2025-10-20

Introduction

Understanding how simple linear regression models a relationship between two variables.

What is Simple Linear Regression?

  • Linear regression models a relationship between a dependent variable \(y\) and an independent variable \(x\).
  • Equation of a line: \[ y = \beta_0 + \beta_1 x + \varepsilon \]

Where: - \(\beta_0\): Intercept - \(\beta_1\): Slope - \(\varepsilon\): Error term

Slide 3: Visualizing the Data with ggplot2

Slide 4: Fitting a Linear Model

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

Slide 5: Add Regression Line to Plot

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

Slide 6: Residual Plot

Slide 7: Interactive Plot with Plotly

R Code Slide

Fitting and predicting from linear model

model <- lm(mpg ~ wt, data = mtcars) predict(model, data.frame(wt = c(2.5, 3.0)))

Summary

  • Simple linear regression helps predict outcomes based on one variable.
  • The model assumes a linear relationship and constant variance.
  • Always check:
    • \(R^2\) to evaluate fit
    • Residual plots to validate assumptions