getGenCov: Pairwise Genetic Covariance

In SFSI: Sparse Family and Selection Index

Pairwise Genetic Covariance

Description

Pairwise genetic covariances for variables with the same experimental design

Usage

getGenCov(y, X = NULL, Z = NULL, K = NULL,
          U = NULL, d = NULL, scale = TRUE,
          pairwise=FALSE, verbose = TRUE, ...)
          

Arguments

y	(numeric matrix) Response variable matrix. It should contain more than one column
X	(numeric matrix) Design matrix for fixed effects. When X=NULL a vector of ones is constructed only for the intercept (default)
Z	(numeric matrix) Design matrix for random effects. When Z=NULL an identity matrix is considered (default) thus G = K; otherwise G = Z K Z' is used
K	(numeric matrix) Kinship relationship matrix
U	(numeric matrix) Eigenvectors from spectral value decomposition of G = U D U'
d	(numeric vector) Eigenvalues from spectral value decomposition of G = U D U'
scale	TRUE or FALSE to scale each column of y by their corresponding standard deviations so the resulting variables will have unit variance
pairwise	TRUE or FALSE to calculate pairwise genetic covariances for all columns in y. When pairwise=FALSE (default) covariances of the first column in y with the remaining columns (2,...,ncol(y)) are calculated
verbose	TRUE or FALSE to whether show progress
...	Other arguments passed to the function 'fitBLUP'

Details

Assumes that both y₁ and y₂ follow the basic linear mixed model that relates phenotypes with genetic values of the form

y₁ = X b₁ + Z u₁ + e₁

y₂ = X b₂ + Z u₂ + e₂

where b₁ and b₂ are the specific fixed effects, u₁ and u₂ are the specific genetic values of the genotypes, e₁ and e₂ are the vectors of specific environmental residuals, and X and Z are common design matrices conecting the fixed and genetic effects with replicates. Genetic values are assumed to follow a Normal distribution as u₁ ~ N(0,σ²_u1K) and u₂ ~ N(0,σ²_u2K), and environmental terms are assumed e₁ ~ N(0,σ²_e1I) and e₂ ~ N(0,σ²_e2I).

The genetic covariance σ²_u1,u2 is estimated from the formula for the variance for the sum of two variables as

σ²_u1,u2 = 1/2(σ²_u3 - σ²_u1 - σ²_u2)

where σ²_u3 is the genetic variance of the variable y₃ = y₁ + y₂ that also follows the same model as for y₁ and y₂.

Likewise, the environmental covariance σ²_e1,e2 is estimated as

σ²_e1,e2 = 1/2(σ²_e3 - σ²_e1 - σ²_e2)

where σ²_e3 is the error variance of the variable y₃.

Solutions are found using the function 'fitBLUP' (see help(fitBLUP)) to sequentialy fit mixed models for all the variables y₁, y₂ and y₃.

Value

Returns a list object that contains the elements:

• varU: (vector) genetic variances.

• varE: (vector) error variances.

• covU: (vector) genetic covariances between response variable 1 and the rest.

• covE: (vector) environmental covariances between response variable 1 and the rest.

When pairwise=TRUE, varU and varE are matrices containing all variances (diagonal) and pairwise covariances (off diagonal)

Examples

  require(SFSI)
  data(wheatHTP)
  
  index = which(Y$trial %in% 1:8)     # Use only a subset of data
  Y = Y[index,]
  M = scale(M[index,])/sqrt(ncol(M))  # Subset and scale markers
  G = tcrossprod(M)                   # Genomic relationship matrix
  y = scale(Y[,4:7])                  # Response variable
  
  # Covariances between the first and the rest: y[,1] and y[,2:4]
  fm = getGenCov(y=y, K=G)
  fm$varU           # Genetic variances
  fm$varE           # Residual variances
  fm$covU           # Genetic covariance between y[,1] and y[,2:4]
  fm$covU+fm$covE   # Phenotypic covariance
  cov(y)[1,-1]      # Sample phenotypic covariance
  
  # Pairwise covariance for all columns in y
  fm = getGenCov(y=y, K=G, pairwise=TRUE)
  fm$varU 
  fm$varE

SFSI documentation built on Nov. 18, 2023, 9:06 a.m.