nadiv (version 2.18.0)

makeAinv: Creates the inverse additive genetic relationship matrix

Description

This returns the inverse of the numerator relationship matrix (inverse additive genetic relatedness matrix). It can also be used to obtain coefficients of inbreeding for the pedigreed population.

Usage

makeAinv(
  pedigree,
  f = NULL,
  ggroups = NULL,
  fuzz = NULL,
  gOnTop = FALSE,
  det = TRUE,
  ...
)
# S3 method for default
makeAinv(
  pedigree,
  f = NULL,
  ggroups = NULL,
  fuzz = NULL,
  gOnTop = FALSE,
  det = TRUE,
  ...
)
# S3 method for fuzzy
makeAinv(
  pedigree,
  f = NULL,
  ggroups = NULL,
  fuzz,
  gOnTop = FALSE,
  det = TRUE,
  ...
)
makeGGAinv(pedigree, f = NULL, ggroups = NULL, det = TRUE, ...)

Value

a list:

Ainv

the inverse of the additive genetic relationship matrix in sparse matrix form

listAinv

the three column list of the non-zero elements for the inverse of the additive genetic relationship matrix with attributes rowNames and geneticGroups. attr(*, "rowNames") links the integer for rows/columns to the ID column from the pedigree. attr(*, "geneticGroups") is a two element vector with the first integer indicating how many genetic groups are included in the pedigree. This last attribute is necessary for some software programs to correctly specify the residual degrees of freedom when calculating the log-likelihood in a model that implicitly fits fixed genetic group effects.

f

the individual coefficients of inbreeding for each individual in the pedigree (matches the order of the first/ID column of the pedigree). If the pedigree contains ‘g’ genetic groups in the first ‘g’ rows, then the first ‘g’ elements of f are assigned 0. If the pedigree contains ‘p’ phantom parents in the first ‘p’ rows, then the first ‘p’ elements of f are assigned 0.

logDet

the log determinant of the A matrix

dii

the (non-zero) elements of the diagonal D matrix of the A=TDT' decomposition. Contains the variance of Mendelian sampling. Matches the order of the first/ID column of the pedigree. If the pedigree contains ‘g’ genetic groups in the first ‘g’ rows, then the first ‘g’ elements of f are assigned 0. If the pedigree contains ‘p’ phantom parents in the first ‘p’ rows, then the first ‘p’ elements of f are assigned 0.

Arguments

pedigree

A pedigree where the columns are ordered ID, Dam, Sire

f

A numeric indicating the level of inbreeding. See Details

ggroups

Either a vector with the unique name of each genetic group, or a numeric indicating the number of unique genetic groups. See Details for different ways to specify. Note, if NULL then the regular A-inverse will be constructed. Also, must be NULL if fuzz is non-NULL.

fuzz

A matrix containing the fuzzy classification of phantom parents into genetic groups. See Details.

gOnTop

A logical indicating if (when including genetic groups) the A-inverse should be constructed with the ‘g’ genetic groups located in the first ‘g’ rows and columns if TRUE, else the ‘g’ genetic groups are located in the last ‘g’ rows and columns of A-inverse

det

Logical, indicating if the (log) determinant of the A matrix should be returned

...

Arguments to be passed to methods

Author

Details

Missing parents (e.g., base population) should be denoted by either 'NA', '0', or '*'.

The functions implement an adaptation of the Meuwissen and Luo (1992) algorithm (particularly, following the description of the algorithm in Mrode 2005) with some code borrowed from the inverseA function by Jarrod Hadfield in the MCMCglmm package. Further, providing a non-NULL argument to ggroups incorporates the Quaas (1988) algorithm for directly obtaining the augmented A-inverse matrix for genetic groups into Meuwissen and Luo's (1992) algorithm, thereby, considering inbreeding during the construction of the A-inverse. Further calculations needed for the algorithm to incorporate inbreeding and genetic groups follow the theory presented in VanRaden (1992). Alternatively, group-specific inverse relatedness matrices can be formed with makeGGAinv, see below.

At the moment, providing the inbreeding level of individuals or the base population has not been implemented. However, this argument is a placeholder for now.

Genetic groups can be incorporated into a single A-inverse by providing a value to the ggroups argument in makeAinv. The value supplied to ggroups can either be (1) a single integer indicating the number of unique genetic groups or (2) a character vector containing the name for each genetic group. These are referred to as pedigree types "A" and "D", respectively, and further details follow below.

(Type="A") the pedigree contains unique IDs for the 'g' genetic groups in the first 'g' lines of the pedigree. The dam and sire of the genetic group rows should contain missing values (e.g., NA, "0", or "*"). All individuals in the pedigree should then have one of the `g' genetic groups instead of an unknown parent. (Type="D") the pedigree contains only individuals in the ID column (no genetic groups have an ID) and there should be no missing values for any dams or sires. Instead, individuals for whom the dam and/or sire is unknown should have one of the genetic groups identified in the vector supplied to ggroups as the dam or sire.

