Simple Linear Regression is used to model linear relationships between two variables, independent (always on x-axis) and dependent (always on y-axis).

It is used in almost any field, typically showing if two variables are correlated with each other (remember: correlation does not equal causation).

Simple linear regression is modeled as any other linear equation, or:

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

Where:

• \(y\) is the dependent variable
• \(x\) is the independent variable
• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope
• \(\varepsilon\) is the error term

Residuals are the differences between observed values and the values predicted by the regression model.

They can be expressed with the following equation:

\[ e_i = y_i - \hat{y}_i = y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_i) \]

These residuals can also be seen as the difference from the expected and actual y values. The closer each residual value is to the line typically means that the two variables are more likely to be correlated. With that, residuals are a great way to show validity of specific models such as linear regression.

We will be using the mtcars dataset for the following graphs.

This is the code used to create the plot for residuals

model <- lm(mpg ~ hp, data = mtcars) residuals_df <- data.frame(hp = mtcars$hp, residuals = resid(model))

rplot <- ggplot(residuals_df, aes(x = hp, y = residuals)) + geom_point(color = “darkseagreen”) + geom_hline(yintercept = 0, linetype = “dashed”) + labs(title = “Residual Plot”, x = “Horsepower”, y = “Residuals”)

print(rplot)

• linear regression models can be helpful with looking at linear correlation
• if two variables do not have linear correlation, that does not automatically mean they are not correlated (they could be exponentially correlated for example)
• once again, correlation does not mean causation! Even if two things have a linear correlation that does not mean one has specifically caused the other