• Simple linear regression: Linear regression analysis is used to predict the value of one variable based on another variable.

Source: IBM, “What is linear regression?”

• The horizontal axis is the predictor \(x\), and the vertical axis is the response \(y\).
• Several possible lines, such as L1, L2, and L3, can be drawn through the data.
• For each line, the vertical distance from a point to the line is known as the error or residual.
• Linear regression chooses the line that makes these prediction errors as small as possible overall.

\[ e = y - \hat{y} \]

\[ e^2 = ( y - \hat{y} )^2 \]

\[ \sum_{ i=1}^{n}(y_i - \hat{y}_i )^2 \]

• Many possible lines can be drawn through the data, such as L1, L2, and L3.
• The residual is the actual value minus predicted value.
• Regression squares each residual so all errors are positive and larger errors count more.
• The least squares regression line is the line with the smallest sum of squared residuals.

\[ b = \text{change in predicted } y \text{ for a 1-unit increase in } x \]

\[ R^2 = \text{proportion of variation in } y \text{ explained by the regression model} \]

• The slope describes how predicted volume changes as girth increases.
• \(R^2\) tells us how much of the variation in volume is explained by girth.
• Extrapolation is the use of known data to make predictions outside the observed range.
• These predictions should be treated carefully as future patterns and results may not continue to trend in this direction.

Source: Eurostat Glossary: Extrapolation

mod <- lm(Volume ~ Girth, data = trees)

ggplot(trees, aes(x = Girth, y = Volume)) +
  geom_point(size = 2) +
  geom_smooth(method = "lm", se = FALSE) +
  theme_minimal()