[R-sig-phylo] Multiple regressions with continuous and categorical data

Simon Blomberg s.blomberg1 at uq.edu.au
Wed Apr 9 01:40:30 CEST 2008


On Tue, 2008-04-08 at 15:20 -0700, Joe Felsenstein wrote:
> On Mon, Apr 07, 2008 at 12:38:59PM +1000, Simon Blomberg wrote:
> > We have very good reasons for suspecting this. Brownian motion
> > (actually, the Weiner process) is almost certainly a poor model for the
> > evolution of any trait you care to name. It assumes a linear
> > relationship between variance and time, there are no bounds to
> > evolution, and although the process is continuous it is nowhere
> > differentiable, making the calculation of instantaneous rates of
> > evolution impossible.
> 
> Genetic drift has much the same effect.  Brownian motion is a poor
> model, and so is Ornstein-Uhlenbeck, but just as democracy is the
> worst method of organizing a society "except for all the others",
> so these two models are all we've really got that is tractable.
> Critics will be admitted to the event, but only if they carry with
> them another tractable model.

This is a line of research that I am pursuing. :-)

Simon.

> 
> > > Anyone can verify that IC and PGLS are the same thing. You can  
> > > translate the matrix equations to equations to look like  
> > > Felsenstein's, you just write out the terms of the matrix as branch  
> > > lengths and multiply them out. They will be the same.
> > 
> > Then please do this! The problem is that Felsenstein presented an
> > algorithm, not a set of equations relating the response and predictor
> > variables. I'm sure you could do what you say, but as it involves the
> > choleski decomposition of the vcv matrix, which is itself a complicated
> > linear transformation of the branch lengths, I don't think it is
> > trivial. But if it is, I would like to see the proof!
> 
> The contrasts can be thought of as matrices of this form:  if you take
> the contrast between species 1 and 2, the transform is the matrix:
> 
>         1    -1    0   0 ...  0  
>         a   (1-a)  0   0 ...  0
>         0     0    1   0 ...  0
>         0     0    0   1 ...  0
>           ...
>         0     0    0   0 ...  1
> 
> (so that there is a little 2x2 box and the rest is the identity matrix).
> Of course I made it look easy by having it be species 1 and 2 but the
> principle is the same.
> 
> The result is that the vector of observations now has item 1 independent
> of the others, with a variance v_1+v_2 (I am using the LaTeX notation which
> indicates subscript by underscore).  The remaining variables 2, 3, ..., n
> have a tree with one fewer tip.  You continue this.  If you call the
> contrast matrices (like the above)  C_1, C_2, ... C_{n-1}
> then the net transform is
>      C_{n-1} C_{n-2} ... C_2 C_1
> and this leaves us with a diagonal covariance matrix.  So getting the
> transform that diagonalizes the covariance matrix is easy.  In these tree
> matrices, they are easy to diagonalize in O(n^2) operations instead of
> O(n^3) as usual.  And yes, they are a nonlinear function of the branch
> lengths which is most easily defined recursively.
> 
> In effect, the contrasts are just a convenient computational scheme for
> diagonalizing the covariance matrix.
> 
> Joe (old enough to remember how to delete lines from a reply)
> ----
> Joe Felsenstein         joe at gs.washington.edu
>  Department of Genome Sciences and Department of Biology,
>  University of Washington, Box 355065, Seattle, WA 98195-5065 USA
-- 
Simon Blomberg, BSc (Hons), PhD, MAppStat. 
Lecturer and Consultant Statistician 
Faculty of Biological and Chemical Sciences 
The University of Queensland 
St. Lucia Queensland 4072 
Australia
Room 320 Goddard Building (8)
T: +61 7 3365 2506
http://www.uq.edu.au/~uqsblomb
email: S.Blomberg1_at_uq.edu.au

Policies:
1.  I will NOT analyse your data for you.
2.  Your deadline is your problem.

The combination of some data and an aching desire for 
an answer does not ensure that a reasonable answer can 
be extracted from a given body of data. - John Tukey.



More information about the R-sig-phylo mailing list