It is possible to transform the branch lengths of a non-ultrametric tree to 
make the OU process equivalent to a BM process (on the transformed branch 
lengths), but there’s an extra correction needed at the tips (i.e. need to 
transform the data too via a diagonal matrix). It makes likelihood calculations 
a lot faster, but I have not seen this used for ancestral state estimation. 
This approach is described in Ho & Ané (2014)
It’s implemented in the R packages 
Rphylopars<>, or 

On Jun 14, 2018, at 8:25 AM, David Bapst 
<<>> wrote:

Simone, others,

Perhaps I misunderstand, but I believe Slater (2014) shows that there is no
correct branch length transformation for a non-ultrametric tree under an OU
process. The correct transformation that Slater applies is Hansen's, yes,
but to the VCV matrix. Slater does point out there is one VCV-free approach
that does work for non-ultrametric trees, and that is the pruning-algorithm
that was at the time implemented in diversitree.

Some quotes from Slater (2014):

"The time-dependent decay in covariance that accrues after divergence under
OU results in the rather unique phenomenon that non-ultrametric trees
cannot be rescaled according to an OU process as ultrametric trees can."

"This is a necessary error, because a treelike structure cannot be
represented with different shared path lengths for pairs of descendents
that branch from the same ancestor. However, using Hansen’s (1997) formula
to directly transform V results in the correct pattern (Fig. 1d); Vt4,t1 is
higher than Vt4,t2 because the phylogenetic distance between t1 and t4, and
thus the time available for covariance decay, is lower as a result of t1
going extinct before the present. It is additionally worth noting that
branch length scaling results in underestimated variances for fossil taxa
(upper left diagonal of 1b, c, and d); branch length transformation reduces
the variance of t1 compared with that of the other species, while direct
matrix scaling increases the variance of the fossil taxon to a similar
value as in the extant species."

"For example, although problems with branch length rescaling render
Freckleton’s (2012) approach inappropriate, it is still possible to use
VCV-free approaches to rapidly compute likelihoods with non-ultrametric
data. The pruning algorithm used by FitzJohn (2012) in his DIVERSITREE
package, which is based on Felsenstein (1973, 1981), can be used to fit OU
models to non-ultrametric trees because it works on per branch basis,
rather than by scaling all branch lengths simultaneously."

Again, this is (I think) common knowledge that OU
branch-length-transformations cannot be safely applied to non-ultrametric
trees, in our post-Slater-2014 world. However, following Lucas's comment, I
wanted to make sure the record is correct for those who stumble on this
thread later. As Slater (2014) points out, us paleontologists are ever
increasingly users of phylogenetic comparative methods, and so as
methods-developers, we should always take into account the possibility that
a user's dataset may include fossil tips.


PS: I've cc'd Graham so he can rebuke me if I'm way off-the-mark here.

On Tue, Jun 12, 2018 at 12:33 AM, Simone Blomberg 

This sounded wrong to me, as the OU process should be agnostic to the
dataset: There are no restrictions inherent in the OU process that apply
particularly to phylogenetic data, whether the tree is ultrametric or not.
I re-read Slater 2014 and it is clear that you can use branch length
transformations with OU, so long as you use the (correct) Hansen formula,
not the Butler-king formula, which does indeed require an ultrametric tree.



Sent from my iPhone

On 12 Jun 2018, at 8:01 am, David Bapst 
<<>> wrote:

Just to follow off what Lucas said, but please note you cannot rescale
branches of a phylogeny using an OU model when the tree is
non-ultrametric (such as when it contains extinct, fossil taxa as
tips). Slater (2014, MEE) discusses this more in a brief correction to
Slater (2013).

I don't know if anyone in this conversation has a non-ultrametric
tree, but I wanted to make that clear for anyone who stumbles on this
thread n the future using a google search.

On Sun, Jun 10, 2018 at 12:25 PM, Lucas Jardim <<>> wrote:
Hi Bruno,

You can transform the branches of your phylogeny using the estimated
parameters of OU models. Then, if those models describe the observed
adequatly, the transformed tree should model the observed data as a
Brownian motion model. So you can use an ancestral state reconstruction
based on Brownian motion model. However, I do not know if that is the
approach as optimum values would not be included into the reconstruction

Lucas Jardim
Doutor em Ecologia e Evolução
Bolsista do INCT-EECBio (Ecologia, Evolução e Conservação da
Instituto de Ciências Biológicas
Laboratório de Ecologia Teórica e Síntese
Universidade Federal de Goiás

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