Gus Gassmann schrieb: > > behaves very differently, essentially filling out a diamond with vertices > at (-2,0), (0,-1), (2,0), (0,1) if plotted against y.
Hi Gus, this was exactly discussed already 1998, but it did not find the way into the CR-papers... > I generated a large sample (1,000,000) of uniform x1 and x2 and computed > y = x1 + x2. If you take it, that y, x1 and x2 are all standardized, then you could also write y = r*x1 + s*x2 where r^2 + s^2 = 1 and share this property with functions like sin and cosine... Precisely, if you create y by simply adding the z-scores of x1 and x2, and rescale y (to be standardized, too), then it is y = 0.707*x1 + 0.707*x2 If you fiddle with all terms, then for the opposite regression you get then y2 = r*x2-s*x1 where again r and s behave like proper sine and cosine of the same angle:45 Deg. Putting it together you have: y = r*x1 + s*x2 y2 = r*x2 - s*x1 (where in the simple examples of CR always r=s=sqrt(0.5)~0.707.) Then you draw your scatterplots for x1/x2 and y/y2 It is obvious then, that the scatterplot for y/y2 is just a rotation of that of x1/x2 by 45 deg, thus building a diamond. If you then take the absolute values to do correlations of x1/x2 and y/y2 then the scatterplot of the new abs(x1)/abs(x2) is still a square but the scatterplot of abs(y)/abs(y2) is a lower triangle. You can see that, because the use of absolute values maps and mirrors all four quadrants only to the first. The regression-line of a lower triangle (abs(y)/abs(y2)) has a negative slope; the one of a square (abs(x1)/abs(x2)) is zero, which reflects the null and negative rde()-values. > The residuals and correlations behave as you predicted, leading > you to the conclusion that y is caused by x1 (and x2). > > Then I selected a subsample from this in such a way that (x2,y) are uniformly > distributed. (This takes some doing, but it is possible.) With this data set > you'd > come to the conclusion that the cause is y! That was also shown already. You can derive the result analytically, too, if you replace the use of absolute values by the use of squared values - which modifies the output little, but preserves the logic. You can derive then, which sampling you need to construct each causality direction you want. I gave three examples already in a posting in 1998 - but don't have them handy. If you like, you can have our exchanges concerning CR as a netscape-mailbox. (whole collection ~ 800 postings from SEMnet, CR-mailinglist and newsgroups-discussions, unsorted, maybe some duplicates) Gottfried Helms -- Univ Kassel . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
