# Re: [R] Sampling distribution (PDF & CDF) of correlation

From: Maria Rizzo <mrizzo7_at_gmail.com>
Date: Thu, 17 Jul 2008 12:00:12 -0400

See Chapter 22 in N.L. Johnson, S. Kotz, and N. Balakrishnan, "Continuous Univariate Distributions", Volume 2, Second Edition, 1995.

Maria

On Thu, Jul 17, 2008 at 11:29 AM, Mike Lawrence <Mike.Lawrence_at_dal.ca> wrote:
> Hi all,
>
> I'm looking for an analytic method to obtain the PDF & CDF of the sampling
> distribution of a given correlation (rho) at a given sample size (N).
>
> I've attached code describing a monte carlo method of achieving this, and
> while it is relatively fast, an analytic solution would obviously be
> optimal.
>
> get.cors <- function(i, x, y, N){
> end=i*N
> .Internal(cor(x[(end-N+1):end] ,y[(end-N+1):end] ,TRUE ,FALSE ))
> }
> get.r.dist <- function(N, rho, it){
> Sigma=matrix(c(1,rho,rho,1),2,2)
> eS = eigen(Sigma, symmetric = TRUE, EISPACK = TRUE)
> ev = eS\$values
> fact = eS\$vectors %*% diag(sqrt(pmax(ev, 0)), 2)
> Z = rnorm(2 * N * it)
> dim(Z) = c(2, N * it)
> Z = t(fact %*% Z)
> x = Z[, 1]
> y = Z[, 2]
> r = sapply(1:it ,get.cors,x, y, N)
> return(r)
> }
>
> #Run 1e3 monte carlo iterations, where each obtains the correlation
> # of 10 pairs of observations from a bivariate normal distribution with
> # a true correlation of .5. Returns 1e3 values for the observed correlation
> mc.rs = get.r.dist( N=10 , rho=.5 , it=1e3 )
>
> #plot the PDF & CDF
> par(mfrow=c(1,2))
> hist(mc.rs,prob=T,xlab='Observed correlation')
> probs = seq(0,1,.01)
> plot(quantile(mc.rs,probs=probs),probs,type='l',xlab='Observed
> correlation',ylab='Cumulative probability')
>
> --
> Mike Lawrence
> Graduate Student, Department of Psychology, Dalhousie University
>
> www.memetic.ca
>
> "The road to wisdom? Well, it's plain and simple to express:
> Err and err and err again, but less and less and less."
> - Piet Hein
>
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