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AGHmatrix Tutorial
Rodrigo Amadeu
2023-10-03
Overview
AGHmatrix software is an R-package to build relationship matrices
using pedigree (A matrix) and/or molecular markers (G matrix) with the
possibility to build a combined matrix of Pedigree corrected by
Molecular (H matrix). The package works with diploid and autopolyploid
data.
Matrices computation implemented in the AGHmatrix
Currently the package computes the following 17 different
relationship matrices:
Pedigree-based relationship matrix (A matrix)
| Diploid |
Henderson (1976) |
Cockerham (1954) |
| Autopolyploid |
Kerr (2012), Slater (2013) |
|
Molecular-based relationship matrix (G matrix)
| Diploid |
Yang (2010), VanRaden (2012), Liu (2020) |
Su (2012), Vitezica (2013) |
| Polyploid |
Slater (2016), de Bem Oliveira (2019) |
Slater (2016), Endelman (2018) |
Combined pedigree and molecular-based relationship matrix (H
matrix)
| Legarra (2009), Munoz (2014), Martini (2018) |
Additionally there is a beta implementation to compute A matrix when
parentage is not deterministic as in a polycross design. See
?AmatrixPolycross.
Installing and loading
Within R:
## Install stable version
install.packages("AGHmatrix")
## Install development version
install.packages("devtools")
devtools::install_github("rramadeu/AGHmatrix")
## Load
library(AGHmatrix)
Relationship matrices using pedigree data - A matrix
Amatrix process the pedigree and build the A-matrix
related to that given pedigree. The matrix is built based in the
recursive method presented in Mrode (2014) and described by Henderson
(1976). This method is expanded for higher ploidies (n-ploidy) as
detailed in Kerr et al. (2012). After loading the package you have to
load your data file into the software. To do this, you can use the
function read.data() or read.csv() for
example. Your data should be available in R as a data.frame
structure in the following order: column 1 must be the
individual/genotype names (id), columns 2 and 3 must be the parent
names. For the algorithm, it does not matter who is the mother and who
is the father (so, no sex column). There is a pedigree data example
(ped.mrode) that can be used to look at the structure and
order the data.
To load ped.mrode:
data(ped.mrode)
ped.mrode
#> Ind Par1 Par2
#> 1 Anc1 0 0
#> 2 Anc2 0 0
#> 3 Var1 Anc1 Anc2
#> 4 Var2 Anc1 0
#> 5 Var3 Var2 Var1
#> 6 Var4 Var3 Anc2
str(ped.mrode) #check the structure
#> 'data.frame': 6 obs. of 3 variables:
#> $ Ind : Factor w/ 6 levels "Anc1","Anc2",..: 1 2 3 4 5 6
#> $ Par1: Factor w/ 4 levels "0","Anc1","Var2",..: 1 1 2 2 3 4
#> $ Par2: Factor w/ 3 levels "0","Anc2","Var1": 1 1 2 1 3 2
The example ped.mrode has 3 columns, column 1 contains
the names of the individual/genotypes, column 2 contains the names of
the first parent, column 3 contains the names of the second parental
(example from Table 2.1 of Mrode 2014). There is no header template, and
the unknown value must be equal 0. Your data has to be in the same
format of ped.mrode.
Internally the algorithm first pre-process the pedigree: the
individuals are numerated \(1\) to
\(n\). Then, it is verified whether the
genotypes in the pedigree are in chronological order (i.e. if the
parents of a given individual are located before to this individual in
the pedigree data set). If this order is not followed, the algorithm
performs the necessary changes to correct them in a iterative way. After
this pre-processing, the matrix computation proceeds as in Henderson
(1976) for diploid - for additive or dominance relationship - and as in
Kerr et al. (2012) for autotetraploids - for additive relationship. For
autotetraploids, there is the option to include double-reduction
fraction. For diploids there is the option to compute the dominant
relationship matrix (Cockerham, 1954).
It follows some usage examples with the ped.mrode.
