Introduction to Categorical Data Analysis

PROBLEMS 133
4.26 Model (4.11) for the probability π of a satellite for horseshoe crabs with color
and width predictors has fit
logit( ˆπ) =−12.715 + 1.330c1 + 1.402c2 + 1.106c3 + 0.468x
Consider this fit for crabs of width x = 20 cm.
a. Estimate π for medium-dark crabs (c3 = 1) and for dark crabs (c1 = c2 =
c3 = 0). Then, estimate the ratio of probabilities.
b. Estimate the odds of a satellite for medium-dark crabs and the odds for
dark crabs. Show that the odds ratio equals exp(1.106) = 3.02. When each
probability is close to zero, the odds ratio is similar to the ratio of probabilities,
providing another interpretation for logistic regression parameters.
For widths at which ˆπ is small, ˆπ for medium-dark crabs is about three
times that for dark crabs.
4.27 The prediction equation for the horseshoe crab data using width and quantitative
color (scores 1, 2, 3, 4) is logit( ˆπ) =−10.071 − 0.509c + 0.458x. Color
has mean = 2.44 and standard deviation = 0.80, and width has mean = 26.30
and standard deviation = 2.11.
a. For standardized versions of the predictors, explain why the estimated coefficients
equal (0.80)(−.509) =−0.41 and (2.11)(.458) = 0.97. Interpret
these by comparing the partial effects on the odds of a one standard deviation
increase in each predictor.
b. Section 4.5.1 interpreted the width effect by finding the change in ˆπ
over the middle 50% of width values, between 24.9 cm and 27.7 cm.
Do this separately for each value of c, and interpret the width effect for
each color.
4.28 For recent General Social Survey data, a prediction equation relating Y =
whether attended college (1 = yes) to x = family income (thousands of dollars,
using scores for grouped categories), m = whether mother attended
college (1 = yes, 0 = no), f = whether father attended college (1 = yes,
0 = no), was logit[ P(Y ˆ = 1)] =−1.90 + 0.02x + 0.82m + 1.33f . To summarize
the cumulative effect of the predictors, report the range of ˆπ values
between their lowest levels (x = 0.5, m = 0, f = 0) and their highest levels
(x = 130, m = 1, f = 1).
4.29 Table 4.20 appeared in a national study of 15- and 16-year-old adolescents.
The event of interest is ever having sexual intercourse. Analyze these data and
summarize in a one-page report, including description and inference about the
effects of both gender and race.
4.30 The US National Collegiate Athletic Association (NCAA) conducted a study
of graduation rates for student athletes who were freshmen during the