The phylogenetic heritability, \(H^2\), is defined as the ratio of the genetic variance over the total phenotypic variance expected at a given evolutionary time t (measured from the root of the tree). Thus, the phylogenetic heritability connects the parameters alpha, sigma and sigmae of the POUMM model through a set of equations. The functions described here provide an R-implementation of these equations.
alpha(H2, sigma, sigmae, t = Inf) sigmaOU(H2, alpha, sigmae, t = Inf) sigmae(H2, alpha, sigma, t = Inf) H2e(z, sigmae, tree = NULL, tFrom = 0, tTo = Inf)
| H2 | Phylogenetic heritability at time t. |
|---|---|
| sigmae | Numeric, environmental phenotypic deviation at the tips. |
| t | Numeric value denoting evolutionary time (i.e. distance from the root of a phylogenetic tree). |
| alpha, sigma | Numeric values or n-vectors, parameters of the OU process; alpha and sigma must be non-negative. A zero alpha is interpreted as the Brownian motion process in the limit alpha -> 0. |
| z | Numerical vector of observed phenotypes. |
| tree | A phylo object. |
| tFrom, tTo | Numerical minimal and maximal root-tip distance to limit the calculation. |
All functions return numerical values or NA, in case of invalid parameters
The function sigmae uses the formula H2 = varOU(t, alpha, sigma) / (varOU(t, alpha, sigma) + sigmae^2)
alpha: Calculate alpha given time t, H2, sigma and sigmae
sigmaOU: Calculate sigma given time t, H2 at time t, alpha
and sigmae
sigmae: Calculate sigmae given alpha, sigma, and H2 at
time t
H2e: "Empirical" phylogenetic heritability estimated
from the empirical variance of the observed phenotypes and sigmae
This function is called sigmaOU and not simply sigma to avoid a conflict with a function sigma in the base R-package.
OU
# At POUMM stationary state (equilibrium, t=Inf) H2 <- H2(alpha = 0.75, sigma = 1, sigmae = 1, t = Inf) # 0.4 alpha <- alpha(H2 = H2, sigma = 1, sigmae = 1, t = Inf) # 0.75 sigma <- sigmaOU(H2 = H2, alpha = 0.75, sigmae = 1, t = Inf) # 1 sigmae <- sigmae(H2 = H2, alpha = 0.75, sigma = 1, t = Inf) # 1 # At finite time t = 0.2 H2 <- H2(alpha = 0.75, sigma = 1, sigmae = 1, t = 0.2) # 0.1473309 alpha <- alpha(H2 = H2, sigma = 1, sigmae = 1, t = 0.2) # 0.75 sigma <- sigmaOU(H2 = H2, alpha = 0.75, sigmae = 1, t = 0.2) # 1 sigmae <- sigmae(H2 = H2, alpha = 0.75, sigma = 1, t = 0.2) # 1