# What is a statistical test of whether a variable is uniformly distributed with kinks?

I am trying to come up with a way to statistically test this question, and could use some input, as it's far afield from what I normally do. I have a hierarchical network, like this one:

And a centrality measurement of each node. What I want to do is test if the measurement for each "layer" (i.e. nodes 1 step from the diamonds, or 2 steps from the diamonds, etc) are uniformly distributed. The problem is that the layers aren't actually independent - the layers closer to the diamond will inherit some of the distortions from the layers below them, so what I actually want to test is that the distribution for layer X is no more distorted than the layer before it.

Any ideas how to go about that?

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I think you need to more precisely define what you mean by "no more distorted"; but one idea is to look at median absolute deviations from the median of each layer. –  Peter Flom Aug 31 '11 at 10:49
@Peter Flom Fair point. When I say "No more distorted" I mean that the distribution of the variable is no less uniform in layer X than it was in the previous layer, which I'll call X+1. So for some hypothetical "uniformness score" where X+1 = 7, the score for X ~ 7. –  EpiGrad Aug 31 '11 at 22:19
I think that, to answer this precisely, you need a "uniformness score" that is not hypothetical. Then you could do some sort of test like bootstrap or permutation test. –  Peter Flom Sep 2 '11 at 11:13
That sounds about right. Time to go invent one of those, unless someone has ideas? –  EpiGrad Sep 2 '11 at 20:36
It would be useful if you could include plots of the distributions you're comparing. But I suspect that the statistic (for comparing the distributions) won't be as big of a problem as the null distribution of the statistic. Do you have just the one network? Do you have a model for how the data arise, whether under the null or under the alternative? –  Karl Sep 3 '11 at 3:03
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