Introduction to Categorical Data Analysis

366 BRIEF SOLUTIONS TO SOME ODD-NUMBERED PROBLEMS
d. Deviance = 3.25, df = 3, P -value = 0.36, so model fits adequately.
e. 1 − cumulative probability for category 2, which is 0.61.
9. a. There are four nondegenerate cumulative probabilities. When all predictor
values equal 0, cumulative probabilities increase across categories, so logits
increase, as do parameters that specify logits.
b. (i) Religion = none, (ii) Religion = Protestant.
c. For Protestant, 0.09. For None, 0.26.
d. (i) e −1.27 = 0.28; that is, estimated odds that Protestant falls in relatively
more liberal categories (rather than more conservative categories) is 0.28 times
estimated odds for someone with no religious preference. (ii) Estimated odds
ratio comparing Protestants to Catholics is 0.95.
11. a. ˆβ =−0.0444 (SE = 0.0190) suggests probability of having relatively less
satisfaction decreases as income increases.
b. ˆβ =−0.0435, very little change. If model holds for underlying logistic latent
variable, model holds with same effect value for every way of defining
outcome categories.
c. Gender estimate of −0.0256 has SE = 0.4344 and Wald statistic = 0.003
(df = 1), so can be dropped.
13. a. Income effect of 0.389 (SE = 0.155) indicates estimated odds of higher of
any two adjacent job satisfaction categories increases as income increases.
b. Estimated income effects are −1.56 for outcome categories 1 and 4, −0.64
for outcome categories 2 and 4, and −0.40 for categories 3 and 4.
c. (a) Treats job satisfaction as ordinal whereas (b) treats job satisfaction as
nominal. Ordinal model is more parsimonious and simpler to interpret,
because it has one income effect rather than three.
17. Cumulative logit model with main effects of gender, location, and seat-belt
has estimates 0.545, −0.773, and 0.824; for example, for those wearing a seat
belt, estimated odds that the response is below any particular level of injury are
e 0.824 = 2.3 times the estimated odds for those not wearing seat belts.
21. For cumulative logit model of proportional odds form with Y = happiness and
x = marital status (1 = married, 0 = divorced), ˆβ =−1.076 (SE = 0.116). The
model fits well (e.g., deviance = 0.29 with df = 1).
CHAPTER 7
1. a. G2 = 0.82, X2 = 0.82, df = 1.
b. ˆλ Y 1 = 1.416, ˆλ Y 2 = 0. Given gender, estimated odds of belief in afterlife equal
e1.416 = 4.1.