[R-sig-phylo] PIC vs. PGLS

Joe Felsenstein joe at gs.washington.edu
Mon Sep 27 20:19:37 CEST 2010

> Simon Blomberg wrote on 27/09/2010 08:10:

(Ted Garland had written:)

> >>Another interesting technical point.  In general, PIC and PGLS are the 
> >>same, especially if you stick with a simple Brownian motion model of 
> >>character evolution.  However, their complete mathematical identity 
> >>has not, to my knowledge, been proven.

(Simon Blomberg:)
> >I have a proof. I have a paper submitted to Sys. Biol. on the topic.
> You can find in my book the R code to get exactly the same coefficients 
> with PICs and PGLS. This works as long as the tree is ultrametric (for 
> equal variance). It's not a formal proof, of course, but a strong suspicion.

I can't speak on behalf of PGLS, but if one uses contrasts,
the ML analysis is of course using an i.i.d. model with zero expectations
of the contrasts.   It is provable that this will be identical (getting
the same P values and estimates) to a REML ("reduced" or "restricted" maximum
likelihood) on the full covarying Brownian motion model.   If PGLS gets
identical results to that, then that proves the identity to contrasts analysis.

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

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