This presentation walks through a simple linear regression example using the trees dataset in R.
The main goal is to examine how tree Volume changes as tree Girth increases.
I will be using:
Girth Height Volume fit resid 1 8.3 70 10.3 5.103149 5.1968508 2 8.6 65 10.3 6.622906 3.6770939 3 8.8 63 10.2 7.636077 2.5639226 4 10.5 72 16.4 16.248033 0.1519667 5 10.7 81 18.8 17.261205 1.5387954 6 10.8 83 19.7 17.767790 1.9322098
I am using the standard simple linear regression model: \[ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i,\quad i = 1,2,\dots,n. \]
Where: - \(Y_i\) is tree Volume
- \(X_i\) is tree Girth
- \(\beta_0, \beta_1\) are the regression parameters
- \(\varepsilon_i\) is the random error term
A common assumption is: \[ \varepsilon_i \sim \text{Normal}(0,\sigma^2) \text{ and independent}. \]
The fitted regression line is: \[ \hat{Y}_i = \hat{\beta}_0 + \hat{\beta}_1 X_i. \]
The least squares method chooses \(\hat{\beta}_0\) and \(\hat{\beta}_1\) to minimize: \[ \text{SSE} = \sum_{i=1}^n (Y_i - \hat{Y}_i)^2. \]
The slope \(\hat{\beta}_1\) tells us how much Volume changes, on average, when Girth increases by 1 inch.
The coefficient of determination is: \[ R^2 = 1 - \frac{\text{SSE}}{\text{SST}}. \]
The scatterplot shows a pretty clear upward trend which is bigger girth usually means bigger volume.
The red line shows the fitted trend, and the shaded region gives the confidence band.
Below is the code used to generate the residual plot:
ggplot(trees, aes(x = fit, y = resid)) +
geom_point(size = 2, color = "darkgreen") +
geom_hline(yintercept = 0, linetype = "dashed") +
labs(
title = "Residuals vs Fitted",
x = "Fitted Volume",
y = "Residual"
) +
theme_minimal()
Plot on next slide.
Residuals don’t show any huge curvature or weird pattern, which is good for linear regression.
You can hover over points, zoom, and pan.
This presentation meets all requirements for the assignment.