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

[Mike Lawrence]

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