this programming exercise. These are already installed for you to use!

Motivation

The National Basketball Association (NBA) is a professional sports league that consists of 30 teams across the United States and Canada. In the early 2000s, NBA teams started to use advanced analytics to gain a competitive edge over their opponents. Teams employ the use of analytics to make data driven decisions on game strategy and business operations. In this programming exercise, we are going to work with NBA data to try and better understand the salary NBA players make.

Data

This dataset consists of player per-game statistics for the NBA’s 2022-23 season with player salary data. We are going to use just a subset of these variables in this programming exercise. The variables we will use can be seen below. The data key for the entire dataset can be revealed below.

Variable Description
Salary Yearly salary a NBA players make in USD
Position Position on the court the NBA player plays: PG, SG, SF, PF, and C
Age Age of the player, rounded to the nearest year

Need to expand

Variable Description
Salary Yearly salary a NBA players make in USD
Position Position on the court the NBA player plays: PG, SG, SF, PF, and C
Age Age of the player, rounded to the nearest year
Team Team of the NBA player
GP Total number of games played
GS Total number of games the player started
MP Average number of minutes played
FG The total number of shots made (2P + 3P)
FGA The total number of shots taken (2PA + 3PA)
FG% The percentage of all shots made (FG / FGA)
3P The total number of 3-point shots made
3PA The total number of 3-point shots taken
3P% The percentage of 3-point shots made (3P / 3PA)
2P Total number of 2-point shots made
2PA Total number of 2-point shots attempted
eFG% Effective Field Goal Percentage ( (FG + 0.5*3P) / FGA) )
FT Total number of free throws made
FTA The total free throws attempted
FT% The percent of FTs made (FT / FTA)
ORB Average offensive rebounds per game
DRB Average defensive rebounds per game
TRB Total rebounds (ORB + DRB) per game
AST Average assists per game
STL Average steals per game played
BLK Average blocks per game played
TOV Average turnovers per game
PF Average fouls accrued per game played
PTS Average points scored per game played

Now, let’s explore these data!

Exploratory data analysis

Before we fit a linear regression model, we are going to explore the data. Let’ start with exploring our response variable of interest, cholesterol (Salary). Specifically, we are interested in the relationship between Salary, Age, and Position. Below, calculate the mean and standard deviation for your quantitative variables. In the same code, produce the count of each position.

nba |>
  group_by(Position) |>
  summarize(
    mean_salary = mean(Salary, na.rm = TRUE),
    sd_salary = sd(Salary, na.rm = TRUE),
    mean_age = mean(Age, na.rm = TRUE),
    sd_age = sd(Age, na.rm = TRUE),
    position_n = n()
  )
# A tibble: 5 × 6
  Position mean_salary sd_salary mean_age sd_age position_n
  <chr>          <dbl>     <dbl>    <dbl>  <dbl>      <int>
1 C           7282722.  8983558.     26.3   4.48         91
2 PF          8885045. 10897789.     26.6   4.78         86
3 PG         11579573. 13889342.     26.2   4.35         77
4 SF          8132253. 11055604.     25.6   3.64         91
5 SG          6681301.  8308597.     24.7   3.85        115

Modeling

Visualizing the model

We are now going to plot our response variable Salary vs Age using a scatterplot. We are also going to color the points based on Position. The plot can be seen below.

From the videos, we learned about two different times of multiple linear regression models that we could fit:

Main effects models: The relationship between x and y does not change based on z

Interaction effects models: The relationship between x and y does change by z

(Thought exercise) Based on these definitions, do you think it would be appropriate to fit an main effects or an interaction effects model?

It is justifiable to fit an interaction effects model instead of a main effects model. Based on the scatterplot, we can see that the relationship between Salary and Age changes depending on the position the player plays.

Multiple linear regression

So we can explore both concepts, regardless of your conclusion in the though exercise, we are first going to fit a main effects model. We are going to this to:

  • Show that R will still fit the model, despite it not being the most appropriate

  • Practice fitting main effects models

  • Practice interpreting main effects model output

Main effects model

Fit the main effects model below. Name this model m1, and wrap this object in the tidy() function. Next, interpret the estimate for PositionPF in the context of the problem.

sal_main_fit <- linear_reg() |>
  fit(Salary ~ Age + Position, data = nba)

tidy(sal_main_fit)
# A tibble: 6 × 5
  term          estimate std.error statistic  p.value
  <chr>            <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept) -20019456.  2996402.    -6.68  6.96e-11
2 Age           1038670.   107316.     9.68  2.88e-20
3 PositionPF    1307286.  1449993.     0.902 3.68e- 1
4 PositionPG    4391276.  1492626.     2.94  3.43e- 3
5 PositionSF    1602852.  1431168.     1.12  2.63e- 1
6 PositionSG    1095288.  1363752.     0.803 4.22e- 1

As an NBA player ages one year, we estimate a mean salary increase of 1,307,286 usd if they play PF.

