Controlling for phylogeny in linear models I have run a linear model to test for a relationship between two continuous traits in a sample of 50 taxa. I'm using the packages phylolm and caper in R (just to check they tell me the same thing, which they do).
I noticed there are different options for fitting the models (e.g. Brownian motion (BM), Pagel's lambda (λ), Pagel's kappa, Pagel's delta, Ornstein–Uhlenbeck fixed root, Ornstein–Uhlenbeck random root). Even with a lot of reading, I'm still not sure which of these I should use.
I first ran the model as the default (Brownian motion), as I thought that was the correct thing to do. My model estimated a strong effect of my predictor variable.
I then noticed that many phylogenetic models use λ in the equation to estimate its maximum likelihood. It seems this is the correct thing to do to ensure model fit. However, I'm not sure why the default is BM if it is not appropriate. I've also seen some critiques of λ, but also of all the other methods.
When I use the λ model, the effect of my predictor variable becomes less strong and is no longer significant. I'm wondering if this is because of a lack of power because my sample size is not great, or if it's actually telling me something about the relationship between x and y.
λ is estimated to be 0.45, which indicates a moderate amount of phylogenetic signal.
When I look at AICc, it tells me the λ model fits better than the BM model. It also says the OU model fits even better, but I've read that is not be trusted at small sample sizes like mine.
I'm not interested in estimating phylogenetic signal - I just want to check that the relationship between x and y is not explained by shared ancestry. Doesn't the BM model do that?
What I don't understand is why my result is showing a significant relationship between x and y at λ = 0 and λ = 1 (which is the BM model), but not when λ = 0.45.
To summarise, my questions are:

*

*Is it necessary to run models of λ?

*If so, what is the BM option for?

*Does estimating λ in the model reduce its power relative to the BM model?


Firstly, I highly recommend reading Revell 2010 Met Ecol Evol for this kind of model.
Second, the lambda estimated in the model is the phylogenetic signal in the residuals (error) of the model, not in the traits you included as variables in the model. Each statistic controls for phylogenetic signal in the residuals differently. For example, Pagel's lambda stretches the tree to better reflect the evolutionary history of traits, instead of species. The value of the statistic is found via ML.
Therefore, estimating lambda in the model doesn't reduce it's power. On the contrary. It's advantage is that it allows fitting a model for traits that do not evolve under BM.
Also, I recommend you to dianose your model,in addition to testing model fit with AICc.
