Objective

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 Visualization

# cars data
head(cars)
##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10
plot(cars)

The simplest regression model is a straight line. It has the mathematical form:

\(\ y = a_{0} + a_{1}x_{1}\)

where \(\ x_{1}\) is the input to the system, \(\ a_{0}\) is the y-intercept of the line, \(\ a_{1}\) is the slope, and y is the output value the model predicts.

The y intercept is \(\ a_{0} = -17.579\)

The slope is \(\ a_{1} = 3.932\)

lm1 = lm(cars$dist~cars$speed)
lm1
## 
## Call:
## lm(formula = cars$dist ~ cars$speed)
## 
## Coefficients:
## (Intercept)   cars$speed  
##     -17.579        3.932
plot(cars$speed, cars$dist)
abline(lm1)

Evaluating the quality of the model

summary(lm1)
## 
## Call:
## lm(formula = cars$dist ~ cars$speed)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -29.069  -9.525  -2.272   9.215  43.201 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -17.5791     6.7584  -2.601   0.0123 *  
## cars$speed    3.9324     0.4155   9.464 1.49e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.38 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
mean(resid(lm1))
## [1] 8.65974e-17

Min = -29.069 Max = 43.201 Median = -2.272 Mean = 8.65974e-17

The line is a good fit with the data, we would expect residual values that are normally distributed around a mean of zero. Giving the mean value of 8.65974e-17 which is quite close to zero, we can conclude this line is a good fit with the data.

Residual Analysis

plot(fitted(lm1),resid(lm1))

We plot the residual values, we would expect to see them distributed uniformly around zero for a well-fitted model. In this scenario, we see the data points distributed uniformly around zero and therefore is a well-fitted model.

qqnorm(resid(lm1))
qqline(resid(lm1))

Using the Q-Q plot for the model, we find that the points plotted in this figure does follow a straight line. This behavior indicates the residuals are normally distributed.