ordinal t_test vs wilcox vs ordinal_regression.R

# t-test vs wilcox vs ordinal

library(tidyverse)
library(multidplyr) # parallelize
library(rms) # ordinal regression


sample_size = 500 # N

# generate paired sets and calculate p-values with different techniques
# reduce the iterations number to reduce time
simulate = function(iterations, seed = NULL) {
  set.seed(seed)
  
  breaks = qnorm(cumsum(c(0, 0.2, 0.25, 0.18, 0.12, 0.1, 0.08, 0.07)))
  
  replicate(iterations, (function() {
    
    #### data with skewed distribution that changes after treatment
    data = tibble(
      subject = paste0("S", 1:sample_size),
      # sample ordinal values with a made-up distribution
      before = rnorm(sample_size),
      #
      after = sqrt(0.8)*before + sqrt(.1)*scale(before^2,center = T) + sqrt(0.1)*rnorm(sample_size, mean=0.15)
    ) %>% 
      # cut into ordinal:
      mutate(after = as.numeric(cut(after,breaks=breaks)),
             before = as.numeric(cut(before,breaks=breaks)))
    
    #### NULL effect
    data_null = tibble(
      subject = paste0("S", 1:sample_size),
      # sample ordinal values with a uniform distribution
      before = rnorm(sample_size),
      # change after treatment is noisy
      after = sqrt(0.9)*before + sqrt(0.1)*rnorm(sample_size)
    ) %>% 
      # cunt into semetric ordinal
      mutate(after = as.numeric(cut(after,breaks=breaks)),
             before = as.numeric(cut(before,breaks=breaks)))
    
    data_long = pivot_longer(data, before:after, names_to = "time", values_to = "measurement")
    data_long_null = pivot_longer(data_null, before:after, names_to = "time", values_to = "measurement")
    
    tibble(
      p_t = t.test(data$before, data$after, paired = TRUE)$p.value,
      p_w = wilcox.test(data$before, data$after, paired = TRUE)$p.value,
      p_o = rms::orm(measurement ~ time, data_long, x=TRUE, y=TRUE) %>% 
        rms::robcov(cluster = data_long$subject) %>% 
        anova() %>%
        as_tibble() %>% mutate(p = as.numeric(P)) %>% pull(p) %>% pluck(1),
      p_t_null = t.test(data_null$before, data_null$after, paired = TRUE)$p.value,
      p_w_null = wilcox.test(data_null$before, data_null$after, paired = TRUE)$p.value,
      p_o_null = rms::orm(measurement ~ time, data_long_null, x=TRUE, y=TRUE) %>% 
        rms::robcov(cluster = data_long_null$subject) %>% 
        anova() %>%
        as_tibble() %>% mutate(p = as.numeric(P)) %>% pull(p) %>% pluck(1)
    )
  })(), simplify = FALSE) %>% bind_rows()
}






# setup for parallel run
my_cluster = multidplyr::new_cluster(parallel::detectCores() - 1)
cluster_library(my_cluster, "rms")
cluster_library(my_cluster, "magrittr")
cluster_library(my_cluster, "dplyr")
cluster_library(my_cluster, "tidyr")
cluster_library(my_cluster, "purrr")

# copy variables to every cluster and run in parallel
start_time = Sys.time()
cluster_copy(my_cluster, "sample_size")
cluster_copy(my_cluster, "simulate")
cluster_assign(my_cluster, iterations = round(10000 / length(my_cluster)))
cluster_assign_each(my_cluster, seed = 8675309 + 1:length(my_cluster))
ps = cluster_call(my_cluster, simulate(iterations, seed))
ps = ps %>% bind_rows()
Sys.time() - start_time
beepr::beep(2)


##### results #####

# what percent of p-values are below 0.05
ps %>% 
  summarise(
    t_test = mean(p_t<0.05) %>% scales::label_percent(.1)(),
    wilcox_test = mean(p_w<0.05) %>% scales::label_percent(.1)(),
    ordinal_regression = mean(p_o<0.05) %>% scales::label_percent(.1)(),
  )
# t_test wilcox_test ordinal_regression
# 73.6%  73.8%       69.7%  



# t-test vs ordinal
ps %>% 
  mutate(t_yes_o_no = (p_t<0.05) & (p_o>0.05)) %>% 
  mutate(t_no_o_yes = (p_t>0.05) & (p_o<0.05)) %>% 
  summarise(
    t_yes_o_no = mean(t_yes_o_no) %>% scales::label_percent(.1)(),
    t_no_o_yes = mean(t_no_o_yes) %>% scales::label_percent(.1)()
  )
# t_yes_o_no t_no_o_yes
# 7.8%       3.9%      

# t-test vs wilcox
ps %>% 
  mutate(t_yes_w_no = (p_t<0.05) & (p_w>0.05)) %>% 
  mutate(t_no_w_yes = (p_t>0.05) & (p_w<0.05)) %>% 
  summarise(
    t_yes_w_no = mean(t_yes_w_no) %>% scales::label_percent(.1)(),
    t_no_w_yes = mean(t_no_w_yes) %>% scales::label_percent(.1)()
  )
# t_yes_w_no t_no_w_yes
# 0.2%       0.5%      

# wilcox vs ordinal
ps %>% 
  mutate(w_yes_o_no = (p_w<0.05) & (p_o>0.05)) %>% 
  mutate(w_no_o_yes = (p_w>0.05) & (p_o<0.05)) %>% 
  summarise(
    w_yes_o_no = mean(w_yes_o_no) %>% scales::label_percent(.1)(),
    w_no_o_yes = mean(w_no_o_yes) %>% scales::label_percent(.1)()
  )
# w_yes_o_no w_no_o_yes
# 7.9%       3.7%      


# t-test vs ordinal EXTREME
ps %>% 
  mutate(t_yes_o_no = (p_t<0.01) & (p_o>0.05)) %>% 
  mutate(t_no_o_yes = (p_t>0.05) & (p_o<0.01)) %>% 
  summarise(
    t_yes_o_no = mean(t_yes_o_no) %>% scales::label_percent(.1)(),
    t_no_o_yes = mean(t_no_o_yes) %>% scales::label_percent(.1)()
  )
# t_yes_o_no t_no_o_yes
# 1.0%       0.1%      




##### NULL results

# NULL: what percent of p-values are below 0.05
ps %>% 
  summarise(
    t_test = mean(p_t_null<0.05) %>% scales::label_percent(.1)(),
    wilcox_test = mean(p_w_null<0.05) %>% scales::label_percent(.1)(),
    ordinal_regression = mean(p_o_null<0.05) %>% scales::label_percent(.1)(),
  )
# t_test wilcox_test ordinal_regression
# 5.1%   5.2%        5.1%              

# NULL: t-test vs ordinal
ps %>% 
  mutate(t_yes_o_no = (p_t_null<0.05) & (p_o_null>0.05)) %>% 
  mutate(t_no_o_yes = (p_t_null>0.05) & (p_o_null<0.05)) %>% 
  summarise(
    t_yes_o_no = mean(t_yes_o_no) %>% scales::label_percent(.1)(),
    t_no_o_yes = mean(t_no_o_yes) %>% scales::label_percent(.1)()
  )
# t_yes_o_no t_no_o_yes
# 1.5%       1.5%