Testing for Pagel's λ < 1.0

A recent commenter on the phytools blog asks:
I’m working on a comparative phylogenetic project and want to show that my data has phylogenetic signal. Using Pagel’s lambda I get lambda – 0.854. Is this value significant enough in that it’s safe to model the variable as Brownian motion?

I interpret this to mean "how can we test the alternative hypothesis that Pagel's λ is less then 1.0; against the null that λ=1.0." This is easy if we accept the contention that our likelihood ratio should be distributed as a Χ² with 1 degree of freedom (for one parameter, λ, estimated in the more complex model), an asymptotic property of likelihoods. (There may be some very good reasons - which we'll set aside for the moment - to question this assertion, for instance see a recent article by Carl Boettiger et al.)

This can be done most naturally using the fitContinuous function in 'geiger' - but we can also do it using entirely phytools functions, and since I developed and maintain phytools (not geiger), I'll demo that option.

First, let's simulate a tree & data for the demo, using the λ=0.854 obtained by the commenter. We will use geiger::lambdaTree for this part:

> library(phytools); library(geiger)
> # simulate tree
> tree<-pbtree(n=50)
> # simulate data with lambda=0.854
> x<-fastBM(lambdaTree(tree,0.854))

OK, now let's fit both models, compute the likelihood ratio, and then compare it to a Χ² distribution with one degree of freedom. We need to use a little bit of a work-around to fit the BM model using brownie.lite:

> # fit lambda model
> lambda<-phylosig(tree,x,method="lambda")
> lambda
$lambda
[1] 0.9497926
$logL
[1] -87.54412
> # get likelihood for bm model
> tree$mapped.edge<-matrix(tree$edge.length, nrow(tree$edge),1,dimnames=list(NULL,"1"))
> bm.logL<-brownie.lite(tree,x)$logL1
> # conduct hypothesis test using chi-square
> LR<--2*(bm.logL-lambda$logL)
> P<-pchisq(LR,df=1,lower.tail=F)
> P
[1] 0.1346832

So, in this simulated case at least, we cannot distinguish our estimated phylogenetic signal from Brownian motion.

An additional word of caution, however. Sampling error in the estimation of species means can also have the effect of depressing phylogenetic signal - so if the species means are estimated with a limited number of samples per species, it might be wise to conduct this analysis including estimated within species sampling variances. These can be supplied using the argument se in both phylosig and brownie.lite.