From: Charles C. Berry <cberry_at_tajo.ucsd.edu>

Date: Thu, 17 Jul 2008 12:57:05 -0700

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https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Thu 17 Jul 2008 - 20:06:05 GMT

Date: Thu, 17 Jul 2008 12:57:05 -0700

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
**>
**> ______________________________________________
**> R-help_at_r-project.org mailing list
**> https://stat.ethz.ch/mailman/listinfo/r-help
**> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
**> 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 Diegohttp://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 92093-0901

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