#Computing additive relationship matrix for diploids (Henderson 1976):
Amatrix(ped.mrode, ploidy=2)
#Computing dominant relationship matrix for diploids (Cockerham 1954):
Amatrix(ped.mrode, ploidy=2, dominance=TRUE)
#Computing additive relationship matrix for autotetraploids (Kerr 2012):
Amatrix(ped.mrode, ploidy=4)
#Computing additive relationship matrix for autooctaploids (Kerr 2012):
Amatrix(ped.mrode, ploidy=8)
#Computing additive relationship matrix for autotetraploids
# and double-reduction of 0.1 (Kerr 2012):
Amatrix(ped.mrode, ploidy=4, w=0.1)
#Computing additive relationship matrix for autotetraploids
# and double-reduction of 0.1 as in Slater et al. (2014):
Amatrix(ped.mrode, ploidy=4, w=0.1, slater = TRUE)
#not recommended, but kept in the package to reproduce some former analysis
#Computing additive relationship matrix for autohexaploids
# and double-reduction of 0.1 (Kerr 2012):
Amatrix(ped.mrode, ploidy=6, w=0.1)
More information about Amatrix can be found with:
Diploid G matrix: relationship matrices using the molecular
data
Gmatrix handles the molecular-marker matrix and builds
the relationship matrix. Molecular markers data should be organized in a
matrix format (individuals in rows and markers in columns) coded as 0,
1, 2 and missing data value (numeric or NA). Import your
molecular marker data into R with the function
read.table() or read.csv() and convert to a
matrix format with the function as.matrix(). The function
Gmatrix can be used to construct the additive relationship
either as proposed by Yang et al. (2010) or the proposed by VanRaden
(2008). The function can also construct the dominance relationship
matrix either as proposed by Su et al. (2012) or as proposed by Vitezica
et al. (2013). As an example, here we build the four matrices using real
data from Resende et al. (2012).
To load snp.pine and to check its structure:
data(snp.pine)
snp.pine[1:5,1:5]
#> 0-10024-01-114 0-10037-01-257 0-10040-02-394 0-10040-02-41
#> 1087120 -9 1 -9 -9
#> 1085618 2 2 2 2
#> 1091040 2 1 2 2
#> 1091686 2 1 2 2
#> 1082624 2 2 0 1
#> 0-10044-01-392
#> 1087120 -9
#> 1085618 2
#> 1091040 2
#> 1091686 2
#> 1082624 1
str(snp.pine)
#> int [1:926, 1:4853] -9 2 2 2 2 2 2 2 2 2 ...
#> - attr(*, "dimnames")=List of 2
#> ..$ : chr [1:926] "1087120" "1085618" "1091040" "1091686" ...
#> ..$ : chr [1:4853] "0-10024-01-114" "0-10037-01-257" "0-10040-02-394" "0-10040-02-41" ...
In this dataset, we have 926 individuals with 4853 markers and the
missing data value is -9.
It follows some examples with the snp.pine data where
the unknown value (missingValue) is -9. Here
we set minimum allele frequency to 0.05, so markers with
minor allele frequency lower than 0.05 are removed from the dataset
prior to the G matrix construction.
#Computing the additive relationship matrix based on VanRaden 2008
G_VanRadenPine <- Gmatrix(SNPmatrix=snp.pine, missingValue=-9,
maf=0.05, method="VanRaden")
#Computing the additive relationship matrix based on Yang 2010
G_YangPine <- Gmatrix(SNPmatrix=snp.pine, missingValue=-9,
maf=0.05, method="Yang")
#Computing the dominance relationship matrix based on Su 2012
G_SuPine <- Gmatrix(SNPmatrix=snp.pine, missingValue=-9,
maf=0.05, method="Su")
#Computing the dominance relationship matrix based on Vitezica 2013
G_VitezicaPine <- Gmatrix(SNPmatrix=snp.pine, missingValue=-9,
maf=0.05, method="Vitezica")
More information about Gmatrix can be found with:
Autopolyploid G matrix: relationship matrices using the molecular
data
Molecular markers data should be organized in a matrix format
(individual in rows and markers in columns) coded according to the
dosage level: 0, 1, 2, …, ploidy level, and missing data value (numeric
user-defined or NA). As an example, an autotetraploid
should be coded as 0, 1, 2, 3, 4, and a missing data value. In
autopolyploids, the function Gmatrix can be used to
construct: i) the additive relationship based on VanRaden (2008) and
extended by Ashraf (2016); ii) the full-autopolyploid including additive
and non-additive model as equations 8 and 9 in Slater et al. (2016);
iii) the pseudo-diploid model as equations 5, 6, and 7 Slater et
al. (2016). iv) the digenic-dominant model based on Endelman et
al. (2018). As an example, here we build the matrices using data from
Endelman et al. (2018) (snp.sol). There is also an option
to build weighted relationship matrices as in Liu et al. (2020).
