Hi Julien, You are right that power is a big (and generally still unexplored in the PCM context) issue. One way to address the question is after estimating the parameters of the BM/OU/mvSLOUCH model you can calculate the covariance/correlation function of the traits, say Cov(Y)(t)/Corr(Y)(t) where Y is the 2D vector trait+environment variable. The mvSLOUCH package returns these, I guess similar as stationary(model_ou) in mvMorph? The of course one can look at the magnitude of the trait+environment correlation coefficient. Regarding power difficult to say, model selection BM v OUBM might be difficult if the sample is small. But on the other hand in the mvSLOUCH paper Bartoszek et. al. (2012) we had a n=33 Cervidae set and a 2dim OU+1dim BM clearly outperformed 3dim BM (by AICc). If the trait is related to the environment then I would hazard that this should be picked up. Estimation of individual parameters in the SDE can be a tough thing but from my experience the function Cov(Y)(now) is decently estimated. Which makes sense - this is a complex combination of the SDE parameters and the ML tries to fit them to the contemporary sample which is described by E(Y)(now), Cov(Y)(now). In the 1dim OU,BM cases this was studied quite well by us (Serik Sagitov) and Cecile Ane et. al. and there seems to be quite a bit of power to get at these parameter combinations.
Another possibility to look at is at the conditional expectation E(trait | environment)(now) - the evolutionary regression or its limit the optimal/limiting regression E(trait | environment)(infinity) . To get significance is not that easy, I guess the best way is a parametric bootstrap as you suggest below. But this depends on the pylogeny size, if it will not take overly long. I raised this point because depending on the model i.e. BM/OU/mvSLOUCH the design matrix of the regression should be different (Hansen et. al. 2008 discuss this at length). And the interpretation of the regression parameters might be different - e.g. you could have a phylogenetic lag in the estimated regression coefficient if the design matrix is just the measured variable. Best wishes Krzysztof > Hi Krzysztof, > > I agree with you that if you want to test explicitly adaptation in a > regression framework (i.e. environmental variables as predictors), > slouch/mvslouch is more sounding than doing a simple regression ((p)GLS) > or multiple regression. > I emphasize here that the coefficients of a regression model and the > correlations are really distincts. It's because all is about the nature > of the variation in the response variable in a regression model (and > it's why that changing the variables in a multiple regression assume > different nature of errors terms). > > The initial question was on how to obtain the correlations between the > traits, and possibly their significance, while CCA provides only the > correlation between the traits sets. > > The significance of the correlations computed through multivariate > approaches may be impeded by the number of parameters to estimate/sample > size (a power issue), but in multiple regression setting if there is > correlated independent variables the standard errors will be impacted > and thus the p-values (in a more misleading way). > It's also true that, as Krzystof pointed out, BM may not be the best > evolutionary model for describing the correlation in the residuals. > However the number of parameters to estimate will be even greater (for > OU multivariate model such as in mvSLOUCH), and we may lack power in the > significance tests of the correlations, no? > > > Julien > > ---------------------------------------- > > Date: Sat, 17 Oct 2015 14:09:25 +0200 > > From: [email protected] > > To: [email protected] > > CC: [email protected] > > Subject: Re: [R-sig-phylo] Question on pCCA (phyl.cca) > > > > Hi Sandra, Julien and Liam! > > One thought that I had is since you have an environmental variable and > one supposedly correlated with it is that then the slouch/mvSLOUCH (on > CRAN) models could be a possibility? In them you can have the > environmental variable(s) evolving as a Brownian motion (i.e. random > drift, no selecetion) and then the responding one(s) as an OU whose > primary optimum depends on the state of the environment. As far as I > followed from the discussion you had either both environment and > responding trait > > following a Brownian motion (meaning there is correlation between them > but not adaptation) or both following an OU process (meaning that the > environment is trying to adapt to something which might not be justified > biologically). > > > > Sandra if you think such a setup - a trait adapting to a primary optimum > defined by a randomly evolving environment (this is not the same as > being correlated with it) is something useful for you I can help you out > with using mvSLOUCH. > > > > You can have a look at: > > * Hansen, T.F.,Pienaar,J.,Orzack,S.H.,2008. A comparative method for > studying adaptation to a randomly evolving environment.Evolution > 62,1965–1977. (for slouch model) > > * Bartoszek, K.,Pienaar,J.,Mostad,P.,Andersson,S.,Hansen,T.F.,2012. A > phylogenetic comparative method for studying multivariate adaptation > .J.Theor.Biol. 314,204–215. (for mvSLOUCH model) > > > > Best wishes > > Krzysztof > > > > _______________________________________________ > > R-sig-phylo mailing list - [email protected] > > https://stat.ethz.ch/mailman/listinfo/r-sig-phylo > > Searchable archive at > http://www.mail-archive.com/[email protected]/ _______________________________________________ R-sig-phylo mailing list - [email protected] https://stat.ethz.ch/mailman/listinfo/r-sig-phylo Searchable archive at http://www.mail-archive.com/[email protected]/
