Overview

Using the “cars” dataset in R, build a linear model for stopping distance as a function of speed and replicate the analysis of your textbook chapter 3 (visualization, quality evaluation of the model, and residual analysis.)

Data Exporation

head(cars)
##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10
summary(cars)
##      speed           dist       
##  Min.   : 4.0   Min.   :  2.00  
##  1st Qu.:12.0   1st Qu.: 26.00  
##  Median :15.0   Median : 36.00  
##  Mean   :15.4   Mean   : 42.98  
##  3rd Qu.:19.0   3rd Qu.: 56.00  
##  Max.   :25.0   Max.   :120.00

Model Building

lm_mod <- lm(speed ~ dist, data = cars)

Visualization

plot(cars$dist, cars$speed, xlab = "Stopping Distance", ylab = "Speed")
abline(lm_mod)

Evaluation

summary(lm_mod)
## 
## Call:
## lm(formula = speed ~ dist, data = cars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.5293 -2.1550  0.3615  2.4377  6.4179 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  8.28391    0.87438   9.474 1.44e-12 ***
## dist         0.16557    0.01749   9.464 1.49e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.156 on 48 degrees of freedom
## Multiple R-squared:  0.6511, Adjusted R-squared:  0.6438 
## F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.49e-12

So, we have a formula

\[ y = 8.28 + 0.16(dist) \]

This simple linear model has R-squared of 0.65 meaning the model explains 65% of the data. Looking at the residual statistics, it has seemingly center around around and the iqr is even.

Residual Analysis

plot(lm_mod$fitted.values, lm_mod$residuals, xlab = "Fitted", ylab = "Residuals")
abline(h = 0)

qqnorm(lm_mod$residuals)
qqline(lm_mod$residuals)

The qq plot tells us that the most of the residual follow the theoretical normal where we see some of the points on top deviate from the line.

par(mfrow = c(2, 2))
plot(lm_mod)