19.12 Group specific parameters

• Climb model 2:

\[ log(\frac{\pi_{ijk}}{1-\pi_{ijk}})=(\beta_0 + b_{0j} + p_{0k}) + \beta_1 X_{ijk1} + \beta_2 X_{ijk2} \]

• For each of the 200 sampled expeditions and 46 sampled peaks, we have associated posterior distributions for the \(b_{0j}\) and \(p_{0k}\)

group_levels_2 <- tidy(climb_model_2, effects = "ran_vals") %>% 
  select(level, group, estimate)

• Example tweaks for peaks:

group_levels_2 %>% 
  filter(group == "peak_name") %>% 
  head(2)

# A tibble: 2 × 3
  level       group     estimate
  <chr>       <chr>        <dbl>
1 Ama_Dablam  peak_name     2.91
2 Annapurna_I peak_name    -2.05

• Example tweaks for expedition:

group_levels_2 %>% 
  filter(group == "expedition_id") %>% 
  head(2)

# A tibble: 2 × 3
  level     group         estimate
  <chr>     <chr>            <dbl>
1 AMAD03107 expedition_id -0.00442
2 AMAD03327 expedition_id  3.36 

• For example, to predict success probability for a climber new expedition to an existing peak (say Ama_Dablam, an ‘easy one’), we would would use \(b=0\) and the appropriate \(p_{0j}=2.91\). If the climber were 30 years old but did not plan to use oxygen:

logit_p = -1.53 +  2.91 - .0474*30 

invlogit(logit_p)

## [1] 0.4895015

49% chance of success. Could also use posterior_prediction

new_expedition <- data.frame(
  age = c(30), oxygen_used = c(FALSE), 
  expedition_id = c("new"), peak_name = c("Ama_Dablam") )

binary_prediction <- posterior_predict(climb_model_2, newdata = new_expedition)

mean(binary_prediction)

# [1] 0.49425

If they did use oxygen:

logit_p = -1.53 +  2.91 - .0474*30  + 6.2

invlogit(logit_p)

## [1] 0.997888

With prediction:

new_expedition <- data.frame(
  age = c(30), oxygen_used = c(TRUE), 
  expedition_id = c("new"), peak_name = c("Ama_Dablam") )

binary_prediction <- posterior_predict(climb_model_2, newdata = new_expedition)

mean(binary_prediction)
# [1] 0.95185

Why different? invlogit is nonlinear (but monotonic)