How do I bootstrap correlations between two data frames?

I have not been able to find an example of bootstrapping correlations between two data frames. Key posts I have looked at are (1) https://stats.stackexchange.com/questions/20701/computing-p-value-using-bootstrap-with-r/ (2) How to bootstrap correlation using vectorised function applied to large matrix?

NB/ p values here need to be obtained through calculating the ratios of how many times the absolute bootstrap test statistics exceeded the theoretical ones,

Fortunately, I have also come across a recent example of using Map to apply the correlations between two data frames. How to use lapply to replace nested for loop to get correlations between two data frames?

I have large datasets that will be run in a unix based HPC and also a Windows OS option for running the calculations on smaller datasets in R.

   D1 <- data.frame(matrix(runif(10*10, 0, 2), ncol=10)) 
   D2 <- data.frame(matrix(runif(10*16, 0, 2), ncol=16)) 

   colnames(D1) <- paste0("a", 1:ncol(D1))
   colnames(D2) <- paste0("b", 1:ncol(D2))

   compare <- expand.grid(colnames(D1), colnames(D2))

  need_modify <- Map(function(x,y) cor.test(D1[, x], D2[, y], method="spearman"), compare$Var1,    compare$Var2) %>% 
  lapply(`[`, c('estimate', 'statistic', 'p.value')) %>% 
  sapply(unlist) %>%  t() %>% as.data.frame() %>% mutate(Var1=compare$Var1, Var2=compare$Var2)

  boot_df <- function(x) x[sample(nrow(x), rep = T), ]

 #number of bootstraps 
 R <- 100

I am stuck on modifying the above so that it run successfully using parallelisation for Unix based OS (mclapply or mcMap) and also a separate one for Windows (clusterMap or future_mapply).

Grateful for any pointers in the right direction or an example elsewhere.

Technically you could resample B times with replacement, extract the "statistic"s,

stopifnot(identical(dim(D1)[1], dim(D2)[1]))  ## check identical nrows
r.names <- Reduce(\(...) paste(..., sep=':'), Cmp)  ## store Cmp names

ncpu <- 7L  ## number of workers
B <- 999L  ## number of bootstrap replications
set.seed(42)  ## for sake of reproducibility
T_star <- parallel::mcmapply(function(x, y) {
  boot <- function() {  ## boot fun
    smp <- sample.int(dim(D1)[1], dim(D1)[1],  ## resample w/ replacement
                      replace=TRUE)
    unlist(cor.test(D1[smp, x], D2[smp, y], method="spearman", exact=FALSE)[
      c('statistic')])  ## extract statistic
  }
  replicate(B, boot())  ## replicate boot fun B times
}, Cmp$Var1, Cmp$Var2, mc.cores=ncpu, mc.set.seed=FALSE, SIMPLIFY=FALSE) |> setNames(r.names)

lapply(T_star, head)[1:3]
# $`a1:b1`
# statistic.S statistic.S statistic.S statistic.S statistic.S statistic.S 
#    212.4375    208.3125    205.7407    278.1132    216.5625    233.5714 
# 
# $`a2:b1`
# statistic.S statistic.S statistic.S statistic.S statistic.S statistic.S 
#   270.18750   195.93750   258.70370   130.75472   216.56250    77.14286 
# 
# $`a3:b1`
# statistic.S statistic.S statistic.S statistic.S statistic.S statistic.S 
#    144.3750    270.1875    220.0000    220.0000    198.0000    158.5714

and according to Davison & Hinkley (1997)'s formula,

calculate a Monte Carlo P-value add up how many times the actual statistic,

T_act <- parallel::mcmapply(\(x, y) cor.test(D1[, x], D2[, y], meth='s'), 
                          Cmp$Var1, Cmp$Var2, mc.cores=ncpu, SIMPLIFY=FALSE) |>
  lapply(`[[`, c('statistic')) |> 
  sapply(unlist) |> setNames(r.names)
T_act[1:3]
# a1:b1 a2:b1 a3:b1 
#   210   156   232 

is exceeded divided by B (using a bias correction by adding +1 to both, the numerator and denominator).

p_values <- parallel::mcmapply(\(x, y) (sum(abs(x) >= abs(y)) + 1)/(B + 1), 
                               T_star, T_act, mc.cores=ncpu) |>
  setNames(r.names) |> unlist()

p_values[1:3]
# a1:b1 a2:b1 a3:b1 
# 0.507 0.477 0.490 

Of course, we can combine everything and simplify to:

ncpu <- 7L  ## number of workers
B <- 999L  ## number of bootstrap replications
set.seed(42)  ## for sake of reproducibility
p_values2 <- parallel::mcmapply(\(x, y) {
  boot <- \() {  ## boot fun
    smp <- sample.int(dim(D1)[1], dim(D1)[1],  ## resample w/ replacement
                      replace=TRUE)
    unlist(cor.test(D1[smp, x], D2[smp, y], method="spearman", exact=FALSE)[
      c('statistic')])  ## extract statistic
  }
  t_star <- replicate(B, boot())  ## replicate boot fun B times
  t_act <- unlist(cor.test(D1[, x], D2[, y], method="spearman", exact=FALSE)[
    c('statistic')])
  (sum(abs(t_star) >= abs(t_act)) + 1)/(B + 1)  ## calc. MC p-value
}, Cmp$Var1, Cmp$Var2, mc.cores=ncpu, mc.set.seed=FALSE) |> unlist() |> 
  setNames(r.names)

head(p_values2, 3)
# a1:b1 a2:b1 a3:b1 
# 0.507 0.477 0.490 

Note, that the answer focuses on how to do it technically. Read Davison & Hinkley (1997) or consult a statistician if you really want it to be sound.

Data:

set.seed(42)
D1 <- data.frame(matrix(runif(10*10, 0, 2), ncol=10)) |> 
  setNames(paste0("a", seq_len(10)))
D2 <- data.frame(matrix(runif(10*16, 0, 2), ncol=16)) |> 
  setNames(paste0("b", seq_len(16)))
Cmp <- expand.grid(colnames(D1), colnames(D2))