When you standardize the regression, aren't you dividing by random variables?
-Dick Startz On 1 Feb 2003 14:56:19 -0800, [EMAIL PROTECTED] (David C. Howell) wrote: >This morning I sent out the message below. I still have basically the same >question, but when I went back and looked at my resampling program, I >discovered an error. When I corrected that, the distribution is normal. But >I still don't know what is wrong with my argument that it shouldn't be normal. > >Sorry for the confusion >***************************************** >This morning's message > >Yesterday I received a query asking about the sampling distribution of >beta, the standardized regression coefficient. I thought that I knew the >answer, but I thought that I should check. Now I am confused and in need of >help. > >Let's simplify the problem to simply look at the sampling distribution of b >for the moment. (One is just a linear transformation of the other.) > >Hogg and Craig(1978, p. 297) state that since a and b are linear functions >of Y, they are each normally distributed. They base this on a variance sum >law argument. Tamhane and Dunlop (2000, p. 358) show a proof using the same >argument. > >We also know, from a million sources, that the sampling distribution of >(b-b*) divided by se(b) follows a t distribution, where b* represents the >parameter. Now the numerator of a t distribution is a normally distributed >variable, so (b - b*) must be normally distributed. Since b* is a constant, >then b must be normally distributed. > >That would seem to settle the question--b is normally distributed. > >BUT, suppose that we standardized our variables. This is a linear >transformation for both X and Y, so would not change the correlation nor >the shape of distributions. With standardized variables the slope is equal >to the standardized slope (beta). And it is easy to show that beta, with >only one predictor, is equal to the correlation coefficient. So when we >take standardized variables, which are just linear transformations of >unstandardized variables, r, b, and beta will all be numerically equal. > >Now, we all know that the sampling distribution for r is skewed when r* >(the parameter) is not 0. If we let r* = .60, the skew is quite noticeable. >But if r and beta are equal, then the sampling distribution of beta will be >equal to the sampling distribution of r, and therefore it will also be >skewed. And if the sampling distribution of beta is skewed, so should be >the sampling distribution of b, because it is simply a linear >transformation on beta. > >So now I have shown that the sampling distribution is skewed, though I >began by quoting experts I respect saying that it is normal. > >Finally, I used Resampling Stats to empirically generate the sampling >distribution. It is very definitely skewed. > >So where did I go wrong? The sampling distribution of b cannot be both >normal and skewed, at the same time. > >Thanks, >Dave Howell > >************************************************************************** >David C. Howell >Professor Emeritus >University of Vermont > >New address: >David C. Howell Phone: (512) 402 1179 >3007 Barton Point Circle Fax: (512) 402-1416 >Austin, Tx 78733 email: [EMAIL PROTECTED] > > >http://www.uvm.edu/~dhowell/StatPages/StatHomePage.html > >http://www.uvm.edu/~dhowell/gradstat/index.html ---------------------- Richard Startz [EMAIL PROTECTED] Lundberg Startz Associates . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
