In this set of Exercises, we will explore linear regression, variable selection, model diagnostics and prediction.
A.1 Open a new R script and save it in the directory you created in Part A.1 of Exercise 1. Then, load the Auto MPG data set with the additional variable diesel using the load() function and file path from the end of Exercise 1.
A.2 Regress MPG on horsepower and look at the model results with the summary() function. Interpret the meaning of the coefficient of horsepower.
A.3 Plot the model diagnostics. Do you think this model fits the data adequately? Why or why not?
B.1 Add in a quadratic term for horsepower and look at the model fit results. HINT: Use the indicator function I() along with update().
B.2 Plot the model diagnostics. Do you think this model fits the data adequately? Why or why not?
B.3 Compare the model from Part A to the model you just fit using an F-test. What model do you conclude fits the data better?
B.4 Make a scatterplot of mpg versus horsepower. Add the estimated regression line from Part A using the abline() function and color it red. Add in the estimated regression line from Part B and color it blue. HINT: You will need to use the predict() and curve() functions, i.e.,
plot()
abline(, col = "red", lwd = 2)
curve(predict(quadFit, data.frame(hp=x)),
add=TRUE, col="blue", lwd=2)
legend("topright", legend=c("Linear Model", "Quadratic Model"),
col=c("red", "blue"), lty=1, lwd=2, bty="n")C.1 Using what you’ve learned so far, fit the best possible linear model you can to predict MPG. Answers will vary. HINT: You can use automatic variable selection methods, or do so manually and compare models via adjusted \(R^2\) and F-tests.
C.2 Using the model you just fit, predict the fuel economy of the 8 vehicles with missing mpg values.
Write a function that performs linear contrasts. E.g., takes in the fitted linear regression object and a vector containing the difference in two predictor vectors and ourputs a point estimate and 95% confidence interval. Specifically, do this comparing (1) two similar vehicles that differ by 10 model years, and (2) two similar vehicles that differ by 500 pounds. NOTE: You may not have included these variables in your final model. If you didn’t, chose two other variables. HINT: This problem is difficult! You will need to get the covariance matrix on the coefficients with vcov() and perform some matrix multiplications using the %*% operator. Ask for help if you make it this far!