Introduction to Categorical Data Analysis

PROBLEMS 123
b. Construct a Wald confidence interval for the odds ratio corresponding to a
1-unit increase in LI. Interpret.
c. Conduct a likelihood-ratio test for the LI effect. Interpret.
d. Construct the likelihood-ratio confidence interval for the odds ratio.
Interpret.
4.3 In the first nine decades of the twentieth century in baseball’s National League,
the percentage of times the starting pitcher pitched a complete game were:
72.7 (1900–1909), 63.4, 50.0, 44.3, 41.6, 32.8, 27.2, 22.5, 13.3 (1980–1989)
(Source: George Will, Newsweek, April 10, 1989).
a. Treating the number of games as the same in each decade, the linear
probability model has ML fit ˆπ = 0.7578 − 0.0694x, where x = decade
(x = 1, 2,...,9). Interpret −0.0694.
b. Substituting x = 12, predict the percentage of complete games for 2010–
2019. Is this prediction plausible? Why?
c. The logistic regression ML fit is ˆπ = exp(1.148 − 0.315x)/[1 +
exp(1.148 − 0.315x)]. Obtain ˆπ for x = 12. Is this more plausible than
the prediction in (b)?
4.4 Consider the snoring and heart disease data of Table 3.1 in Section 3.2.2.
With scores {0, 2, 4, 5} for snoring levels, the logistic regression ML fit is
logit( ˆπ) =−3.866 + 0.397x.
a. Interpret the sign of the estimated effect of x.
b. Estimate the probabilities of heart disease at snoring levels 0 and 5.
c. Describe the estimated effect of snoring on the odds of heart disease.
4.5 For the 23 space shuttle flights before the Challenger mission disaster in 1986,
Table 4.10 shows the temperature ( ◦ F) at the time of the flight and whether at
least one primary O-ring suffered thermal distress.
a. Use logistic regression to model the effect of temperature on the probability
of thermal distress. Interpret the effect.
b. Estimate the probability of thermal distress at 31 ◦ F, the temperature at the
time of the Challenger flight.
c. At what temperature does the estimated probability equal 0.50? At that
temperature, give a linear approximation for the change in the estimated
probability per degree increase in temperature.
d. Interpret the effect of temperature on the odds of thermal distress.
e. Test the hypothesis that temperature has no effect, using (i) the Wald test,
(ii) the likelihood-ratio test.
4.6 Refer to Exercise 3.9. Use the logistic regression output reported there to (a)
interpret the effect of income on the odds of possessing a travel credit card,
and conduct a (b) significance test and (c) confidence interval about that effect.