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

[Charles C. Berry]

On Thu, 17 Jul 2008, Mike Lawrence 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).


See Fisher (1915) Biometrika.

It is non-central t up to a monotone transformation.

Finding the value of the non-centrality parameter and that 
transformation is left as an exercise for the reader.

Chuck


>
> 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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> and provide commented, minimal, self-contained, reproducible code.
>

Charles C. Berry                            (858) 534-2098
                                             Dept of Family/Preventive Medicine
E mailto:cberry at tajo.ucsd.edu	            UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/  La Jolla, San Diego 92093-0901