The argument ploidy.correction defines the denominator
of the formula for the VanRaden method. If
ploidy.correction=TRUE, it uses the parametric correction
as \(\sum_i p f_i(1-f_i)\), where \(p\) is the ploidy level and \(f_i\) is the minor allele frequency of the
\(i_th\) marker. If
ploidy.correction=FALSE, it uses the sampling variance
correction as \(\sum_i \frac{1}{p}
s^2(m_i)\), where \(s^2(m_i)\)
is the sampling variance of the \(i_th\) marker. Both corrections are
equivalent when sampling size goes to the infinity. The default is to
use the sampling variance as the correction (i.e.,
ploidy.correction=FALSE).
#Loading the data
data(snp.sol)
str(snp.sol)
#Computing the additive relationship matrix based on VanRaden 2008
# adapted by Ashraf 2016
G_VanRaden <- Gmatrix(snp.sol, method="VanRaden", ploidy=4)
#Computing the dominance (digenic) matrix based on Endelman 2018 (Eq. 19)
G_Dominance <- Gmatrix(snp.sol, method="Endelman", ploidy=4)
#Computing the full-autopolyploid matrix based on Slater 2016 (Eq. 8
#and 9)
G_FullAutopolyploid <- Gmatrix(snp.sol, method="Slater", ploidy=4)
#Computing the pseudodiploid matrix based on Slater 2016 (Eq. 5, 6,
#and 7)
G_Pseudodiploid <- Gmatrix(snp.sol, method="VanRaden", ploidy=4, pseudo.diploid=TRUE)
#Computing G matrix with specific weight for each marker as
# in Liu et al. (2020).
Gmatrix_weighted <- Gmatrix(snp.sol, method="VanRaden", weights = runif(3895,0.001,0.1), ploidy=4)
More information about Gmatrix can be found with:
Ratio (non-dosage) G matrix: relationship matrices using the
molecular data without dosage calling
Molecular markers data should be organized in a matrix format
(individual in rows and markers in columns) coded according to a
fraction that represents its molecular information, this can be any
number between 0 to 1. Such ratio can represent the count of alternative
alleles over the read depth for each individual-marker combination
(GBS-like technique). It can be the signal of the alternative allele
over the sum of the signals of the alternative and reference alleles
(GCMS-like technique). It can also be used for family-pool
genotypes.
#Loading the data
library(AGHmatrix)
data(snp.sol)
snp.sol.ratio = snp.sol/4 #transforming it in a ratio of the minor allele frequency
Gmatrix <- Gmatrix(snp.sol, method="VanRaden", ploidy=4, ratio=FALSE)
Gmatrix.ratio <- Gmatrix(snp.sol.ratio, method="VanRaden", ploidy=4, ratio=TRUE)
Gmatrix[1:5,1:5]==Gmatrix.ratio[1:5,1:5]
## it also has the ploidy.correction option
Gmatrix.alternative <- Gmatrix(snp.sol,
method="VanRaden",
ploidy=4,
ratio=FALSE,
ploidy.correction=TRUE)
Gmatrix.ratio.alternative <- Gmatrix(snp.sol.ratio,
method="VanRaden",
ploidy=4,
ratio=TRUE,
ploidy.correction=TRUE)
Gmatrix[1:5,1:5]==Gmatrix.alternative[1:5,1:5]
Gmatrix.alternative[1:5,1:5]==Gmatrix.ratio.alternative[1:5,1:5]
Combined relationship matrix - H matrix
H matrix is the relationship matrix using combined information from
the pedigree and genomic relationship matrices. First, you need to
compute the matrices separated and then use them as input to build the
combined H matrix. Two methods are implemented: Munoz
shrinks the G matrix towards the A matrix scaling the molecular
relatadness by each relationship classes; Martini is a
modified version from Legarra et al. 2009 where combines A and G matrix
using scaling factors. As an example, here we build the matrices using
data from Endelman et al. (2018) (ped.sol and
snp.sol).
