• Simple linear regression is a statistical method that uses a linear equation to predict the value of one dependent variable based on another independent variable.
• It helps us understand how one variable (predictor) affects another (outcome)
• Let’s explore this with an example.
• Example: How does House area (sq ft) influence house price ($k)?
• We will:
  • Simulate a small housing data
  • Fit a linear model
  • Visualize results with ggplot2 and Plotly

set.seed(42)
n <- 120
area <- round(rnorm(n, mean = 1800, sd = 450))
area[area < 600] <- 600
price <- 100 + 0.15 * area + rnorm(n, 0, 30)

housing <- data.frame(area, price)
head(housing)

  area    price
1 2417 417.7412
2 1546 287.7869
3 1963 398.1911
4 2085 382.8508
5 1982 397.2453
6 1752 349.9522

Let’s fit a simple linear regression:

\[ \text{Price}_i = \beta_0 + \beta_1 \,\text{Area}_i + \varepsilon_i, \quad \varepsilon_i \sim \mathcal{N}(0,\sigma^2) \]

• \(\beta_0\): intercept (price when area = 0)
  
• \(\beta_1\): slope (change in price per 1 sq ft)

A scatter plot is a graph that displays the relationship between two quantitative variables using points. One variable is on the x-axis and the other on the y-axis.

m <- lm(price ~ area, data = housing)

ggplot(housing, aes(x = area, y = price)) +
geom_point(color = "#e5a0c6") + 
geom_smooth(method = "lm", se = TRUE, color = "#7f3667") +
labs(title = "Housing Prices vs Area",
x = "Area (sq ft)", y = "Price ($K)") +
theme_classic()

A residual plot shows the difference between the observed and predicted values (residuals) on the y-axis and the fitted values on the x-axis, helping us check how well our regression model fits the data.

housing$fitted <- fitted(m)
housing$resid  <- resid(m)

ggplot(housing, aes(x = fitted, y = resid)) +
geom_point(color = "#7f3667") +
geom_hline(yintercept = 0, linetype = "dashed") +
labs(title = "Residuals vs Fitted Values",
x = "Fitted Price ($K)", y = "Residuals") +
theme_minimal()

An interactive plot lets users explore data by hovering, zooming, or rotating elements, making it easier to visualize and understand relationships dynamically.

plot_ly(housing, x = ~area, y = ~price,
type = "scatter", mode = "markers",
marker = list(color = '#e5a0c6', size = 8)) %>%
add_lines(x = housing$area,
y = fitted(m),
line = list(color = '#7f3667')) %>%
layout(title = "Interactive Price vs Area",
xaxis = list(title = "Area (sq ft)"),
yaxis = list(title = "Price ($K)"))

The slope’s \(100(1-\alpha)\%\) confidence interval:

\[ \hat\beta_1 \pm t_{n-2,\,1-\alpha/2}\,\mathrm{SE}(\hat\beta_1) \]

Test \(H_0:\ \beta_1=0\) with the \(t\)-statistic: \[ t = \frac{\hat\beta_1}{\mathrm{SE}(\hat\beta_1)} \]