What, Why Model

• Predicts numeric \(y\) from numeric \(x\)
• Interprets the slope as expected change in \(y\) per +1 in \(x\)

  

\[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \qquad \varepsilon_i^{\text{iid}} \sim \mathcal{N}(0,\sigma^2). \]

\[ \hat{\beta}_1 = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})} {\sum_{i=1}^{n}(x_i-\bar{x})^2}, \qquad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}. \]

For the dataset I chose the “diamonds” dataset used in previous lectures. Here I fit in a SLR of price of carat and the price increase.

data("diamonds")
smaller <- subset(diamonds, carat <= 3)
mod_d <- lm(price ~ carat, data = smaller)
summary(mod_d)

## 
## Call:
## lm(formula = price ~ carat, data = smaller)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14644.6   -813.9     -9.8    560.6  12711.3 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2301.41      12.97  -177.4   <2e-16 ***
## carat        7819.33      14.03   557.3   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1531 on 53906 degrees of freedom
## Multiple R-squared:  0.8521, Adjusted R-squared:  0.8521 
## F-statistic: 3.106e+05 on 1 and 53906 DF,  p-value: < 2.2e-16

Here I showed the relationship between carat and price using a scatter plot with a linear regression line.

ggplot(smaller, aes(x = carat, y = price)) +
geom_point(alpha = 0.25) +
geom_smooth(method = "lm", se = FALSE, color = "darkred") +
labs(title = "Diamonds: Price vs Carat",
x = "Carat", y = "Price (USD)")

With residual standard error \(s\) and \(SS_x=\sum_{i=1}^{n}(x_i-\bar{x})^2\):

\[ SE(\hat{\beta}_1)=\sqrt{\frac{s^2}{SS_x}}, \qquad t=\frac{\hat{\beta}_1}{SE(\hat{\beta}_1)}, \qquad \text{CI: } \hat{\beta}_1 \pm t_{1-\alpha/2,\,n-2}\, SE(\hat{\beta}_1). \]

y <- smaller$price
x <- smaller$carat
n  <- length(y)
SSx <- sum( (x - mean(x))^2 )
b1  <- sum( (x - mean(x)) * (y - mean(y)) ) / SSx
b0  <- mean(y) - b1*mean(x)
e   <- y - (b0 + b1*x)
s2  <- sum(e^2)/(n-2); s <- sqrt(s2)
SEb1 <- sqrt(s2/SSx)
tcrit <- qt(p = 0.05, df = n-2, lower.tail = FALSE)  # ~95% two-sided
CI_b1 <- c(b1 - tcrit*SEb1, b1 + tcrit*SEb1)
### Print
b0; b1; SEb1; CI_b1

## [1] -2301.412

## [1] 7819.326

## [1] 14.03087

## [1] 7796.247 7842.405

ggplot(smaller, aes(carat, price)) +
geom_point(alpha = 0.25) +
geom_smooth(se = TRUE) +
labs(title = "Visual check: is the linear trend reasonable? Carat vs Price",
x = "Carat", y = "Price (USD)")

data(trees)
m <- lm(Volume ~ Girth, data = trees)
plot_ly(trees, x = ~Girth, y = ~Volume,
type = "scatter", mode = "markers", name = "data") %>%
add_lines(x = ~Girth, y = ~fitted(m), name = "fitted") %>%
layout(title = "Trees: Volume vs Girth (interactive)")

Assumptions:

Knowing gold carats and prices going up as the carat went up, I assumed the same for diamonds and so wanted to put it on the test.

• Linearity: the mean of \(y\) changes linearly with \(x\).

• Constant variance: residual spread is roughly constant across carat.

Takeaways:

The carat and price relationship is strong, positive, and approximately linear in the subset. The slope with the CI justified the assumptions

• Price–carat relationship is strong and positive (R² ≈ 0.85).

• Slope \(\hat\beta_1 \approx 7819\): ~$7.8k per +1 carat (subset).