Henry , today I checked the posting, that you mentioned, and I ran some quick tests to see, what happens, in answering your posting in a bit more detail:
Gottfried Helms wrote: > > Henry wrote: > > > > I don't know which group of the 3 sci.stat.* groups you were talking > > about, but I remember looking at double exponential distributions > > (with positive kurtosis, rather than a uniform distributions negative > > kurtosis) and getting a reverse CR effect. > > <http://groups.google.com/groups?selm=391b2a44.679260%40news.btinternet.com> > > This relation may exist, because the uniform-distribution looks somehow like > a limit for kurtosis. (kurtosis->0) So I guess that in a CR-model > that variable with a lower kurtosis will be declared to be the "cause". > (but I didn't check it really) > Just some quick examples to check about kurtosis. I draw some randomdata (first example: uniform, second example normal distributed), and artificially compressed resp. expanded the values by exponentiating, so that the density changes to different kurtosises. The different compressions/expansions are documented sideways, the used exponent heads the resp. columns in the first line. For exponent 1 the computed data equal the real data. Always the "causes" are uncorrelated. The cause-indicator x1 is simply computed as x1=cause1. The "effect" y1 is computed by y1=0.707*(cause1+cause2) x2 and y2 are computed via regression (although their values are identical with x2=cause2 and y2 = 0.707*(cause2-cause1) by construction. Result: * obviously the CR-effect is strongly dependent on kurtosis of the causes-measures. This does not only mean the absolute value of the d-statistic (the direction- indicator) but even the direction itself. (*see table below) I was talking about refinement of CR in a previer post. For instance you can observe, that the rde-coefficient for x1/x2 is alway low in absolute value and the same coefficient for y1/y2 is always high in absolute value. This can be easily understood, if you look at the scatterplots. The changing of the sign of the d-statistic is then determined by the change of the rdy-part of the coefficient. May be here is an aspect, how to improve the method. In another previous post I also talked about rotation. The Quartimax-rotation of the x1',x2',y1'y2'-set localizes the factors always at the x1'/x2'-variables, which have the least correlation, and it does not suffer from that problem which spoils the d-statistic (which is simply a difference of the rde-values). The non-correlation of the uniformely distributed x1/x2 survives exponentiation and thus dominates in a subsequent quartimax-orientated factorsystem. May be, this is a better path to proceed in enhancing CR - also it seems quite natural to try and expand it to a more multivariate approach. But that needs definitely some more research. Gottfried Helms -------- [3] causes = randomu(maxv,n) // 2x200 uniform-randomdata (...) [133] rde = rde||{{power},schiefe,exzess,rdexy,dstat} CR-checks (basing on uniform-distributed data) ------------------------------------------------------------------------------------------------- where x' means that its values are the signed powers of the absolute values of x x'[i] = sign(x[i]) * ( abs(x[i])^exponent ) exponent | 1.00 2.00 3.00 4.00 0.50 | ------------ |---------------|--------------|------------|------------|-------------- skewness x1 | 0.10 0.15 0.23 0.32 0.05 | x2 | -0.17 -0.50 -0.93 -1.39 -0.07 | ------------ |---------------|--------------|------------|------------|-------------- kurtosis x1 | -1.19 -0.19 0.95 2.17 -1.66 | x2 | -1.07 0.27 1.69 3.24 -1.64 | ------------ |---------------|--------------|------------|------------|-------------- rde: correlations (x1',x2') | -0.06 -0.04 -0.03 -0.03 -0.06 | (y1',y2') | -0.45 0.21 0.54 0.70 -0.84 | ------------ |---------------|--------------|------------|------------|-------------- CR-D-statistic (x,y) | -0.39 0.25 0.57 0.73 -0.78 | ------------ |---------------|--------------|------------|------------|-------------- (note: the true "causal" direction is found, when the d-statistics has negative values) =========================================================================================== [3] causes = randomn(maxv,n) // 2x200 normal-randomdata (...) [133] rde = rde||{{power},schiefe,exzess,rdexy,dstat} CR-checks basing on normal-distributed data ------------------------------------------------------------------------------------------------- where x' means that its values are the signed powers of the absolute values of x x'[i] = sign(x[i]) * ( abs(x[i])^exponent ) exponent | 1.00 2.00 3.00 4.00 0.50 | ------------ |---------------|--------------|------------|------------|-------------- skewness x1 | 0.17 0.87 2.16 3.62 0.07 | x2 | 0.15 0.61 1.63 3.14 0.07 | ------------ |---------------|--------------|------------|------------|-------------- kurtosis x1 | 0.14 7.37 19.90 34.11 -1.40 | x2 | -0.10 6.27 18.85 38.08 -1.45 | ------------ |---------------|--------------|------------|------------|-------------- rde: correlations (x1',x2') | 0.08 0.10 0.12 0.12 0.08 | (y1',y2') | -0.05 0.64 0.82 0.88 -0.70 | ------------ |---------------|--------------|------------|------------|-------------- CR-D-statistic (x,y) | -0.13 0.54 0.70 0.76 -0.78 | ------------ |---------------|--------------|------------|------------|-------------- (note: the true "causal" direction is found, when the d-statistics has negative values) =========================================================================================== -- ---------------------------------------------------------------- Gottfried Helms Soz.Paed./Soz.Arb. 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