Pairwise

library(tidyverse)
library(MASS)
library(magrittr)

set.seed(41)

d <- 200
n <- 6

dists <-
  map_df(seq(1000), function(i) {
    a <- mvrnorm(n = n, rnorm(d), diag(rnorm(d) ** 2))
    b <- mvrnorm(n = n, rnorm(d) + d*rnorm(1), diag(rnorm(d) ** 2))
    
    diff <- as.matrix(colMeans(a) - colMeans(b))
    
    dist_bw_means <- norm(diff, "2")
    
    mean_of_dists <-
      mean(as.vector(as.matrix(dist(rbind(
        a, b
      )))[(n + 1):(2 * n), 1:n]))
    
    cor_bw_means <- cor(colMeans(a), colMeans(b))

    cm <- cor(t(a), t(b))

    mean_of_cors <- mean(cm[upper.tri(cm)])
    
    data.frame(dist_bw_means, mean_of_dists, cor_bw_means, mean_of_cors)
    
  })

ggplot(dists, 
       aes(dist_bw_means, mean_of_dists)) + 
  geom_point(alpha = 0.1) + 
  coord_equal() + 
  geom_abline(slope = 1, color = "red")

ggplot(dists, 
       aes(cor_bw_means, mean_of_cors)) + 
  geom_point(alpha = 0.1) + 
  geom_abline(slope = 1, color = "red")