Table 9.7 displays associations among smoking status (S), breathing test results (B), and age (A) for workers in certain industrial plants. Treat B as a response.

a. Specify a baseline-category logit model with additive factor effects of S and A. This model has deviance G² = 25.9. Show that df = 4, and explain why this model treats all variables as nominal.

b. Treat B as ordinal and S as ordinal in terms of how recently one was a smoker, with scores {s_i}. Consider the model

with a₁ = 0 and a₂ = 1. Show that this assumes a linear effect of S with slope β₁ for age < 40 and β₁ + β₃ for age 40 €“ 59. Using {s_i = i}, β = 0.115, β̂₂ = 0.311, and β̂₃ = 0.663 (SE = 0.164). Interpret the interaction.

c. From part (b), for age 40 €“ 59 show that the estimated odds of abnormal rather than borderline breathing for current smokers are 2.18 times those for former smokers and exp(2 × 0.778) = 4.74 times those for never smokers. Explain why the squares of these values are estimated odds of abnormal rather than normal breathing.

Table 9.7:

P(B = k + 1|S = i, A = j) P(B = k|S - i, A = j) az + B,s, + Bza, + B35,a; log Breathing Test Results Smoking Status Never smoked Former smoker Current smoker Never smoked Former smoker Current smoker Normal 577 192 Abnormal Age Borderline 27 20 < 40 3 11 46 4 15 47 682 40-59 164 145 245 27