Bootstrapped p-value for a correlation coefficient

On R, I used the boostrap method to get a correlation coefficient estimation and the confidence intervals. To get the p-value, I thought, I can calculate the proportion of the confidence intervals which do not contain zero. But this is not the solution.

How can I get the p-value in this case ?

I am using cor.test to get the coefficient estimation. cor.test may also gives me the p-value from every test. But how can I get the bootstrapped p-value ?

Thank you very much !

Below an example :

n=30
data = matrix (data = c (rnorm (n), rnorm (n),rnorm (n), rpois(n,1), 
rbinom(n,1,0.6)), nrow =  n, byrow = F)
data= as.data.frame(data)
z1  = replicate( Brep, sample(1:dim(data)[1], dim(data)[1], replace = T))
res = do.call  ( rbind, apply(z1, 2, function(x){ res=cor.test(data$V1[x], data$V2[x]) ; return ((list(res$p.value,res$estimate))) }))

 coeffcorr  = mean(unlist(res[,2]), na.rm = T) #bootstrapped coefficient
 confInter1 = quantile(unlist(res[,2]), c(0.025, 0.975), na.rm = T)[1] #confidence interval 1
 confInter2 = quantile(unlist(res[,2]), c(0.025, 0.975), na.rm = T)[2] #confidence interval 2  
 p.value    =  mean    (unlist(res[,1]), na.rm = T )  # pvalue

The standard way of bootstrapping in R is to use base package boot. You start by defining the bootstrap function, a function that takes two arguments, the dataset and an index into the dataset. This is function bootCorTest below. In the functionyou subset the dataset selecting just the rows defined by the index.

The rest is straightforward.

library(boot)

bootCorTest <- function(data, i){
    d <- data[i, ]
    cor.test(d$x, d$y)$p.value
}


# First dataset in help("cor.test")
x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
y <- c( 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)
dat <- data.frame(x, y)

b <- boot(dat, bootCorTest, R = 1000)

b$t0
#[1] 0.10817

mean(b$t)
#[1] 0.134634

boot.ci(b)

For more information on the results of functions boot and boot.ci see their respective help pages.

EDIT.

If you want to return several values from the boot statistic function bootCorTest, you should return a vector. In the following case it returns a named vector with the values required.

Note that I set the RNG seed, to make the results reproducible. I should already have done it above.

set.seed(7612)    # Make the results reproducible

bootCorTest2 <- function(data, i){
    d <- data[i, ]
    res <- cor.test(d$x, d$y)
    c(stat = res$statistic, p.value = res$p.value)
}


b2 <- boot(dat, bootCorTest, R = 1000)

b2$t0
#  stat.t  p.value 
#1.841083 0.108173


colMeans(b2$t)
#[1] 2.869479 0.133857