In Example 7.5 (attitudes to working mothers) compare inferences from the residuals W_i − X_iβ with those based on Monte Carlo estimates of the conditional predictive ordinates (harmonic means of the sampled normal likelihoods for each subject). In Example 7.5 apply an ordered logistic model by data augmentation by direct sampling from a logistic and by sampling from a normal using scale mixing with an appropriate degrees of freedom.

Augmented data model for attitudes to working mothers Long (1997) presents maximum likelihood ordinal probit and ordinal logit analysis of data from two US General Social Surveys (1977 and 1989). The response relates to the question ‘A working mother can establish just as warm and secure a relationship with her children as a mother who does not work’, with responses y_i ∈ (1,… 4), namely, 1 = stronglydisagree,2 = disagree, 3 = agree, and 4 = strongly agree. Predictors are yr89(= 1 for later survey), gender (1 = male), ethnicity (1 = white, 0 = other), age, years of education and occupational prestige. Here an ordinal probit model using data augmentation is applied with a constant included in the regression and so only two free cutpoints. Priors on the latter are appropriately constrained to reflect sampled W_i values. The input data are ordered by values of y so that the constraints can be easily expressed. N(0, 10) priors are assumed on the intercept and binary predictor coefficients, but N(0, 1) priors taken on the coefficients of the continuous predictors (which are centred) to avoid numerical problems. The second half of a two chain 5000 iteration run produces similar estimates to those reported by Long (1997, p. 127), except that there seems to be a only a small gap between the first two cutpoints. The negative intercept is equivalent to κ₁ and has posterior mean −0.66(−0.86, −0.46), while κ₂ has mean −0.64(−0.83, −0.45), and κ₃ has mean 2 (0.1, 4.0). Less favourable attitudes to mothers working occur among men and older people, while favourable attitudes increase with education and prestige. A significant effect for prestige of 0.0057 (0.0015, 0.01) contrasts with the marginally significant effect reported by Long (1997), while the effect of white ethnicity is not quite significant whereas Long (1997) finds it to be a significantly negative predictor of favourable attitude.