Re: stan error of r
David and others, Some comments follow which I hope will be of some interest. On 28 Mar 2001 15:52:49 -0800, [EMAIL PROTECTED] (David C. Howell) wrote: > >Dennis, > >The closest answer that I can find as an answer to your question is that "it is >very complicated." The exact distribution under bivariate normality is given in Kendall and Stuart, "The Advanced Theory of Statistics". You are absolutely right. It is very complicated. Computing the cumulative distribution accurately is very tricky in C or Fortran, requiring extensive, careful programming. My computer program, Statistica Power Analysis, computes the exact cumulative distribution of r for any population correlation rho. (under the traditional, unrealistic assumption of bivariate normality) > But the first question is what you would do with the answer >if you had it? Because the distribution of r is skewed when rho is not equal to >0, you wouldn't want to use that standard error to create a confidence >interval--it would be too wide on one side, and not wide enough on the other. In a sense this is right, but in another sense it really isn't. The exact confidence interval can be computed, and is, by Statistica Power Analysis. If you mean that you cannot compute a simple confidence interval, using a formula like r (plus or minus) 1.96 * SE(r) where SE(r) is the standard error of r, of course you are correct. But this method for computing confidence intervals is only a special case of a much more general method, called "the inversion method," discussed in great detail by Steiger and Fouladi (1997) in the Erlbaum book, "What if there were no significance tests?" This "inversion" method, using advanced software, can compute such an exact confidence interval in many cases where simpler approaches cannot. The use of this method opens up many new opportunities for using confidence intervals in place of hypothesis tests. The Steiger and Fouladi (1997) article not only describes how to compute exact confidence intervals on rho, the population correlation, it also describes how to do so for the squared multiple correlation, and many other interesting quantities, such as the root mean square standardized effect in ANOVA. This latter confidence interval replaces the F test, including all the information available in an ANOVA F-test, and more. BTW, the ANOVA confidence intervals are also computed in Statistica Power Analysis. > >There are two other answers. I'm sure that you are aware that we can convert r >to r' (or z' as Fisher called it), and that its standard error is estimated >well by 1/sqrt(N-3). The other approach would be to estimate the standard error >by bootstrapping. That is actually a relatively simple process, but, again, I >don't know what I would do with the answer once I found it. The exact standard error of r is not very valuable, although it can be computed directly using advanced symbolic software like Mathematica. Using Mathematica, you can compute it in a few lines of code. The standard error of r can be approximated by the square root of the asymptotic formula for the variance of r, which is Var(r) = (1/N)* (1-rho^2)^2 This formula, by the way, is not always given correctly. For example, the classic book by Glass and Stanley has it wrong in several places. Interestingly, when rho is zero, the above formula reduces to 1/N, and leads to a very simple, oft-forgotten formula for a "quick and dirty" significance test for a single correlation. By standard asymptotic theory, the test statistic Z = r / Sqrt(1/N) = Sqrt(N) r has an asymptotically N(0,1) distribution if rho=0. Moreover, the distribution of r is rather symmetric and close to normal when rho=0, so this formula is quite accurate. Consequently, as a quick and dirty test at the .05 level, simply examine whether the absolute value of the above statistic is less than 1.96. |Z| < 1.96 This, in turn, leads to "quick and dirty" "significance points" for r, because |Z| < 1.96 is the same as |r| < 1.96/Sqrt(N), or, roughly, |r| < 2/Sqrt(N) = Sqrt(4/N). I call the formula Sqrt(4/N) the "blunt method." Compare 1.96/Sqrt(N) with the tabled "critical values" of r at the .05 level. N Exact Quick Blunt -- 200.139 .139 .141 100.197 .196 .2 50.278 .277 .283 25.396 .392.4 - Having the quick and dirty formula or blunt formula in the back of one's mind allows one to have an intuitive appreciation for sample sizes necessary for reasonable correlational analysis, without having to carry around a textbook. I quickly add that all the above formulas depend, more or less, on the highly questionable assumption of bivariate normality. So bootstrapping may be an excellent idea in any case. --Jim James H. Steiger, Professor Dept. of
Re: stan error of r - Virus