‘Fuzzy classification’ of genetic groups (Fikse 2009) can be implemented if a ‘matrix’ (of class matrix or Matrix) is supplied to the fuzzy argument. The fuzzy classification matrix must have row names matching all of the phantom parents in the pedigree and the column names must be present and specify the genetic groups. The fuzzy classification matrix essentially contains probability of group membership for each phantom parent. Therefore, each row should sum to 1. The pedigree must have an identity in a unique row for every phantom parent and cannot have genetic groups as either identities (in the first column) or as dam or sire (second and third columns). Further, if fuzzy classification is desired, the function must specify ggroups = NULL.

When genetic groups (including the case of fuzzy classification of genetic groups) are included in the A-inverse matrix, the argument to gOnTop specifies if the genetic group elements in the A-inverse should occupy the top-left (gOnTop = TRUE) or bottom-right (gOnTop = FALSE) of the matrix. Depending on how the software implementing an animal model solves the mixed model equations, the equations for the genetic groups (and thus the elements in the augmented A-inverse) should be the first or last set of equations.

References

Fikse, F. 2009. Fuzzy classification of phantom parent groups in an animal model. Genetics Selection Evolution 41:42.

Meuwissen, T.H.E & Luo, Z. 1992. Computing inbreeding coefficients in large populations. Genetics, Selection, Evolution. 24:305-313.

Mrode, R.A. 2005. Linear Models for the Prediction of Animal Breeding Values, 2nd ed. Cambridge, MA: CABI Publishing.

Quaas, R.L. 1988. Additive genetic model with groups and relationships. Journal of Dairy Science. 71:1338-1345.

VanRaden, P.M. 1992. Accounting for inbreeding and crossbreeding in genetic evaluation of large populations. Journal of Dairy Science. 75:3136-3144.

See Also

makeAstarMult, makeA

Examples

 ##  Without genetic groups  ##
 makeAinv(Mrode2)
 
 ##  With genetic groups  ##
  ## Type A
   typeAped <- Q1988[-c(3:7), c("id", "damGG", "sireGG")]
   AstarA <- makeAinv(typeAped, ggroups = 2, gOnTop = FALSE)$Ainv
  ## Type D
   typeDped <- Q1988[-c(1:7), c("id", "damGG", "sireGG")]
   AstarD <- makeAinv(typeDped, ggroups = c("g1", "g2"), gOnTop = FALSE)$Ainv
  stopifnot(identical(AstarA, AstarD))
  
  # Show that the augmented A-inverse with genetic groups
  # contains the normal A-inverse (i.e., without genetic groups)
   ## Augmented A-inverse with genetic groups
    ggAinv <- makeAinv(Mrode3[-c(1,2), c("calf", "damGG", "sireGG")],
	ggroups = c("g1", "g2"), gOnTop = FALSE)$Ainv
    noggAinv <- makeAinv(Mrode3[-c(1,2), c("calf", "dam", "sire")],
	ggroups = NULL)$Ainv
    # First 8 rows & columns of ggAinv are same as A-inverse without 
    ## genetic groups
    ggAinv[1:8, 1:8]
    noggAinv
   stopifnot(all.equal(ggAinv[1:8, 1:8], structure(noggAinv, geneticGroups = NULL)))
   
 ##  With fuzzy classification of genetic groups  ##
  ## example in Fikse (2009)
  Fped <- F2009[-c(1:3), c("id", "phantomDam", "phantomSire")]
    Fped$id <- factor(Fped$id, levels = as.character(unique(Fped$id)))
  Ffuzz <- as.matrix(F2009[4:10, c("g1", "g2", "g3")])
    dimnames(Ffuzz)[[1]] <- as.character(F2009[4:10, 1])
  AstarF <- makeAinv(Fped, fuzz = Ffuzz, gOnTop = FALSE)$Ainv

  ## Show that A-inverse with fuzzy classification of genetic groups
  ### can be the same as genetic group A-inverse without fuzzy classification
  ### Create a 'null' fuzzy classification matrix for Q1988 pedigree
  QfuzzNull <- matrix(c(1,0,0,1,0, 0,1,1,0,1), nrow = 5, ncol = 2,
	dimnames = list(letters[1:5], c("g1", "g2")))
  typeFped <- Q1988[-c(1:2), c("id", "phantomDam", "phantomSire")]
  AstarNullFuzzy <- makeAinv(typeFped, fuzz = QfuzzNull, gOnTop = FALSE)$Ainv
  # Same as above using either pedigree type 'A' or 'D'
  stopifnot(identical(AstarNullFuzzy, AstarA),
	identical(AstarNullFuzzy, AstarD))

 ##  With genetic groups  ##
  ## Type A
   typeAped <- Q1988[-c(3:7), c("id", "damGG", "sireGG")]
   (AinvOutA <- makeGGAinv(typeAped, ggroups = 2)$Ainv)
  ## Type D
   typeDped <- Q1988[-c(1:7), c("id", "damGG", "sireGG")]
   (AinvOutD <- makeGGAinv(typeDped, ggroups = c("g1", "g2"))$Ainv)
  stopifnot(identical(AinvOutA, AinvOutD))