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)

## Arguments

H2 Phylogenetic heritability at time t. Numeric, environmental phenotypic deviation at the tips. Numeric value denoting evolutionary time (i.e. distance from the root of a phylogenetic tree). 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. Numerical vector of observed phenotypes. A phylo object. Numerical minimal and maximal root-tip distance to limit the calculation.

## Value

All functions return numerical values or NA, in case of invalid parameters

## Details

The function sigmae uses the formula H2 = varOU(t, alpha, sigma) / (varOU(t, alpha, sigma) + sigmae^2)

## Functions

• 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

## Note

This function is called sigmaOU and not simply sigma to avoid a conflict with a function sigma in the base R-package.

OU

## Examples

# 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