statisticalrethinkin..

170 5. MULTIVARIATE LINEAR MODELS5.7.7. m3. It is sometimes observed that the best predictor of fire risk is the presence of firefighters—States and localities with many fire fighters also have more fires. Presumably firefighters do not cause fires. Nevertheless, this is not a spurious correlation. Instead fires causefire fighters. Consider the same reversal of causal inference in the context of the divorce andmarriage data. How might a high divorce rate cause a higher marriage rate? Can you thinkof a way to evaluate this relationship, using multiple regression?5.7.8. m4. In the divorce data, States with high numbers of Mormons (members of eChurch of Jesus Christ of Latter-day Saints) have much lower divorce rates than the regressionmodels expected. Find a list of Mormon population by State and use those numbersas a predictor variable, predicting divorce rate using marriage rate, median age at marriage,and percent Mormon population (possibly standardized). You may want to consider transformationsof the raw percent Mormon variable. [I actually don’t know how this regressionturns out. I suspect getting it right requires interacting Mormon population with medianage marriage, which will have to wait until Chapter 7.]5.7.9. m5. One way to reason through multiple causation hypotheses is to imagine detailedmechanisms through which predictor variables may influence outcomes. For example, it issometimes argued that the price of gasoline (predictor variable) is positively associated withlower obesity rates (outcome variable). However, there are at least two important mechanismsby which the price of gas could reduce obesity. First, it could lead to less driving andtherefore more exercise. Second, it could lead to less driving, which leads to less eating out,which leads to less consumption of huge restaurant meals. Can you outline one or more multipleregressions that address these two mechanisms? Assume you can have any predictordata you need.HardAll three exercises below use the same data, data(foxes) (part of rethinking). 69 eurban fox (Vulpes vulpes) is a successful exploiter of human habitat. Since urban foxes movein packs and defend territories, data on habitat quality and population density is also included.e data frame has five columns:(1) group: Number of the social group the individual fox belongs to(2) avgfood: e average amount of food available in the territory(3) groupsize: e number of foxes in the social group(4) area: Size of the territory(5) weight: Body weight of the individual fox5.7.10. Bivariate entrée. Fit two bivariate Gaussian regressions, using map: (1) body weightas a linear function of territory size (area), and (2) body weight as a linear function ofgroupsize. Plot the results of these regressions, displaying the MAP regression line and the95% confidence interval of the mean. Is either variable important for predicting fox bodyweight?5.7.11. Multivariate course. Now fit a multiple linear regression with weight as the outcomeand both area and groupsize as predictor variables. Plot the predictions of the modelfor each predictor, holding the other predictor constant at its mean. What does this modelsay about the importance of each variable? Why do you get different results than you got inexercise just above?