Definition

• Simple linear regression is a statistics method used to model the relationship between two variables, the independent (x) and the dependent (y). The goal is to find a line that best describes how y changes as x changes

Equation

• The linear regression equation is as follows:

\[ y = \beta_0 + \beta_1 x + \epsilon \]

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

Least Squares

• The regression line is chosen using the least squares method, which is minimizing the sum of square residuals:

\[ \sum (y_i - \hat{y_i})^2 \]

• Residual is the difference between the observed and predicted values.

mtcars Example

• using the built-in dataset mtcars, the two variables used for linear regression will be:

• wt: weight of car

• mpg: miles per gallon

ggplot scatterplot

Here is a ggplot showing that as weight goes up, mpg goes down

ggplot(mtcars, aes(x = wt, y = mpg))+geom_point()

ggplot with regression line

• Linear regression can be done in R using ggplot, as shown in the following slide

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

Plotly: Interactive model of linear regression