Suppose F(x, y; rho) is the cdf of a bivariate normal distribution, with standardized marginals and correlation parameter rho. For any fixed x and y, I wonder if F(x, y; rho) is a monotone increasing function of rho, i.e., there is a 1 to 1 map from rho to F(x, y; rho). I explored it using the function pmvnorm in package mvtnorm with different x and y. The plot suggests the statement may be true. But I can not prove it analytically after a few days thinking. I would appreciate if people on the list can point me to the references. Jun -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html Send "info", "help", or "[un]subscribe" (in the "body", not the subject !) To: r-help-request at stat.math.ethz.ch _._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._
On Wed, 1 May 2002, Jun Yan wrote:> Suppose F(x, y; rho) is the cdf of a bivariate normal distribution, with > standardized marginals and correlation parameter rho. For any fixed x and > y, I wonder if F(x, y; rho) is a monotone increasing function of rho, > i.e., there is a 1 to 1 map from rho to F(x, y; rho). >It is true. You want references to the tetrachoric correlation. The tetrachoric correlation is the estimate of rho based on F(x,y), and it wouldn't be any use if it weren't 1-1. I don't have any specific references. I tried Kendall & Stuart, but it doesn't give details. -thomas Thomas Lumley Asst. Professor, Biostatistics tlumley at u.washington.edu University of Washington, Seattle -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html Send "info", "help", or "[un]subscribe" (in the "body", not the subject !) To: r-help-request at stat.math.ethz.ch _._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._
Dear Thomas and Jun Yan, The Encyclopedia of Statistical Sciences (Kotz and Johnson, eds.) has an article on tetrachoric correlation, including many references. John At 08:39 AM 5/1/2002 -0700, Thomas Lumley wrote:>On Wed, 1 May 2002, Jun Yan wrote: > > > Suppose F(x, y; rho) is the cdf of a bivariate normal distribution, with > > standardized marginals and correlation parameter rho. For any fixed x and > > y, I wonder if F(x, y; rho) is a monotone increasing function of rho, > > i.e., there is a 1 to 1 map from rho to F(x, y; rho). > > > >It is true. You want references to the tetrachoric correlation. The >tetrachoric correlation is the estimate of rho based on F(x,y), and it >wouldn't be any use if it weren't 1-1. I don't have any specific >references. I tried Kendall & Stuart, but it doesn't give details.----------------------------------------------------- John Fox Department of Sociology McMaster University Hamilton, Ontario, Canada L8S 4M4 email: jfox at mcmaster.ca phone: 905-525-9140x23604 web: www.socsci.mcmaster.ca/jfox ----------------------------------------------------- -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html Send "info", "help", or "[un]subscribe" (in the "body", not the subject !) To: r-help-request at stat.math.ethz.ch _._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._
A while back, I asked the following quesiton:> Suppose F(x, y; rho) is the cdf of a bivariate normal distribution, with > standardized marginals and correlation parameter rho. For any fixed x and > y, I wonder if F(x, y; rho) is a monotone increasing function of rho, > i.e., there is a 1 to 1 map from rho to F(x, y; rho).Actually there is a beautiful result on this and Professor S. Le Cessie from Netherlands kindly gave me an elegant proof of the following: d F(x, y; rho) / d rho = f(x, y; rho), where f is the standard bivariate normal density. It can be proved by showing that d f(x, y; rho) / d rho = d f(x, y; rho) / dx dy. My thanks to Prof. Thomas Lumley, Chong Gu, and John Fox for pointing me to the references. Jun -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html Send "info", "help", or "[un]subscribe" (in the "body", not the subject !) To: r-help-request at stat.math.ethz.ch _._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._