Return your attention to the cats data frame in package MASS. In the first few problems in Exercise 21.1, you fitted the main-effect-only model to predict the heart weights of domestic cats by total body weight and sex.
a. Fit the model again, and this time include an interaction between the two predictors. Inspect the model summary. What do you notice in terms of the parameter estimates and their significance when compared to the earlier main-effect-only version?
b. Produce a scatterplot of heart weight on body weight, using different point characters or colors to distinguish the observations according to sex. Use abline to add two lines denoting the fitted model. How does this plot differ from the one in Exercise 21.1 (d)?
c. Predict the heart weight of Tilman’s cat using the new model (remember that Sigma is a 3.4 kg female) accompanied by a 95 percent prediction interval. Compare it to the main-effectsonly model from the earlier exercise.
you accessed the trees data frame in the contributed faraway package. After loading the package, access the ?trees help file; you’ll find the volume and girth measurements you used earlier, as well as data on the height of each tree.
d. Without using any transformations of the data, fit and inspect a main-effects-only model for predicting volume from girth and height. Then, fit and inspect a second version of this model including an interaction.
e. Repeat (d), but this time use the log transformation of all variables. What do you notice about the significance of the interaction between the untransformed and transformed models? What does this suggest about the relationships in the data?
Turn back to the mtcars data set and remind yourself of the variables in the help file ?mtcars.
f. Fit a linear model for mpg based on a two-way interaction between hp and factor(cyl) and their main effects, as well as a main effect for wt. Produce a summary of the fit.
g. Interpret the estimated coefficients for the interaction between horsepower and the (categorical) number of cylinders. h. Suppose you’re keen on purchasing a 1970s performance car. Your mother advises you to purchase a “practical and economical” car that’s capable of an average MPG value of at least 25. You see three vehicles advertised: car 1 is a four-cylinder, 100 horsepower car that weighs 2100 lbs; car 2 is an eight-cylinder, 210 horsepower car that weighs 3900 lbs; and car 3 is a six-cylinder, 200 horsepower car that weighs 2900 lbs.
i. Use your model to predict the mean MPG for each of the three cars; provide 95 percent confidence intervals. Based on your point estimates only, which car would you propose to your mother?
ii. You still want the most gas-guzzling car you can own with your mother’s blessing, so you decide to be sneaky and base your decision on what the confidence intervals tell you instead. Does this change your choice of vehicle?