This question was asked by Peter Smits and by Edwin Lebrija Trejos. In
Peter Smit's question it was this:
> I have a similar question to Edwin. I too am working with a hierarchical
> Bayesian model, though I've implemented it in stan. I've included a
> phylogenetic random effect term, which is modeled as being distributed as
> multivariate normal with known covariance matrix up to a constant,
> sigma_{phy}. This follows Lynch '91 Am Nat and Housworth et al. '04 Am Nat
> by drawing on the similarity with the "animal model" from quantitative
> genetics.
>
> My question is about the scaling of the covariance matrix: is it necessary
> to have the the diagonal terms satisfy x <= 1 and all the off diagonals to
> be 0 <= x < 1? I have a time scaled phylogeny from which I have my
> covariance matrix, so currently all elements of the matrix are not scaled
> so that the greatest distance from the root to a tip is 1. Currently, the
> elements of the matrix are just the sum of the shared branch lengths. Is
> this appropriate? Why or why not?
>
>
I hope people will correct me on this, but my take is:
1. To infer a variance term for the "phylogenetic random term"
one has to scale that term somehow, and then the variance
will specify how many of those scaling units are in this term. If
you had the tree depth be 2 instead of 1, the variance inferred
would then be half as great, because it would be saying
how many multiples of 2 rather than how many multiples of 1.
2. I have not had time to read through all of Edwin's code to
see exactly what the model is. However, note that there is a
distinction between "individual" effects that are on a whole
species, and "individual" effects that are on a single sampled
individual. The latter are taking into account that the mean
phenotype of each species is only known from a finite sample
of individuals. So it is taking into account within-species
phhenotypic variation and finiteness of sample sizes. The
relevant methods there are by Ives, Midford and Garland
(2007) and by me (2008). Ives et al. assume within-species
phenotypic variances and covariances are known, I give
methods for inferring them.
The methods of Lynch and Housworth are in effect
either assuming a sample size of 1 for each species,
or are considering their "individual" effect to be on the
whole species.
Within-species phenotypic variation can be a substantial
problem, as Ricklefs and Starck (1996) noted. In their
example, the largest contrasts were between closely-related
species. They suspected that this was due to within-species
phenotypic variation (which shows up as variance due to
sampling of the individual specimens measured). It
causes trouble because the small branch lengths between
closely-related species are an inadequate predictor of
how different the species means will be. In effect, the
model is wrong so some of the changes attributed to
between-species evolution are actually within-species
sampling variation (phenotypic variance).
Joe
----
Joe Felsenstein j...@gs.washington.edu
Department of Genome Sciences and Department of Biology,
University of Washington, Box 355065, Seattle, WA 98195-5065 USA
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