data(ped.sol)
data(snp.sol)
#Computing the numerator relationship matrix 10% of double-reduction
Amat <- Amatrix(ped.sol, ploidy=4, w = 0.1)
Gmat <- Gmatrix(snp.sol, ploidy=4,
maf=0.05, method="VanRaden")
Gmat <- round(Gmat,3) #see appendix
#Computing H matrix (Martini)
Hmat_Martini <- Hmatrix(A=Amat, G=Gmat, method="Martini",
ploidy=4, missingValue=-9, maf=0.05)
#Computing H matrix (Munoz)
Hmat_Munoz <- Hmatrix(A=Amat, G=Gmat, markers = snp.sol,
ploidy=4, method="Munoz",
missingValue=-9, maf=0.05)
Covariance matrices due to epistatic terms
Here we present how to compute the epistasis relationship matrices
using Hadamard products (i.e. cell-by-cell product), denoted by
*. For more information please see Munoz et al. (2014). In
this example we are using the molecular-based relationship matrices.
First, build the additive and dominance matrices:
data(snp.pine)
A <- Gmatrix(SNPmatrix=snp.pine, method="VanRaden", missingValue=-9, maf=0.05)
D <- Gmatrix(SNPmatrix=snp.pine, method="Vitezica", missingValue=-9,maf=0.05)
For the first degree epistatic terms:
#Additive-by-Additive Interactions
A_A <- A*A
#Dominance-by-Additive Interactions
D_A <- D*A
#Dominance-by-Dominance Interactions
D_D <- D*D
For the second degree epistatic terms:
#Additive-by-Additive-by-Additive Interactions
A_A_A <- A*A*A
#Additive-by-Additive-by-Dominance Interactions
A_A_D <- A*A*D
#Additive-by-Dominance-by-Dominance Interactions
A_D_D <- A*D*D
#Dominance-by-Dominance-by-Dominance Interactions
D_D_D <- D*D*D
And so on…
Relationship matrices using pedigree data for polycrosses - A matrix
(beta)
Creates an additive relationship matrix A based on a
non-deterministic pedigree with 4+ columns where each column represents
a possible parent. This function was built with the following designs in
mind. 1) A mating design where you have equally possible parents. For
example, a generation of insects derived from the mating of three
insects in a cage. All the insects in this generation will have the same
expected relatedness with all the possible parents (1/3). If there are
only two parents in the cage, the function assumes no-inbreeding and the
pedigree is deterministic (the individual is offspring of the cross
between the two parents). Another example, a population of 10
open-pollinated plants where you harvest the seeds without tracking the
mother. 2) When fixedParent is TRUE: a mating design where
you know one parent and might know the other possible parents. For
example, a polycross design where you have seeds harvested from a mother
plant and possible polén donors.
The following pedigree has the id of the individual followed by
possible parents. The possible parents are filled from left to right, in
the pedigree data frame: id 1,2,3,4 have unknown parents
and are assumed unrelated; id 5 has three possible parents (1,2,3); id 6
has three possible parents (2,3,4); id 7 has two parents (deterministic
case here, the parents are 3 and 4); id 8 has four possible parents
(5,6,7,1).
pedigree = data.frame(id=1:8,
parent1 = c(0,0,0,0,1,2,3,5),
parent2 = c(0,0,0,0,2,3,4,6),
parent3 = c(0,0,0,0,3,4,0,7),
parent4 = c(0,0,0,0,0,0,0,1),
parent5 = 0)
print(pedigree)
#> id parent1 parent2 parent3 parent4 parent5
#> 1 1 0 0 0 0 0
#> 2 2 0 0 0 0 0
#> 3 3 0 0 0 0 0
#> 4 4 0 0 0 0 0
#> 5 5 1 2 3 0 0
#> 6 6 2 3 4 0 0
#> 7 7 3 4 0 0 0
#> 8 8 5 6 7 1 0
AmatrixPolyCross(pedigree)
#> Your data was chronologically organized with success.
