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User Manual for Gmat64 Chao Ning China Agricultural University email@example.com; firstname.lastname@example.org Introduction A piece of cake to build different kinds of kinship matrix. Gmat64 is written by C to build different kinds of kinship matrix including additive, dominant and epistatic. The program start from PLINK binary file, and five command-line arguments are needed. Note: Missing genotype is replaced by a random value in the program, which has little influence on results for relative low missing rate (eg. Less than 0.05 for each SNP). Dependencies Intel® Math Kernel Library (Intel MKL) Using the binary of REMMA for Linux, users do need to install Intel MKL ( https://software.intel.com/en-us/intel-mkl ). However, it is recommended to install Intel® Parallel Studio XE ( https://software.intel.com/en-us/intel-parallel-studio-xe ), which simplifies the progress of compiling the source code. Quick start Gmat64 --help Gmat64 --bfile plink_file_prefix --inv 1 --normMat 1 --outformat 1 -Gmat addGmat Theory We introduce basic theory of different kinds of kinship matrix in this part. Additive kinship Additive kinship matrix is built with the method by VanRaden (2008). Let Z be the standardized additive marker matrix. Matrix Z is constructed as follows, Z ( z1, z2 ,, z j ,, zm ) / 2 p j (1 p j ) , (1) where pj is the allele frequency of allele A for the jth SNP, and zj is the jth SNP vector with elements defined as 2 2 p j z j 1 2 p j 0 2p j AA Aa . (2) aa Then the additive kinship is the matrix product Ka = ZZ’. Dominant kinship There are two kinships of dominant kinship matrices in the program and please refer to Vitezica, et al. (2013) for detailed theory. (1) Let H be the standardized dominant marker matrix. Matrix H is constructed as follows, 2 p q (1 2 p q ) , H (h1 , h 2 , , h j , , h m ) / j j j j (3) where pj is the allele frequency of allele A for the jth SNP, and hj is the jth SNP vector with elements defined as 0 2 p j q j h j 1 2 p j q j 0 2p q j j AA (4) Aa . aa Then the dominant kinship is the matrix product Kd1 = HH’. (2) Let W be the standardized dominant marker matrix. Matrix W is constructed as follows, W ( w1 , w 2 , , w j , , w m ) / (2 p q ) j j 2 , (5) where pj is the allele frequency of allele A for the jth SNP, and wj is the jth SNP vector with elements defined as 2q 2j w j 2 p j q j 2 p2 j AA Aa. aa Then the dominant kinship is the matrix product Kd2 = WW’. Note: The second method is recommended as the first method underestimates the additive genetic variance and overestimates the dominance variance (Vitezica, et al., 2013). Additive-by-additive (AxA) epistatic kinship The simplest method to build AxA epistatic kinship matrix is Ka # Ka, (6) where # represents the Hadamard matrix product (Henderson, 1985). However, the equation included interactions of loci with themselves. The accurate and efficient form (Jiang and Reif, 2015) used in the program is K aa K a # K a (Z# Z)(Z# Z)'. (7) Additive-by- dominant (AxD) epistatic kinship Two kinds of epistatic kinship are provided. (1) K ad1 K a # K d 1 (Z# H)(Z# H)'. (8) (2) K ad 2 K a # K d 2 (Z# W)(Z# W)'. (9) Note: Similar to dominant kinship, the second method is recommended. Dominant -by- dominant (DxD) epistatic kinship Two kinds of epistatic kinship are provided. (1) K d 1d 1 K d 1 # K d 1 (H# H)(H# H)'. (10) (2) K d 2d 2 K d 2 # K d 2 ( W# W)( W# W)'. (11) Note: Similar to dominant kinship, the second method is recommended. Parameters --bfile: string; The path of the prefix for the PLINK binary file --inv: 0 or 1; whether to calculate and output the inversion of kinship matrix. 0 means NO while 1 means YES. --normMat: 0 or 1; whether to output the standardized additive (equation 1) or dominant (equation 3 or 5) marker matrix. 0 means NO while 1 means YES. The parameter only works for building additive and dominant kinship matrix. It must be provided for other kinds of kinship although it means nothing. --outformat: 0, 1 or 2; the output format of kinship matrix. When the value is 0, the output format is square matrix of order n (the number of individuals). When the value is 1, the program outputs the lower triangle elements of kinship matrix. The output file includes three columns of row, column, and value. It is sorted column within row. When the value is 2, the program outputs the lower triangle elements of kinship matrix. The output file includes three columns of ID, ID, and value. --Gmat: string; the type of kinship matrix. Please refer to Theory part for detailed information. addGmat: additive kinship matrix; domGmat_AS: the first kinship of dominant kinship in Theory part; domGmat_GS: the second (recommended) kinship of dominant kinship; epiGmatAA: Additive-by-additive (AxA) epistatic kinship; epiGmatAD_AS: the first kinship of additive-by-dominant (AxD) epistatic kinship in Theory part; epiGmatAD_GS: the second (recommended) kinship of additive-by- dominant (AxD) epistatic kinship; epiGmatDD_AS: the first kinship of dominant-by-dominant (DxD) epistatic kinship in Theory part; epiGmatDD_GS: the second (recommended) kinship of dominant-bydominant (DxD) epistatic kinship. Output files The program generates 1-3 output files depend on parameters. FILE 1: kinship matrix; prefix.type.grm0/1/2. The ‘prefix’ is the same to the prefix of PLINK binary file; ‘type’ is the type of kinship matrix defined by parameter --Gmat; ‘grm0’, ‘grm1’ or ‘grm2’ means different kinds of output format defined by parameter --outformat. FILE 2: The standardized marker matrix; optional; generate when -normMat 1; Binary format of double precision. prefix. addMarkerMat.bin, prefix.domMarkerMat_AS.bin or prefix.domMarkerMat_GS.bin. FILE 3: The inversion of kinship matrix; optional; generate when --inv 1. It has similar format as FILE1. prefix.type.giv0/1/2. Reference Henderson, C. (1985) Best linear unbiased prediction of nonadditive genetic merits, J. Anim. Sci, 60, 111-117. Jiang, Y. and Reif, J.C. (2015) Modeling Epistasis in Genomic Selection, Genetics, 201, 759-768. VanRaden, P.M. (2008) Efficient methods to compute genomic predictions, Journal of dairy science, 91, 4414-4423. Vitezica, Z.G., Varona, L. and Legarra, A. (2013) On the additive and dominant variance and covariance of individuals within the genomic selection scope, Genetics, 195, 1223-1230.
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