Hi, Agus. The variance-covariance matrix comes from the tree and the
evolutionary model, not the data. Each entry between taxa A and B in the
VCV is how much covariance I should expect between data for taxa A and B
simulated up that tree using that model. I don't want to be *that guy*, but
O'Meara et al. (2006)
https://onlinelibrary.wiley.com/doi/10.1111/j.0014-3820.2006.tb01171.x has
a fairly accessible explanation of this (largely b/c I was just learning
about VCVs when working on that paper). Hansen and Martins (1996)
https://onlinelibrary.wiley.com/doi/10.1111/j.1558-5646.1996.tb03914.x have
a much more detailed description of how you get these covariance matrices
from microevolutionary processes.
Typically, ape::vcv() is how you get a variance covariance for a phylogeny,
assuming Brownian motion and no measurement error. It just basically takes
the history two taxa share to create the covariance (or variance, if the
two taxa are the same taxon). A different approach, which seems to be what
you're doing, would be to simulate up a tree many times, and then for each
pair of taxa (including the pair of a taxon with itself, the diagonal of
the VCV), calculate the covariance. These approaches should get the same
results, though the shared history on the tree approach is faster.
Best,
Brian
_______________________________________________________________________
Brian O'Meara, http://www.brianomeara.info, especially Calendar
<http://brianomeara.info/calendars/omeara/>, CV
<http://brianomeara.info/cv/>, and Feedback
<http://brianomeara.info/teaching/feedback/>
Associate Professor, Dept. of Ecology & Evolutionary Biology, UT Knoxville
Associate Head, Dept. of Ecology & Evolutionary Biology, UT Knoxville
On Sat, Aug 25, 2018 at 1:16 PM Agus Camacho <agus.cama...@gmail.com> wrote:
Dear list users,
I am trying to make an easy R demonstration to teach the
variance-covariance matrix to students. However, After consulting the
internet and books, I found myself facing three difficulties to understand
the math and code behind this important matrix. As this list is answered by
several authors of books of phylocomp methods, thought this might make an
useful general discussion.
Here we go,
1) I dont know how to generate a phyloVCV matrix in R (Liams kindly
described some options here
<
http://blog.phytools.org/2013/12/three-different-ways-to-calculate-among.html
but I cannot tell for sure what is X made of. It would seem a dataframe of
some variables measured across species. But then, I get errors when I
write:
tree <- pbtree(n = 10, scale = 1)
tree$tip.label <- sprintf("sp%s",seq(1:n))
x <- fastBM(tree)
y <- fastBM(tree)
X=data.frame(x,y)
rownames(X)=tree$tip.label
## Revell (2009)
A<-matrix(1,nrow(X),1)%*%apply(X,2,fastAnc,tree=tree)[1,]
V1<-t(X-A)%*%solve(vcv(tree))%*%(X-A)/(nrow(X)-1)
## Butler et al. (2000)
Z<-solve(t(chol(vcv(tree))))%*%(X-A)
V2<-t(Z)%*%Z/(nrow(X)-1)
## pics
Y<-apply(X,2,pic,phy=tree)
V3<-t(Y)%*%Y/nrow(Y)
2) The phyloVCV matrix has n x n coordinates defined by the n species, and
it represents covariances among observations made across the n species,
right?. Still, I do no know whether these covariances are calculated over
a) X vs Y values for each pair of species coordinates in the matrix, across
the n species, or b) directly over the vector of n residuals of Y, after
correlating Y vs X, across all pairs of species coordinates. I think it may
be a) because, by definition, variance cannot be calculated for a single
value. I am not sure though, since it seems the whole point of PGLS is to
control phylosignal within the residuals of a regression procedure, prior
to actually making it.
3) If I create two perfeclty correlated variables with independent
observations and calculate a covariance or correlation matrix for them, I
do not get a diagonal matrix, with zeros at the off diagonals (ex. here
<
https://www.dropbox.com/s/y8g3tkzk509pz58/vcvexamplewithrandomvariables.xlsx?dl=0
),
why expect then a diagonal matrix for the case of independence among the
observations?
Thanks in advance and sorry if I missed anything obvious here!
Agus
Dr. Agustín Camacho Guerrero. Universidade de São Paulo.
http://www.agustincamacho.com
Laboratório de Comportamento e Fisiologia Evolutiva, Departamento de
Fisiologia,
Instituto de Biociências, USP.Rua do Matão, trav. 14, nº 321, Cidade
Universitária,
São Paulo - SP, CEP: 05508-090, Brasil.
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