#> Your data was chronologically organized with success.
#> Your data was chronologically organized with success.
#> Your data was chronologically organized with success.
#> Constructing matrix A using ploidy = 2
#> Completed! Time = 0 minutes
#> 1 2 3 4 5 6 7
#> 1 1.0000000 0.0000000 0.0000000 0.0000000 0.3333333 0.0000000 0.0000000
#> 2 0.0000000 1.0000000 0.0000000 0.0000000 0.3333333 0.3333333 0.0000000
#> 3 0.0000000 0.0000000 1.0000000 0.0000000 0.3333333 0.3333333 0.5000000
#> 4 0.0000000 0.0000000 0.0000000 1.0000000 0.0000000 0.3333333 0.5000000
#> 5 0.3333333 0.3333333 0.3333333 0.0000000 1.0000000 0.2222222 0.1666667
#> 6 0.0000000 0.3333333 0.3333333 0.3333333 0.2222222 1.0000000 0.3333333
#> 7 0.0000000 0.0000000 0.5000000 0.5000000 0.1666667 0.3333333 1.0000000
#> 8 0.3333333 0.1666667 0.2916667 0.2083333 0.4305556 0.3888889 0.3750000
#> 8
#> 1 0.3333333
#> 2 0.1666667
#> 3 0.2916667
#> 4 0.2083333
#> 5 0.4305556
#> 6 0.3888889
#> 7 0.3750000
#> 8 1.0879630
If fixedParent=TRUE, the above pedigree will be
interpreted with the possible parents are filled from left to right
after the known parent, in the pedigree data frame: id
1,2,3,4 have unknown parents and are assumed unrelated; id 5 is
offspring of parent 1 in a deterministic way and two other possible
parents (2,3); id 6 is offspring of parent 2 in a deterministic way and
two other possible parents (3,4); id 7 has two parents (deterministic
case here, the parents are 3 and 4); as before; id 8 is offspring of
parent 5 in a deterministic way and has three other possible parents
(6,7,1).
AmatrixPolyCross(pedigree,fixedParent=TRUE)
#> Your data was chronologically organized with success.
#> Your data was chronologically organized with success.
#> Your data was chronologically organized with success.
#> Your data was chronologically organized with success.
#> Completed! Time = 0 minutes
#> 1 2 3 4 5 6 7 8
#> 1 1.0000000 0.0000000 0.00 0.000 0.5000000 0.0000000 0.0000000 0.4166667
#> 2 0.0000000 1.0000000 0.00 0.000 0.2500000 0.5000000 0.0000000 0.2083333
#> 3 0.0000000 0.0000000 1.00 0.000 0.2500000 0.2500000 0.5000000 0.2500000
#> 4 0.0000000 0.0000000 0.00 1.000 0.0000000 0.2500000 0.5000000 0.1250000
#> 5 0.5000000 0.2500000 0.25 0.000 1.0000000 0.1875000 0.1250000 0.6354167
#> 6 0.0000000 0.5000000 0.25 0.250 0.1875000 1.0000000 0.2500000 0.3020833
#> 7 0.0000000 0.0000000 0.50 0.500 0.1250000 0.2500000 1.0000000 0.2708333
#> 8 0.4166667 0.2083333 0.25 0.125 0.6354167 0.3020833 0.2708333 1.1354167
Amatrix() benchmark
It follows a small memory and computational time profiling for the
Amatrix() function. The required RAM was computed based on
the peak of the process for different pedigree sizes (based on
/usr/bin/time -v output). The time profiling was done using AMD Milan
2.95GHz, so it might be an underestimated value when compared with lower
speed processors. Numerator relationship matrices for pedigrees with
less than 20,000 rows can built with low-specs user-end machines
(<8GB RAM) using our package.
