- Simple linear regression models the relationship between two (or more) variables.
- Goal: Predict a response \(Y\) from a predictor \(X\).
2025-04-12
Simple linear regression
Regression Model Equation
The simple linear regression model:
\[ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i \]
Fitted model (based on sample data):
\[ \hat{Y}_i = \hat{\beta}_0 + \hat{\beta}_1 X_i \]
Use Case Example - mtcars
We’ll use the mtcars dataset to explore the relationship between car weight (wt) and fuel efficiency (mpg).
## mpg cyl disp hp drat wt qsec vs am gear carb ## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4 ## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4 ## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1 ## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1 ## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2 ## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1
Scatter Plot (ggplot2)
Summary of Linear Regression Line
model = lm(mpg ~ wt, data = mtcars) summary(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
Scatter Plot (ggplot2 with linear regression line)
Graph with fitted line
Plotly - Weight vs Miles Per Gallow vs Horsepower
Slide with R Output
summary(mtcars)
## mpg cyl disp hp ## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0 ## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5 ## Median :19.20 Median :6.000 Median :196.3 Median :123.0 ## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7 ## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0 ## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0 ## drat wt qsec vs ## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000 ## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000 ## Median :3.695 Median :3.325 Median :17.71 Median :0.0000 ## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375 ## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000 ## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000 ## am gear carb ## Min. :0.0000 Min. :3.000 Min. :1.000 ## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000 ## Median :0.0000 Median :4.000 Median :2.000 ## Mean :0.4062 Mean :3.688 Mean :2.812 ## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000 ## Max. :1.0000 Max. :5.000 Max. :8.000