12.11 Negative Binomial Regression

\[\begin{array}{rcl} Y_{i} | \beta_{0}, \beta_{1}, \beta_{2}, r & \sim & \text{NegBin}(\mu_{i}) \\ \beta_{0c} & \sim & \text{N}(2, 0.5^{2}) \\ \beta_{1} & \sim & \text{N}(0, 0.15^{2}) \\ \beta_{2} & \sim & \text{N}(0, 5.01^{2}) \\ r & \sim & \text{Exp}(1) \\ \end{array}\]

books_negbin_sim <- stan_glm(
  books ~ age + wise_unwise, 
  data = pulse, family = neg_binomial_2,
  prior_intercept = normal(0, 2.5, autoscale = TRUE),
  prior = normal(0, 2.5, autoscale = TRUE), 
  prior_aux = exponential(1, autoscale = TRUE),
  chains = 4, iter = 5000*2, seed = 84735)

12.11.1 Prior Distribution

prior_summary(books_negbin_sim)

## Priors for model 'books_negbin_sim' 
## ------
## Intercept (after predictors centered)
##  ~ normal(location = 0, scale = 2.5)
## 
## Coefficients
##   Specified prior:
##     ~ normal(location = [0,0], scale = [2.5,2.5])
##   Adjusted prior:
##     ~ normal(location = [0,0], scale = [0.15,5.01])
## 
## Auxiliary (reciprocal_dispersion)
##  ~ exponential(rate = 1)
## ------
## See help('prior_summary.stanreg') for more details

12.11.2 Posterior Predictive Check

pp_check(books_negbin_sim) + 
  xlim(0, 75) + 
  xlab("books")

12.11.3 Interpretation

tidy(books_negbin_sim, conf.int = TRUE, conf.level = 0.80)

## # A tibble: 3 × 5
##   term                        estimate std.error conf.low conf.high
##   <chr>                          <dbl>     <dbl>    <dbl>     <dbl>
## 1 (Intercept)                 2.23       0.131    2.07      2.41   
## 2 age                         0.000365   0.00239 -0.00270   0.00339
## 3 wise_unwiseWise but Unhappy 0.266      0.0798   0.162     0.368