Homework 4

Introduction

This report examines the relationship between mpg and horsepower using the Auto dataset.

Setup

Load Data

library(ISLR)
data(Auto)
Auto <- na.omit(Auto)

Linear Model

model <- lm(mpg ~ horsepower, data = Auto)
summary(model)

## 
## Call:
## lm(formula = mpg ~ horsepower, data = Auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.5710  -3.2592  -0.3435   2.7630  16.9240 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 39.935861   0.717499   55.66   <2e-16 ***
## horsepower  -0.157845   0.006446  -24.49   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.906 on 390 degrees of freedom
## Multiple R-squared:  0.6059, Adjusted R-squared:  0.6049 
## F-statistic: 599.7 on 1 and 390 DF,  p-value: < 2.2e-16

Residuals vs Fitted Plot

plot(model$fitted.values, model$residuals,
     xlab = "Fitted Values",
     ylab = "Residuals",
     main = "Residuals vs Fitted Plot")

abline(h = 0, col = "red", lwd = 2)

Interpretation

• Random scatter → Linear relationship
• U-shape → Nonlinear relationship

Polynomial Model

model_poly <- lm(mpg ~ horsepower + I(horsepower^2), data = Auto)
summary(model_poly)

## 
## Call:
## lm(formula = mpg ~ horsepower + I(horsepower^2), data = Auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.7135  -2.5943  -0.0859   2.2868  15.8961 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     56.9000997  1.8004268   31.60   <2e-16 ***
## horsepower      -0.4661896  0.0311246  -14.98   <2e-16 ***
## I(horsepower^2)  0.0012305  0.0001221   10.08   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.374 on 389 degrees of freedom
## Multiple R-squared:  0.6876, Adjusted R-squared:  0.686 
## F-statistic:   428 on 2 and 389 DF,  p-value: < 2.2e-16

Residual Plot (Polynomial)

plot(model_poly$fitted.values, model_poly$residuals,
     xlab = "Fitted Values",
     ylab = "Residuals",
     main = "Residuals vs Fitted (Polynomial Model)")

abline(h = 0, col = "blue", lwd = 2)

Conclusion

Residual plots help determine whether a linear model is appropriate or if a nonlinear model is needed.