Ross,

An interesting question. I understand it as that you want to test if
the trait is overdispersed relative to phylogeny, which still makes me
think that measures of 'phylogenetic signal' might be still be useful,
even though the typical interpretation is 'signal' as 'heritability'.
I would try some toy examples with smallish trees and artificial data
and play with different signal measures; particularly your idea
regarding that the variance is high at the level of closest
relatedness suggests that you perhaps should investigate Blomberg's K
as a measure, rather than Pagel's lambda:

Blomberg, S. P., T. Garland, and A. R. Ives. 2003. Testing for
phylogenetic signal in comparative data: behavioral traits are more
labile. Evolution 57.

However, your soft polytomies are worrisome; I suggest using the MPT
or posterior tree sample, if such exists, or considering resolving
those polytomies somehow.

Cheers,
-Dave

On Tue, Mar 14, 2017 at 5:45 AM, Ross Mounce <ross.mou...@gmail.com> wrote:
> Hi all,
>
> I'm interested in the distribution of a non-heritable binary
> trait/observation across a large tree 1000+ tip tree. The tree is
> non-distinct in shape and balance, it is neither fully pectinate nor fully
> balanced. It has many soft polytomies too.
>
> I believe the distribution of this trait to be significantly stratified
> such that just for the sake of explanation, every other tip is "present"
> for the trait. So essentially I'm interested in testing the evenness of
> distribution of "present" tips across the tree.
>
> In this instance it doesn't seem to me that I should be testing for
> "phylogenetic signal" or using models that do that, nor am I testing the
> randomicity of distribution of the trait.
> Specifically, I want to test if the observed distribution is significantly
> close to "perfect" stratification for the given number of "presences"
> (which is ~33% of the tips of the tree), on the given fixed tree shape.
>
> TL;DR
>
> How can I meaningfully test the evenness of the distribution of a binary
> trait across a tree, with R?
>
>
> Any ideas?
>
> Thanks,
>
> Ross
>
>
> --
> --
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> Ross Mounce, PhD
> Software Sustainability Institute Fellow
> Dept. of Plant Sciences, University of Cambridge
> www.rossmounce.co.uk <http://rossmounce.co.uk/>
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-- 
David W. Bapst, PhD
Adjunct Asst. Professor, Geology and Geol. Eng.
South Dakota School of Mines and Technology
501 E. St. Joseph
Rapid City, SD 57701

http://webpages.sdsmt.edu/~dbapst/
http://cran.r-project.org/web/packages/paleotree/index.html

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