evolvabilityMeansMCMC {evolvability}

Calculate posterior distribution of average evolvability parameters of a G-matrix

Description

evolvabilityMeans calculates the average (unconditional) evolvability (e), respondability (r), conditional evolvability (c), autonomy (a) and integration (i) given the posterior distribution of a additive-genetic variance matrix using the approximation formulas described in Hansen and Houle (2008, 2009).

Usage

evolvabilityMeansMCMC(G_mcmc)

Arguments

G_mcmc

the posterior distribution of a variance matrix in the form of a table. Each row in the table must be one iteration of the posterior distribution (or bootstrap distribution). Each iteration of the matrix must be on the form as given by c(x), where x is a matrix. A posterior distribution of a matrix in the slot VCV of a object of class MCMCglmm is by default on this form.

Details

The equations for calculating the evolvability parameters are approximations, except for the minimum, maximum and unconditional evolvability which are exact. The bias of the approximations depends on the dimensionality of the G-matrix, with higher bias for few dimensions (see Hansen and Houle 2008). For low dimensional G-matrices, we recommend estimating the averages of the evolvability parameters using evolavbilityBetaMCMC over many random selection gradients ( randomBeta). The maximum and minimum evolvability, which are also the maximum and minimum respondability and conditional evolvability, equals the largest and smallest eigenvalue of the G-matrix, respectively.

Value

An object of class 'evolvabilityMeansMCMC', which is a list with the following components:

post.dist				The posterior distribution of the average evolvability parameters.
post.medians				The posterior medians and HPD interval of the average evolvability parameters.

Author(s)

Geir H. Bolstad

References

Hansen, T. F. & Houle, D. (2008) Measuring and comparing evolvability and constraint in multivariate characters. J. Evol. Biol. 21:1201-1219.
Hansen, T. F. & Houle, D. (2009) Corrigendum. J. Evol. Biol. 22:913-915.

Examples

# Simulating a posterior distribution
# (or bootstrap distribution) of a G-matrix:
G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3)
G_mcmc <- sapply(c(G), function(x) rnorm(10, x, 0.01))
G_mcmc <- t(apply(G_mcmc, 1, function(x) {
  G <- matrix(x, ncol = sqrt(length(x)))
  G[lower.tri(G)] <- t(G)[lower.tri(G)]
  c(G)
}))

# Simulating a posterior distribution
# (or bootstrap distribution) of trait means:
means <- c(1, 1.4, 2.1)
means_mcmc <- sapply(means, function(x) rnorm(10, x, 0.01))

# Mean standardizing the G-matrix:
G_mcmc <- meanStdGMCMC(G_mcmc, means_mcmc)

# Estimating average evolvability paramters:
evolvabilityMeansMCMC(G_mcmc)