• A regression model that intends to determine a linear relationship or correlation between two quantitative variables

  • Examples:
    • Points vs. Field Goal Attempts
    • Points vs. Minutes Played
    • Personal Fouls vs. Turnovers
      
      

• A linear relationship is expressed by a best fit line

The graph suggests a positive correlation between field goal attempts and points, such that an increase in field goal attempts correspond to an increase in points.

• Model: \(\displaystyle {y = \beta_0 + \beta_1 \cdot x + \varepsilon}\), where \(\varepsilon\) represents the noise of the data
• Fitted Model: \(\displaystyle {y = \beta_0 + \beta_1 \cdot x}\), where the equation represents the best fit line

Equation of Fitted Model Line for Points vs. Field Goal Attempts Plot:

\(\small{\beta_1 = slope = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{22.47488-10.69017}{17.8-8.5} = \frac{11.78471}{9.3} = 1.26717}\)

\(\small{\beta_0 = y-intercept = 22.474885-1.2671731182\cdot17.8 = -0.08080}\)

\(\small{y = -0.08080 + 1.26717 \cdot x}\)

• The best fit line expresses the trend of the data
  • Line does not necessarily hold true for each and every observation, but can be used to predict values in the data
    
    
• The best fit line criteria:
  1. Line contains a lot of points
  2. Points are as close to the line as possible
• The most common method for determining the best fit line is by using the sum of squares of errors: \(\displaystyle {SSE = \Sigma^{n}_{i=1}(y_{i} - \hat y_{i})^2}\)

fig2 <- ggplot(df, aes(x=MP, y=PTS)) +
  geom_point(aes(color=Pos)) + 
  geom_smooth(method="lm", color="black", se=FALSE) +
  labs(x="Minutes Played", y="Points", color="Position", 
       title="Points vs. Minuted Played in NBA Playoffs")

The Points vs. Minutes Played in NBA Playoffs plot suggests that there is a linear trend in the data. It seems to show that an increase in minutes played corresponds to an increase in points.

fig3 <- ggplot(df, aes(x=TOV, y=PF)) +
  geom_point(aes(color=Pos)) +
  geom_smooth(method="lm", color="black", se=FALSE) +
  labs(x="Turnovers", y="Personal Fouls", 
       title="Personal Fouls vs. Turnovers in NBA Playoffs")

The plot for Personal Fouls vs. Turnovers in NBA Playoffs by Position suggests a slightly positive trend in the data. It shows that generally as turnovers increase so do personal fouls, especially when a player has committed less than 3 turnovers.

• There may or may not always be a visible trend in the plot of the data
• Sometimes the Best Fit Line is not always linear
• Most of the relationships between two or more variables in real-life are not linear but rather much more complex

• Bevans, R. (2023, June 22). Simple Linear Regression | An Easy Introduction & Examples. Scribbr. Retrieved April 3, 2024, from http://www.scribbr.com/statistics/simple-linear-regression/

• Samara, M., personal communication - lecture, March-April 2024

• Dataset found on Kaggle by Vinco, V. 2021-2022 NBA Player Stats - Playoffs.csv.