Example 3: Phylogenetic Signal
We may test for phylogenetic signal in the data by finding the tree transformation parameter lambda that maximizes the likelihood of the comparative model (Pagel 1997,1999; Revell 2010).
Again we use a simulated phylogeny and dataset with missing observations from the example in Example 1: Getting Started):
library(Rphylopars)
library(phytools) # for simulating pure-birth phylogenies
set.seed(21) # Set the seed for reproducible results
trait_cov <- matrix(c(4,2,2.2,2,3,1.5,2.2,1.5,2.5),nrow = 3,ncol = 3)
tree <- pbtree(n = 20)
To estimate Pagel's lambda, simply include the argument model="lambda":
p_lambda <- phylopars(trait_data = sim_data$trait_data,
tree = sim_data$tree,model = "lambda")
p_lambda # Estimated trait covariance and Pagel's lambda
Now we fit a model assuming a star phylogeny, which is the null hypothesis when testing for significant phylogenetic signal under Pagel's lambda (i.e., lambda=0).
p_star <- phylopars(trait_data = sim_data$trait_data,tree = sim_data$tree,model = "star")
p_star # Estimated trait covariance, fixed Pagel's lambda = 0
Now we use the likelihood ratio to test for phylogenetic signal:
chi_square <- as.double(2*(logLik(p_lambda) - # 2*(logLik_alt - logLik_null)
logLik(p_star)))
degrees_freedom <- p_lambda$npars - p_star$npars # df = difference in # model parameters
pchisq(q = chi_square,df = degrees_freedom,lower.tail = FALSE) # p-value
The test is significant (p<0.05), suggesting significant phylogenetic signal in the phylogenetic residuals (Revell 2010). NOTE: this method jointly fits a single lambda parameter for all traits (and their covariances), as opposed to fitting a different lambda for each trait (Freckleton 2012). When applied to a single trait, this method is identical to the phylosig function in phytools, except that 1) phytools fits models by maximum likelihood whereas Rphylopars uses restricted maximum likelihood (which can be overridden by setting REML=FALSE in the phylopars function), and 2) lambda is constrained between 0 and 1 in Rphylopars whereas phytools does not set the upper limit of Pagel's lambda to 1.