Welcome to this presentation on Simple Linear Regression — a fundamental statistical technique for understanding the relationship between two quantitative variables.

We will use the mtcars dataset to explore the link between horsepower (hp) and miles per gallon (mpg).

The simple linear regression model assumes a straight-line relationship between \(Y\) and \(X\):

\[ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i \]

where
\(\beta_0\) = intercept, \(\beta_1\) = slope, \(\varepsilon_i\) = random error.

We’ll inspect the variables of interest from the built-in mtcars dataset.

summary(mtcars[, c("mpg", "hp", "wt")])

##       mpg              hp              wt       
##  Min.   :10.40   Min.   : 52.0   Min.   :1.513  
##  1st Qu.:15.43   1st Qu.: 96.5   1st Qu.:2.581  
##  Median :19.20   Median :123.0   Median :3.325  
##  Mean   :20.09   Mean   :146.7   Mean   :3.217  
##  3rd Qu.:22.80   3rd Qu.:180.0   3rd Qu.:3.610  
##  Max.   :33.90   Max.   :335.0   Max.   :5.424

library(ggplot2) library(plotly)

ggplot(mtcars, aes(x = hp, y = mpg)) +
geom_point(color = "darkred", size = 2) +
labs(title = "MPG vs Horsepower",
x = "Horsepower", y = "Miles per Gallon") +
theme_minimal()

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

ggplot(mtcars, aes(x = hp, y = mpg)) +
geom_point(color = "steelblue") +
geom_smooth(method = "lm", se = FALSE, color = "red") +
labs(title = "Regression Line for MPG vs Horsepower",
x = "Horsepower", y = "Miles per Gallon") +
theme_classic()

## `geom_smooth()` using formula = 'y ~ x'

plot_ly(mtcars, x = ~hp, y = ~mpg, z = ~wt,
type = "scatter3d", mode = "markers",
marker = list(size = 5, color = ~mpg, colorscale = "Viridis")) %>%
layout(scene = list(
xaxis = list(title = "Horsepower"),
yaxis = list(title = "MPG"),
zaxis = list(title = "Weight")))

##Slide 8: Diagnostics

par(mfrow = c(2,2))
plot(model)