library(performance)
## Warning: package 'performance' was built under R version 4.1.3

Instructions

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.)

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

To start, we plot speed on the x-axis and stopping distance on the y-axis for a quick check on a potential linear relationship.

plot(cars[,"speed"],cars[,"dist"], main="cars",
xlab="Speed", ylab="Stopping Distance")

We see a linear relationship, if an imperfect one. Let’s construct a linear model:

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

Initial look at the model:

cars_lm
## 
## Call:
## lm(formula = speed ~ dist, data = cars)
## 
## Coefficients:
## (Intercept)         dist  
##      8.2839       0.1656

Let’s add the abline from our metric to that initial plot we generated:

plot(speed ~ dist, data=cars)
abline(cars_lm)

Model performance metrics:

summary(cars_lm)
## 
## 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

Our coefficient for our stopping distance is roughly 10x the standard error, which is a good sign, as is the y-intercept coefficient and SE. The very low p-value shows very strong evidence of a linear relationship between speed and stopping distance. the R-squared value shows that roughly 65% of variation in stopping distance can be explained by speed.

Now we analyze residuals and other assumptions for effective linear regression.

plot(fitted(cars_lm),resid(cars_lm))

Residuals seem to vary uniformly - there’s no clear patter of variance.

qqnorm(resid(cars_lm))
qqline(resid(cars_lm))

Q-Q plot’s straight line shows that residuals are normally distributed around a mean of 0.

The performance package in R includes a nice check_model function that outputs performance metrics like discussed above, and they automatically vary based on the time of model used.

check_model(cars_lm)

For a real world model, we would have obviously done a train-test split and saved sample data for testing the model built on our training data.