Interspecies correlation for neutrally evolving traits

A simple way to model phenotypic evolution is to assume that after splitting, the trait values of the sister species diverge as independent Brownian motions. Relying only on a prior distribution for the underlying species tree (conditioned on the number, n  , of extant species) we study the random vector (X1,…,Xn)(X1,…,Xn) of the observed trait values. In this paper we derive compact formulae for the variance of the sample mean and the mean of the sample variance for the vector (X1,…,Xn)(X1,…,Xn).The key ingredient of these formulae is the correlation coefficient between two trait values randomly chosen from (X1,…,Xn)(X1,…,Xn). This interspecies correlation coefficient takes into account not only variation due to the random sampling of two species out of n and the stochastic nature of Brownian motion but also the uncertainty in the phylogenetic tree. The latter is modeled by a (supercritical or critical) conditioned branching process. In the critical case we modify the Aldous–Popovic model by assuming a proper prior for the time of origin.