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Below is the graph of two variables, X and Y, each representing count data. N=348. Note the scales of the axes:
http://i.imgur.com/tNGyTX5.jpg

Y is very approximately lognormal, but X has no decent fit (including Poisson, negative binomial, lognormal and gamma of the log transform).
Spearman coefficient between X and Y is close to 0, and p-value to reject no correlation is very high.

From the plot, there appears to be no combinations of extreme values of both x and y.

When I log transform both X and Y, the following plot results:
enter image description here
Clearly any appearance of pattern has disappeared.

My questions are:

  • Why is there a lack of combinations of "extreme" values on the linear scale, but not on the log scale?
  • Is there any significance to the lack of combination of extreme values on the linear scale, and is there anyway to investigate further?

The purpose of this study is exploratory.

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1 Answer 1

up vote 13 down vote accepted

Consider flipping your questions around.

Begin with uncorrelated data - I generated this data randomly, so these variables are independent; my y is normal and my x is log(1+X1) where X1 is a mixture of several geometric distributions chosen to give a roughly similar appearance to your plot:

enter image description here

The y variable is symmetric and the x-variable is mildly skew, but critically, neither of these variables is very long-tailed.

You then get many cases of relatively large values of X and Y together, because the probability that either is above its midrange (the center of the plot) is high, so the probability that both will be is also reasonably high (say, somewhere around 0.15-0.25), the product in this case of 0.5 for the y-variable and something a bit less that 0.5 for the x-variable.

What happens if you exponentiate two such variables, which are simply independent, shortish-tailed variables:

enter image description here

You get something broadly similar to your first plot. It makes both variables - though still independent - strongly right skew (long tailed to the right), ... and that's where the appearance comes from.

Why does it look "L" shaped? Simply because high values of the X and Y variables are both relatively rare, and the combination of the two (due to their independence) rarer still. Because for each variable, almost all the other variable's values are far below its midrange, an extreme X or Y is likely to be associated with values of the other variable below/left-of the middle of the plot.

e.g. if 5% of each distribution is above the midrange, then about 0.25% of values (i.e. not quite 1 on average) will be in the upper right quadrant.

You see pretty much the same phenomenon with any two sufficiently right-skew variables that are independent, and in many that are close to independent. Here are two independent random variables (the absolute values of t-distributed random variables with 1.5 d.f. and different scales):

enter image description here

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Thanks for the great answer; how you presented it was very understandable. –  TLJ Dec 16 '13 at 6:13

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