In Example 4.3, apply the existing ZIP regression, but using a mixed predictive check based on replicate \(t_{r e p, i} \sim \operatorname{Bern}(\zeta)\). This should show the proportion of underpredicted cases to be around \(4 \%\).

Data from Example 4.3

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Data analysed by Deb and Trivedi (1997) obtained from National Medical Expenditure Survey (NMES) have been shown to contain excess zeroes; see also Zeileis et al. (2008). The num- ber of hospitalizations by n = 4406 subjects is the dependent variable (3541 subjects have response 0), and explanatory variables relate to health status namely excellent health (binary); poor health (binary), and number of chronic conditions; demographic status (age and male); and socio-economic status, namely years of education and private health insurance (binary). The zero inflated Poisson regression (without regression to predict subject varying ;) has core code model { for (i in 1:4406) (y[i] ~dpois (mu.c[i]); mu.c[i] < (1-t [i]) *mu [i]; t[i] ~ dbern (zeta) log (mu [i])