Hi Marguerie, Simone and everyone,

Like Marguerie says there is no need to transform the OU process. If you have 
fitted an OU process, then you know the joint  distribution of all the the 
species and nodes. It is a multivariate normal, and you can use standard 
formulas for prediction of the nodes. We gave a general formula for such 
prediction in Martins and Hansen (1997. Am. Nat. 149: 646-667), but it is just 
standard prediction theory. The one thing to add to Marguerite's descriptions 
is that you need to include the fixed effects (predicted optima) for the nodes 
in your prediction. 

While I am at it, let me echo Simone and Marguerite's warnings. The predicted 
ancestral states will reflect the process you assumed to predict them. Hence, 
if you  use them to make inferences about evolution, you will recover your own 
assumptions. I.e. if you predict from a model with no trend, you will find no 
trend, etc. Many comparative studies are flawed for this reason. 



> On Jun 12, 2018, at 09:43, Simone Blomberg <s.blombe...@uq.edu.au> wrote:
> I would add an extra caveat to Marguerite’s excellent post: Most researchers 
> work with extant taxa only, ignoring extinction. This causes a massive 
> ascertainment bias, and the character states of the extinct taxa can often be 
> very different to the ancestral state reconstructions, particularly if the 
> evolutionary model is wrong. Eg. there has been an evolutionary trend for 
> example. Ancestral state reconstructions based only on extant taxa should be 
> treated as hypotheses to be tested with fossil data. I wouldn’t rely on them 
> for much more.
> Cheers,
> Simone.
> Sent from my iPhone
> On 12 Jun 2018, at 4:59 pm, Marguerite Butler <mbutler...@gmail.com 
> <mailto:mbutler...@gmail.com>> wrote:
>> Aloha all,
>> There is no requirement for an ultra metric tree in the formulae reported in 
>> Butler-King 2004. Interested investigators should in particular read the 
>> supplementary materials where the mathematical details are worked out. 
>> We do generally use ultrametric trees because as comparative biologists, it 
>> is more straightforward to think about evolution in units of time rather 
>> than in terms of mutational units, etc. However this is by choice, not any 
>> methodological limitation. 
>> Once the model parameters are found, the phylogenetic variance-covariance 
>> matrix defined by the alpha, thetas, and sigmas can be used to compute 
>> ancestral states using a weighted least squares reconstruction method 
>> (instead of the typical BM var-cov matrix). The mapping of the alphas, 
>> thetas, and sigmas onto the tree are incorporated into this V-COV matrix, so 
>> that accounts for the OU model. 
>> NOTES: 
>> 1) without knowing why you are doing this, I do feel compelled to warn you 
>> that it is unclear why one would want to estimate ancestral states for 
>> poorly-fitting models. Be careful!
>> 2) I hope you realize that ancestral states are in general poorly estimated, 
>> even assuming the “correct” model. This is because there is less and less 
>> information to anchor the values as you get farther from the tips, similar 
>> to the root estimation problem described below.  This issue was clearly 
>> exposed in Schluter et al 1997 (and less famously so in Butler and Losos 
>> 1997). These depressing results were among the motivations for developing 
>> model-fit approaches in the first place. 
>> 3) In 2008/2009 the algorithms in OUCH, SLOUCH, and possibly other methods 
>> have changed in the estimation of the value of the root state (X0) which is 
>> an internal calculation in fitting the model.  Ho and Ane 2013, and Hansen 
>> et al 2008 both reported that the root state X0 is ill-defined (unless there 
>> are fossil data to anchor the value). This makes sense intuitively, as all 
>> of the information is from the tips, and the root is very far down the tree. 
>> A reasonable assumption is that it is distributed according to the 
>> stationary distribution of the OU process (X0 ~ N(theta(0), sigma^2/2*alpha) 
>> and this assumption is what these methods now employ. 