Bibliography
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Amadeu, RR, et al. 2016 AGHmatrix: R package to construct
relationship matrices for autotetraploid and diploid species: a
blueberry example. The Plant Genome 9, 4.
Ashraf, BH, et al. 2016. Estimating genomic heritabilities at the
level of family-pool samples of perennial ryegrass using
genotyping-by-sequencing. Theoretical and Applied Genetics 129,
45-52.
Cockerham, CC. 1954 An extension of the concept of partitioning
hereditary variance for analysis of covariances among relatives when
epistasis is present. Genetics 39, 859–882.
de Bem Oliveira, I, et al. 2019 Genomic prediction of
autotetraploids; influence of relationship matrices, allele dosage, and
continuous genotyping calls in phenotype prediction. G3: Genes, Genomes,
Genetics, 9(4), pp.1189-1198.
Endelman, JB, et al. 2018 Genetic variance partitioning and
genome-wide prediction with allele dosage information in autotetraploid
potato. Genetics 209, 77-87.
Hamilton, MG, et al. 2017 Computation of the inverse additive
relationship matrix for autopolyploid and multiple-ploidy populations.
Theoretical and Applied Genetics 131, 851-890.
Henderson, C. 1976 A simple method for computing the inverse of a
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Kerr, RJ, et al., 2012 Use of the numerator relation ship matrix in
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and genomic information. Journal of Dairy Science 92, 4656–4663.
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unbiased prediction integrating variants selected from sequencing data
by association and bioinformatics analyses. Genet Sel Evol 52, 48.
Martini, JW, et al. 2018, The effect of the H\(^{1}\) scaling factors \(\tau\) and \(\omega\) on the structure of H in the
single-step procedure. Genetics Selection Evolution 50(1), 16.
Mrode, RA. 2014 Linear models for the prediction of animal
breeding values. Cabi. 3rd ed.
Munoz, PR, et al., 2014 Unraveling additive from nonadditive effects
using genomic relationship matrices. Genetics 198, 1759–1768.
R Core Team, 2016 R: A Language and Environment for
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Austria.
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standard data set of loblolly pine (Pinus taeda l.). Genetics
190, 1503–1510.
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complex traits for enhanced genetic gain in potato. Theoretical and
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in autotetraploid potato. The Plant Genome 9.
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variances and predicting genetic merits using genome-wide dense single
nucleotide polymorphism markers. PloS one 7, e45293. https://doi.org/10.1371/journal.pone.0045293
VanRaden, P. 2008 Efficient methods to compute genomic predictions.
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sessionInfo()
#> R version 4.2.3 (2023-03-15 ucrt)
#> Platform: x86_64-w64-mingw32/x64 (64-bit)
#> Running under: Windows 10 x64 (build 19044)
#>
#> Matrix products: default
#>
#> locale:
#> [1] LC_COLLATE=C
#> [2] LC_CTYPE=English_United States.utf8
#> [3] LC_MONETARY=English_United States.utf8
#> [4] LC_NUMERIC=C
#> [5] LC_TIME=English_United States.utf8
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] AGHmatrix_2.1.4 rmarkdown_2.23 knitr_1.43
#>
#> loaded via a namespace (and not attached):
#> [1] codetools_0.2-19 lattice_0.20-45 zoo_1.8-11 digest_0.6.31
#> [5] grid_4.2.3 R6_2.5.1 jsonlite_1.8.7 evaluate_0.21
#> [9] highr_0.10 cachem_1.0.7 rlang_1.1.0 cli_3.6.1
#> [13] rstudioapi_0.15.0 jquerylib_0.1.4 Matrix_1.6-0 bslib_0.5.0
#> [17] tools_4.2.3 xfun_0.39 yaml_2.3.7 fastmap_1.1.1
#> [21] compiler_4.2.3 htmltools_0.5.5 sass_0.4.7
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.