Simple Linear Regression is used to model the relationship between a dependent variable (Y) and an independent variable (X).

The equation for a simple linear regression is:

\[ Y = \beta_0 + \beta_1 X + \epsilon \]

Where: - \(Y\) = dependent variable - \(X\) = independent variable - \(\beta_0\) = intercept - \(\beta_1\) = slope - \(\epsilon\) = error term

We use the mtcars dataset in R.

data(mtcars)
head(mtcars)

##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

library(ggplot2)
ggplot(mtcars, aes(x=hp, y=mpg)) +
  geom_point(color='blue') +
  geom_smooth(method='lm', se=FALSE, color='red') +
  labs(title='Horsepower vs MPG', x='Horsepower', y='Miles Per Gallon')

model <- lm(mpg ~ hp, data=mtcars)
summary(model)

## 
## Call:
## lm(formula = mpg ~ hp, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.7121 -2.1122 -0.8854  1.5819  8.2360 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 30.09886    1.63392  18.421  < 2e-16 ***
## hp          -0.06823    0.01012  -6.742 1.79e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.863 on 30 degrees of freedom
## Multiple R-squared:  0.6024, Adjusted R-squared:  0.5892 
## F-statistic: 45.46 on 1 and 30 DF,  p-value: 1.788e-07

library(plotly)
p <- plot_ly(mtcars, x = ~hp, y = ~mpg, z = ~wt, type = 'scatter3d', mode='markers')
p

• The slope (\(\beta_1\)) tells us how much mpg changes for a 1-unit increase in hp.
• The intercept (\(\beta_0\)) is the expected mpg when hp = 0.

Simple Linear Regression is a powerful tool to model relationships between variables and make predictions.

• R Documentation
• https://ggplot2.tidyverse.org/
• https://plotly.com/r/