Simple linear regression models the relationship between one independent variable (x) and one dependent variable (y) using a straight-line equation.

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

• \(\beta_0\): the intercept
• \(\beta_1\): the slope
• \(\epsilon\): the random error

The goal is to find the line of best fit that minimizes the sum of squared residuals between observed and predicted values.

We use linear regression to:

• Understand relationships between variables.
• Make predictions.
• Identify trends in data.

The trees data set in R contains 31 observations and three variables:

• Girth: Tree diameter in inches.
• Height: Tree height in feet.
• Volume: Timber volume in cubic feet.

We want to investigate the linear relationships between girth and volume, and height and volume.

data(trees)
head(trees)

##   Girth Height Volume
## 1   8.3     70   10.3
## 2   8.6     65   10.3
## 3   8.8     63   10.2
## 4  10.5     72   16.4
## 5  10.7     81   18.8
## 6  10.8     83   19.7

We consider two simple linear regression models.

Model 1: Girth vs Volume

• Independent variable (X): Girth
• Dependent variable (Y): Volume

Model 2: Height vs Volume

• Independent variable (X): Height
• Dependent variable (Y): Volume

p1 = ggplot(trees, aes(x = Girth, y = Volume)) +
     geom_point(color = "steelblue", size = 3) +
     geom_smooth(method = "lm", se = FALSE, color = "red") +
     labs(title = "Girth vs Volume",
        x = "Girth (inches)",
        y = "Volume (cubic feet)")

ggplotly(p1)

model1 = lm(Volume ~ Girth, data = trees)
model1

## 
## Call:
## lm(formula = Volume ~ Girth, data = trees)
## 
## Coefficients:
## (Intercept)        Girth  
##     -36.943        5.066

The fitted regression equation is:

\[ \widehat{Volume} = -36.943 + 5.066(Girth) \]

p2 = ggplot(trees, aes(x = Height, y = Volume)) +
     geom_point(color = "steelblue", size = 3) +
     geom_smooth(method = "lm", se = FALSE, color = "red") +
     labs(title = "Height vs Volume",
        x = "Height (feet)",
        y = "Volume (cubic feet)")

ggplotly(p2)

model2 = lm(Volume ~ Height, data = trees)
model2

## 
## Call:
## lm(formula = Volume ~ Height, data = trees)
## 
## Coefficients:
## (Intercept)       Height  
##     -87.124        1.543

The fitted regression equation is:

\[ \widehat{Volume} = -87.124 + 1.543(Height) \]

Model 1: Girth vs Volume

• Girth and volume have a strong, positive linear relationship.
• The observed points are relatively close to the fitted regression line.
• There are no obvious outliers.

Model 2: Height vs Volume

• Height and volume have a weaker, positive linear relationship.
• The observed points are more widely scattered around the fitted regression line.
• Some observations have relatively large residuals.

cor(trees$Height, trees$Girth)

## [1] 0.5192801

We find that height and girth has a moderate positive relationship.

• Both girth and height have positive linear relationships with volume. Girth has a stronger linear relationship with volume than height, suggesting that girth is a better predictor of tree volume.

• Height and girth have a moderate positive correlation (r = 0.519). This suggests a possible indirect association between height and volume through girth. However, correlation does not imply causation.

• To improve prediction accuracy, we could explore more advanced models, such as multiple linear regression, models with interaction terms, or higher-order polynomial regression.