>> 4) Whatever you end up doing, do check for the robustness of your results 
>> with parametric bootstrap on your fitted models (a la Boettinger et al 
>> 2012). As many investigators have reported, these parameters can have large 
>> confidence intervals, and can covary with one another (being on a likelihood 
>> ridge, etc.). But do note that even when parameters may not be uniquely 
>> identifiable, it may still be possible to have robust model selection (see 
>> Cressler et al 2015).  So perhaps you want to fit ancestral states to see if 
>> the different models give you the same states? IDK?
>> So in short, yes, you can do it, with any number of methods. But why? If you 
>> can answer your biological question with methods that do not involve 
>> estimation of a parameter that is inherently fraught with error, it might be 
>> better to go another way. Bottom line - use caution and be thoughtful!
>> I am sure if I have made any errors Aaron, Clay, or Thomas will help.
>> Hope this helps
>> Marguerite
>> Schluter, D., T. Price, A. O. Mooers, and D. Ludwig. 1997. Likelihood of 
>> ancestor states in adaptive radiation. Evolution 51:1699–1711.
>> Butler, M. A., and J. B. Losos. 1997. Testing for unequal amounts of 
>> evolution in a continuous character on dif- ferent branches of a 
>> phylogenetic tree using linear and squared-change parsimony: an example 
>> using Lesser Antillean Anolis lizards. Evolution 51:1623–1635. 
>> Hansen T.F., Pienaar J., Orzack S.H. 2008. A comparative method for studying 
>> adaptation to a randomly evolving environment. Evolution 62:1965–1977.
>> Ho L.S.T., Ané C.. 2014. Intrinsic inference difficulties for trait 
>> evolution with Ornstein-Uhlenbeck models. Methods Ecol. Evol. 2:1133–1146.
>> Cressler C., Butler M.A., and King A. A. (2015) Detecting adaptive evolution 
>> in phylogenetic comparative analysis using the Ornstein-Uhlenbeck model.  
>> Sys. Bio. 64(6):953-968. DOI: 10.1093/sysbio/syv043
>> Boettiger C., Coop G., Ralph P. 2012. Is your phylogeny informative? 
>> Measuring the power of comparative methods. Evolution 66: 2240–2251. 
>> ____________________________________________
>> Marguerite A. Butler
>> Professor
>> Department of Biology 
>> 2538 McCarthy Mall, Edmondson Hall 216 
>> Honolulu, HI 96822
>> Office: 808-956-4713
>> Dept: 808-956-8617
>> Lab:  808-956-5867
>> FAX:   808-956-4745
>> http://butlerlab.org <http://butlerlab.org/>
>> http://manoa.hawaii.edu/biology/people/marguerite-butler 
>> <http://manoa.hawaii.edu/biology/people/marguerite-butler>
>> http://www2.hawaii.edu/~mbutler <http://www2.hawaii.edu/~mbutler>
>>> On Jun 11, 2018, at 7:33 PM, Simone Blomberg <s.blombe...@uq.edu.au 
>>> <mailto:s.blombe...@uq.edu.au>> wrote:
>>> 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.
>>> Cheers,
>>> Simone.
>>> Sent from my iPhone
>>>> On 12 Jun 2018, at 8:01 am, David Bapst <dwba...@tamu.edu 
>>>> <mailto:dwba...@tamu.edu>> 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.
>>>> -Dave
>>>>> On Sun, Jun 10, 2018 at 12:25 PM, Lucas Jardim <lucas.ljard...@gmail.com 
>>>>> <mailto:lucas.ljard...@gmail.com>> 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 data
>>>>> 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 best
>>>>> approach as optimum values would not be included into the reconstruction
>>>>> process.
>>>>> Best,
>>>>> --
>>>>> Lucas Jardim
>>>>> Doutor em Ecologia e Evolução
>>>>> Bolsista do INCT-EECBio (Ecologia, Evolução e Conservação da
>>>>> Biodiversidade)
>>>>> Instituto de Ciências Biológicas
>>>>> Laboratório de Ecologia Teórica e Síntese
>>>>> Universidade Federal de Goiás
>>>>> http://dinizfilho.wix.com/dinizfilholab 
>>>>> <http://dinizfilho.wix.com/dinizfilholab>
>>>>>       [[alternative HTML version deleted]]
>>>>> _______________________________________________
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>>>> -- 
>>>> David W. Bapst, PhD
>>>> Asst Research Professor, Geology & Geophysics, Texas A & M University
>>>> https://github.com/dwbapst/paleotree <https://github.com/dwbapst/paleotree>
>>>> Google Calendar: https://goo.gl/EpiM4J <https://goo.gl/EpiM4J>
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