Dr. Knodt, Perhaps you should make an appointment with some nerdy geek in the IR department who can explain to you how viruses get promulgated. It is usually through executables (.exe) files or microsoft macros. Otherwise you will continue to be needlessly worried about non-existent threats. :-) Tom G. [EMAIL PROTECTED] wrote: > This might be a great way to spread virus. > > With all the virus going around, please do not post e-mail with attachments > to the mailing list. > > Send attachments only to those who request them. > > Thanks for your understanding > > Dr. Robert C. Knodt > [EMAIL PROTECTED] > > = > Instructions for joining and leaving this list and remarks about > the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ > = = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: stan error of r - Virus
This might be a great way to spread virus. With all the virus going around, please do not post e-mail with attachments to the mailing list. Send attachments only to those who request them. Thanks for your understanding Dr. Robert C. Knodt [EMAIL PROTECTED] = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: stan error of r
Dennis, Elliot Cramer gave the standard error of r as [(1/n)* (1 - rho^2)^2]. For rho = .80, and n = 100, this would come to .036. The attached jpeg file is the result of bootstrapping 10,000 resamples from a data set where r = .801. You will see that the standard error there, which is simply the st. dev. of those 10,000 r's, is also .036, as it should be. Note that the 95% confidence limits are .718 and .862, which are asymmetric, as they should be. If we naively took r +/- 1.984(.036) we would get .730 and .872, which are symmetric, but wrong. Notice that the bootstrap distribution looks just like the textbooks say it should. Dave Howell bootstrapCorr.jpeg David C. Howell Phone: (802) 656-2670 Dept of Psychology Fax: (802) 656-8783 University of Vermont email: [EMAIL PROTECTED] Burlington, VT 05405 http://www.uvm.edu/~dhowell/StatPages/StatHomePage.html http://www.uvm.edu/~dhowell/gradstat/index.html
Re: stan error of r
Elliot Cramer <[EMAIL PROTECTED]> wrote: : dennis roberts <[EMAIL PROTECTED]> wrote: : : anyone know off hand quickly ... what the formula might be for the standard : : error for r would be IF the population rho value is something OTHER than zero? correctiont: the variance is (1/n)*(1-rho^2)^2 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: stan error of r
dennis roberts <[EMAIL PROTECTED]> wrote: : anyone know off hand quickly ... what the formula might be for the standard : error for r would be IF the population rho value is something OTHER than zero? It's (1/n)*(1-rho^2)^2 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
RE: stan error of r
dennis, you can use fisher's z-transformation, and compute a confidence interval (or a z-test), given that the standard error is (approximately) 1/sqrt(n-3) ethan -Original Message- From: dennis roberts [mailto:[EMAIL PROTECTED]] Sent: Wednesday, March 28, 2001 1:18 PM To: [EMAIL PROTECTED] Subject: stan error of r anyone know off hand quickly ... what the formula might be for the standard error for r would be IF the population rho value is something OTHER than zero? _ dennis roberts, educational psychology, penn state university 208 cedar, AC 8148632401, mailto:[EMAIL PROTECTED] http://roberts.ed.psu.edu/users/droberts/drober~1.htm = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ = = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: stan error of r
Fisher z transform is normally distributed with variance 1/(N-3). z=0.5*/ln((1+r)/(1-r)). Will At 4:18 PM -0500 28/3/01, dennis roberts wrote: >anyone know off hand quickly ... what the formula might be for the >standard error for r would be IF the population rho value is >something OTHER than zero? = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: stan error of r
Dennis, The closest answer that I can find as an answer to your question is that "it is very complicated." But the first question is what you would do with the answer if you had it? Because the distribution of r is skewed when rho is not equal to 0, you wouldn't want to use that standard error to create a confidence interval--it would be too wide on one side, and not wide enough on the other. There are two other answers. I'm sure that you are aware that we can convert r to r' (or z' as Fisher called it), and that its standard error is estimated well by 1/sqrt(N-3). The other approach would be to estimate the standard error by bootstrapping. That is actually a relatively simple process, but, again, I don't know what I would do with the answer once I found it. Dave .At 04:18 PM 3/28/01 -0500, dennis roberts wrote: >anyone know off hand quickly ... what the formula might be for the standard >error for r would be IF the population rho value is something OTHER than zero? > >_ >dennis roberts, educational psychology, penn state university >208 cedar, AC 8148632401, mailto:[EMAIL PROTECTED] >http://roberts.ed.psu.edu/users/droberts/drober~1.htm > > > >= >Instructions for joining and leaving this list and remarks about >the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ >= David C. Howell Phone: (802) 656-2670 Dept of Psychology Fax: (802) 656-8783 University of Vermont email: [EMAIL PROTECTED] Burlington, VT 05405 http://www.uvm.edu/~dhowell/StatPages/StatHomePage.html http://www.uvm.edu/~dhowell/gradstat/index.html
