Phylogenetic signal with K and λ

A recent Google search string that led a reader to my blog was the following:

different phylogenetic signal with K vs lambda.

Phylogenetic signal is generally recognized to be the tendency of related species to resemble one another; and Blomberg et al.'s (2003) K and Pagel's (1999) λ are two quantitative measures of this pattern. The metrics are quite different from one another. λ is a scaling parameter for the correlations between species, relative to the correlation expected under Brownian evolution. K is a scaled ratio of the variance among species over the contrasts variance (the latter of which will be low if phylogenetic signal is high). λ has a nice natural scale between zero (no correlation between species) and 1.0 (correlation between species equal to the Brownian expectation). λ itself is not a correlation, but a scaling factor for a correlation, so λ>1.0 is theoretically possible. However, depending on the structure of the tree, λ>>1.0 is usually not defined. K, a variance ratio, is rescaled by dividing by the Brownian motion expectation. This gives it the property of having an expected value of 1.0 under Brownian evolution, but K for empirical and simulated datasets can sometimes be >>1.0. Since they measure the qualitative tendency towards similarity of relatives in entirely different ways, we have no real expectation that they will be numerically equivalent - except (by design) under Brownian motion evolution.

So, accepting that λ & K are different measures of phylogenetic signal, we might reasonably ask - do statistical tests based on each measure tend to find significant phylogenetic signal for the same datasets? One way to crudely test this would be simulate datasets for various generating λ and then ask if significant tests based on λ and K tend to be associated.

require(phytools); require(geiger)
nrep<-1000
P.K<-P.l<-l<-vector()
for(i in 1:nrep){
   tree<-pbtree(n=50)
   l[i]<-runif(n=1)
   x<-fastBM(lambdaTree(tree,l[i]))
   K<-phylosig(tree,x,test=T)
   lambda<-phylosig(tree,x,method="lambda",test=T)
   P.K[i]<-K$P
   P.l[i]<-lambda$P
}

Compute the fraction of significant tests using K & λ, and the fraction expected to be the same (either both significant or both non-significant) if they are independent (this we compute as the product of the preceding fractions plus the product of 1 minus each fraction):

> pK<-mean(P.K<=0.05); pK
[1] 0.417
> pL<-mean(P.l<=0.05); pL
[1] 0.514
> pK*pL+(1-pK)*(1-pL)
[1] 0.497676

Now, the fraction of times that tests on λ & K yield the same result (i.e., either both significant or both non-significant):

> mean(((P.K<=0.05)==(P.l<=0.05)))
[1] 0.765

Obviously, there is some relation between the tests of the two metrics, but it is not extremely strong. Let's also ask if the average generating value of λ is higher when either or both λ and K produce a significant result:

> # first lambda
> mean(l[P.l<=0.05]) # lambda significant
[1] 0.6671113
> mean(l[P.l>0.05]) # not significant
[1] 0.2884
> # then K
> mean(l[P.K<=0.05]) # K significant
[1] 0.6842788
> mean(l[P.K>0.05]) # not significant
[1] 0.339131
> # then P & lambda
> mean(l[(P.K<=0.05)*(P.l<=0.05)==1]) # lambda & K significant
[1] 0.7352098
> mean(l[(P.K<=0.05)*(P.l<=0.05)==0]) # neither significant
[1] 0.3484733