(Thought exercise) What position do you not see a term for? Where is it?

We don’t see a specific term for C. That is because it’s our (Intercept) term! We can interpret the intercept as: For an age of, we estimate the mean salary of an NBA C to be -2,0019,456 usd. Each other position estimate is the estimated difference in Salary relative to the C, after controlling for Age!

We can change the intercept. See the following code that changes the intercept to PG, and notice how the estimates change based on what our intercept term is! Note: Age does not change, because the relationship between Salary and Age does not depend on Position.

nba_diff <- nba |>
  mutate(Position = fct_relevel(Position, "PG"))

sal_main_2_fit <- linear_reg() |>
  fit(Salary ~ Age + Position, data = nba_diff)

tidy(sal_main_2_fit)
# A tibble: 6 × 5
  term          estimate std.error statistic  p.value
  <chr>            <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept) -15628180.  3018134.     -5.18 3.38e- 7
2 Age           1038670.   107316.      9.68 2.88e-20
3 PositionC    -4391276.  1492626.     -2.94 3.43e- 3
4 PositionPF   -3083990.  1512886.     -2.04 4.21e- 2
5 PositionSF   -2788424.  1494146.     -1.87 6.27e- 2
6 PositionSG   -3295988.  1429037.     -2.31 2.15e- 2

Interaction effects model

Now, it’s time to fit the interaction effects model between Salary, Age, and Position. Please do so below, and interpret the Age:PositionPF interaction term.

sal_int_fit <- linear_reg() |>
  fit(Salary ~ Age * Position, data = nba)

tidy(sal_int_fit)
# A tibble: 10 × 5
   term             estimate std.error statistic  p.value
   <chr>               <dbl>     <dbl>     <dbl>    <dbl>
 1 (Intercept)     -3607189.  5950994.    -0.606 0.545   
 2 Age               414290.   223212.     1.86  0.0641  
 3 PositionPF     -13289313.  8318463.    -1.60  0.111   
 4 PositionPG     -29720679.  8918015.    -3.33  0.000931
 5 PositionSF     -26887800.  9258556.    -2.90  0.00386 
 6 PositionSG     -11312489.  8280299.    -1.37  0.173   
 7 Age:PositionPF    556044.   310153.     1.79  0.0737  
 8 Age:PositionPG   1300074.   335283.     3.88  0.000121
 9 Age:PositionSF   1096922.   353991.     3.10  0.00207 
10 Age:PositionSG    461940.   321063.     1.44  0.151

For a Power Forward, a one-year increase in age, we estimate on average an increase in salary of approximately 970,334 usd.

Why 970,334? The term Age:PositionPF has a coefficient of 556,044. This is the estimated difference in the slope coefficient vs the Center position. Thus, the slope coefficient specifically for the PF position is estimated to be 414,290 larger than the baseline.

Note: This is the mathematical representation to the plot created above!

If we do change the baseline term, the Age coefficient will also change, because the interaction term allows for the relationship between Age and Salary to change based on Position. Let’s see this below.

sal_int_2_fit <- linear_reg() |>
  fit(Salary ~ Age * Position, data = nba_diff)

tidy(sal_int_2_fit)
# A tibble: 10 × 5
   term             estimate std.error statistic  p.value
   <chr>               <dbl>     <dbl>     <dbl>    <dbl>
 1 (Intercept)    -33327869.  6642038.    -5.02  7.54e- 7
 2 Age              1714364.   250181.     6.85  2.40e-11
 3 PositionC       29720679.  8918015.     3.33  9.31e- 4
 4 PositionPF      16431366.  8826050.     1.86  6.33e- 2
 5 PositionSF       2832880.  9717161.     0.292 7.71e- 1
 6 PositionSG      18408190.  8790091.     2.09  3.68e- 2
 7 Age:PositionC   -1300074.   335283.    -3.88  1.21e- 4
 8 Age:PositionPF   -744030.   330094.    -2.25  2.47e- 2
 9 Age:PositionSF   -203152.   371586.    -0.547 5.85e- 1
10 Age:PositionSG   -838134.   340365.    -2.46  1.42e- 2

Summary

  • Multiple linear regression allows for a single quantitative variable to be modeled by > 1 predictor variable.

  • An main effects model has the restriction of keeping the relationship between x and y consistent , regardless of the values of the other variables in the model.

  • An interaction effects model relaxes this restriction, and allows the relationship between x and y to change based on values of z.

  • Regression output has a baseline group when one predictor variable is categorical. This can be found in the Intercept term, with the other categorical level coefficients representing the deviation from the baseline.

Your turn: Challenge

Use this space to fit more complicated models, and explore different relationships that model our response variable Salary! As a reminder, the complete data dictionary can be found above.