Reacnorm (version 0.2.1)

rn_gen_decomp: Compute the decomposition of the additive genetic variance and their \(\gamma\)- and \(\iota\)-decomposition

Description

This function calculates the total additive genetic variance of the reaction norm (\(V_{\text{Add}}\), and its \(\gamma\)-decomposition), the marginal additive genetic variance of the trait (\(V_{\text{A}}\)) and the additive genetic variance of plasticity (\(V_{\text{A}\times\text{E}}\), and its \(\iota\)-decomposition).

Usage

rn_gen_decomp(theta, G_theta, env = NULL, shape = NULL, X = NULL, fixed = NULL,
              wt_env = NULL, compute_gamma = TRUE, compute_iota  = TRUE,
              add_vars = NULL, width = 10)

Value

This function yields the values for \(V_{\text{Add}}\), \(V_{\text{A}}\) and \(V_{\text{A}\times\text{E}}\), as well as the \(\gamma\)- and \(\iota\)-decomposition if requested, formatted as a 1-row data.frame (data.frame, all numeric).

Arguments

theta

Average parameters of the shape function. It must be a named vector, with the names corresponding to the parameters in the shape expression. (numeric)

G_theta

Genetic variance-covariance matrix of the parameters. It can be of lesser dimensions than theta, see fixed parameter. (numerical matrix)

env

Vector of environmental values (numeric).

shape

Expression providing the shape of the reaction where x is the environment. For example: expression(a + b * x + c * x^2).

X

If the model used was linear in the parameters, the design matrix X of the model (numeric, incompatible with the arguments env and shape).

fixed

If some parameters of shape, included in theta are not included in the G_theta matrix, then those dimensions are considered as genetically "fixed". Hence, fixed should contain a vector of the index of those parameters. Otherwise (if all parameters vary genetically), and by default, fixed should be set as NA. (integer)

wt_env

Weights to apply to the env vector values, providing an information regarding their relative probability in the biological context. The weights must non-negative, and at least one must non-zero. The vector wt_env must be the same length as env. By default, no weighting is applied. (numeric)

compute_gamma

Should the \(\gamma\)-decomposition be performed? Default is TRUE. (boolean)

compute_iota

Should the \(\iota\)-decomposition be performed? Default is TRUE. (boolean)

add_vars

Optional, provide additive genetic variances \(V_{\text{Add}}\), \(V_{\text{A}}\) and \(V_{\text{A}\times\text{E}}\) (in that exact order!) computed from another model, e.g. a character-state model. These values will be used to scale the \(\gamma\)'s and \(\iota\)'s and be returned by the function for the respective additive genetic variances. (numeric vector of length 3)

width

Parameter for the integral computation. The integral is evaluated from mu - width * sqrt(var) to mu + width * sqrt(var). The default value is 10, which should be sensible for most models. (numeric)

Author

Pierre de Villemereuil

Details

The variance \(V_{\text{Add}}\) is the total additive genetic variance in the reaction norm. It can be decomposed according to each of the parameters in G_theta (and their pairwise covariance), using the \(\gamma\)-decomposition, if the compute_gamma is activated. It can also be decomposed into the marginal additive genetic variance of the trait (\(V_{\text{A}}\), the additive genetic variance of the trait that would be computed if plasticity were ignored), and the additive genetic variance of plasticity itself (\(V_{\text{A}\times\text{E}}\), i.e. the genetic-by-environment interaction variance). This last component can also be decomposed according to each of the parameters in G_theta (and their pairwise covariance), using the \(\iota\)-decomposition (for interaction).

It is very important that the parameters are in the same order in the argument of the shape function, in theta and in G_theta.

See Also

rn_vgen, rn_gen_decomp

Examples

# Some environment vector
vec_env <- seq(-2, 2)

# Shape function
expr <- expression(
     cmax * exp(
         - exp(rho * (x - xopt) - 6) -
             sigmagaus * (x - xopt)^2
     ))

# Theta
theta <- c(cmax = 1, xopt = 0.9, rho = 8, sigmagaus = 0.4)
# G, only for cmax and xopt
G     <- matrix(c(0.1,      0.01,
                  0.01,     0.05),
                ncol = 2)

# Computing V_gen
out <- 
  rn_gen_decomp(theta   = theta,
                G_theta = G,
                env     = vec_env,
                shape   = expr,
                fixed   = c(3, 4))
# Note that fixed is set for the third and forth parameters than are not in G

# Checking that V_Add = V_A + V_AxE
out$V_Add == out$V_A + out$V_AxE
# Checking that the sum of \eqn{\gamma}'s is 1
out$Gamma_cmax + out$Gamma_xopt + out$Gamma_cmax_xopt
# Checking that the sum of \eqn{\iota}'s is 1
out$Iota_cmax + out$Iota_xopt + out$Iota_cmax_xopt

# Applying some weighting to the environmental values according to a normal distribution
rn_gen_decomp(theta   = theta,
              G_theta = G,
              env     = vec_env,
              shape   = expr,
              fixed   = c(3, 4),
              wt_env  = dnorm(vec_env))
              
# If a polynomial was used, it is possible to use the linear modeling rather having
# to compute integrals
theta <- c(a = 1.5, b = 0.5, c = -0.5)
X     <- cbind(1, vec_env, (vec_env - mean(vec_env))^2)
G     <- 0.1 * diag(3)
rn_gen_decomp(theta   = theta,
              G_theta = G,
              X        = X)