ASREML user guide release 3.0

A reviewer for the Journal of the Royal Statistical Society of England comments \"This is the first book covering in one volume all important topics in genetical statistics.\" Written to provide basic probability ideas in terms of genetic situations, since the theory of genetics is a probability theory; to give a definitive treatment of applications of these ideas to genetic theory; and to describe statistical methods appropriate to the data models that are developed.

An Introduction to Genetic Statistics

Linear models

The incorporation of genomic coefficients into the numerator relationship matrix allows estimation of breeding values using all phenotypic, pedigree and genomic information simultaneously. In such a single-step procedure, genomic and pedigree-based relationships have to be compatible. As there are many options to create genomic relationships, there is a question of which is optimal and what the effects of deviations from optimality are. Data of litter size (total number born per litter) for 338,346 sows were analyzed. Illumina PorcineSNP60 BeadChip genotypes were available for 1,989. Analyses were carried out with the complete data set and with a subset of genotyped animals and three generations pedigree (5,090 animals). A single-trait animal model was used to estimate variance components and breeding values. Genomic relationship matrices were constructed using allele frequencies equal to 0.5 (G05), equal to the average minor allele frequency (GMF), or equal to observed frequencies (GOF). A genomic matrix considering random ascertainment of allele frequencies was also used (GOF*). A normalized matrix (GN) was obtained to have average diagonal coefficients equal to 1. The genomic matrices were combined with the numerator relationship matrix creating H matrices. In G05 and GMF, both diagonal and off-diagonal elements were on average greater than the pedigree-based coefficients. In GOF and GOF*, the average diagonal elements were smaller than pedigree-based coefficients. The mean of off-diagonal coefficients was zero in GOF and GOF*. Choices of G with average diagonal coefficients different from 1 led to greater estimates of additive variance in the smaller data set. The correlation between EBV and genomic EBV (n = 1,989) were: 0.79 using G05, 0.79 using GMF, 0.78 using GOF, 0.79 using GOF*, and 0.78 using GN. Accuracies calculated by inversion increased with all genomic matrices. The accuracies of genomic-assisted EBV were inflated in all cases except when GN was used. Parameter estimates may be biased if the genomic relationship coefficients are in a different scale than pedigree-based coefficients. A reasonable scaling may be obtained by using observed allele frequencies and re-scaling the genomic relationship matrix to obtain average diagonal elements of 1.

/pdf/different-genomic-relationship-matrices-for-single-step-388safpn88.pdf

Different genomic relationship matrices for single-step analysis using phenotypic, pedigree and genomic information

Genetic correlations can have profound effects on evolutionary change (Lande and Arnold, 1983; Mitchell-Olds and Rutledge, 1986). Present patterns of genetic correlations in an organism may be caused by pre-existing pleiotropic and developmental relationships among traits and may produce constraints on evolution by natural selection (Cheverud, 1984; Maynard Smith et aI., 1985; Via and Lande, 1985; Futuyma, 1986; Mitchell-Olds and Rutledge, 1986; Barker and Thomas, 1987; Clark, 1987a; Zeng, 1988). Alternatively, selection may directly alter the patterns of genetic correlations, especially in cases in which two or more traits interact to perform a given function (Cheverud, 1984; Lande, 1984; Clark, 1987a. 1987b). In this paper we examine the patterns of phenotypic and additive genetic correlations among 10 morphological and life-history traits in wild radish plants. We hypothesized that some ofthese correlations have been influenced by selection. We predicted that floral and vegetative traits would be uncorrelated and that correlations among the lengths of the corolla tube, pistil and stamens would be higher than the rest of the floral correlations. The results were consistent with most of these predictions. Wild radish, Raphanus raphanistrum (Brassicaceae), is an annual weed of disturbed areas. The hermaphroditic flowers of wild radish are almost entirely selfincompatible (Sampson, 1964; Stanton et aI., 1989) and this species does not propagate vegetatively, so virtually all reproduction depends on successful insect pollination. Wild radish is pollinated by a variety of insects, mainly bees, butterflies, and flies (Kay, 1976;

Patterns of phenotypic and genetic correlations among morphological and life-history traits in wild radish, raphanus raphanistrum.

Covariance between relatives in a multibreed population was derived for an additive model with multiple unlinked loci. An efficient algorithm to compute the inverse of the additive genetic covariance matrix is given. For an additive model, the variance for a crossbred individual is a function of the additive variances for the pure breeds, the covariance between parents, and segregation variances. Provided that the variance of a crossbred individual is computed as presented here, the covariance between crossbred relatives can be computed using formulae for purebred populations. For additive traits the inverse of the genotypic covariance matrix given here can be used both to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood in multibreed populations. For nonadditive traits, the procedure currently used to analyze multibreed data can be improved using the theory presented here to compute additive covariances together with a suitable approximation for nonadditive covariances.

Covariance between relatives in multibreed populations: additive model.

At present, genetic evaluation in livestock using best linear unbiased prediction (BLUP) assumes autosomal inheritance. There is evidence, however, of X-chromosomal inheritance for some traits of economic importance. BLUP can accommodate models that include X-chromosomal in addition to autosomal inheritance. To obtain BLUP with autosomal and X-chromosomal additive inheritance for a population in which allelic frequency is equal in the sexes, and that is in gametic equilibrium, we write y
i = x′iβ + ai + si + ei, where y
i is the phenotypic value for individual i, x′i, is a vector of constants relating y
i to fixed effects, β is a vector of fixed effects, a
i is the additive genetic effect for autosomal loci, S
i is the additive genetic effect for X-chromosomal loci, and e
i is random error. The covariance matrix of a
i's is Aσ

 A
 2
 , where A is the matrix of twice the co-ancestries between relatives for autosomal loci, and σ

 A
 2
 is the variance of additive genetic effects for autosomal loci. The covariance matrix of s
i's is Sσ

 F
 2
 , where S is a matrix of functions of co-ancestries between relatives for X-chromosomal loci and σ

 F
 2
 is the variance of additive genetic effects for X-chromosomal loci for noninbred females. Given the covariance matrices of random effects a
i, si, and e
i, BLUPs of autosomal and of X-chromosomal additive effects can be obtained using mixed model equations. Recursive rules to construct S and an efficient algorithm to compute its inverse are given.

Genetic evaluation with autosomal and X-chromosomal inheritance.

This paper presents theory and methods to compute genotypic means and covariances in a two breed population under dominance inheritance, assuming multiple unlinked loci. It is shown that the genotypic mean is a linear function of five location parameters and that the genotypic covariance between relatives is a linear function of 25 dispersion parameters. Recursive procedures are given to compute the necessary identity coefficients. In the absence of inbreeding, the number of parameters for the mean is reduced from five to three and the number for the covariance is reduced from 25 to 12. In a two-breed population, for traits exhibiting dominance, the theory presented here can be used to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood.

Theory for modelling means and covariances in a two-breed population with dominance inheritance.

Effects of data imbalance on bias, sampling variance and mean square error of heritability estimated with variance components were examined using a random two-way nested classification. Four designs, ranging from zero imbalance (balanced data) to “low”, “medium” and “high” imbalance, were considered for each of four combinations of heritability (h2=0.2 and 0.4) and sample size (N=120 and 600). Observations were simulated for each design by drawing independent pseudo-random deviates from normal distributions with zero means, and variances determined by heritability. There were 100 replicates of each simulation; the same design matrix was used in all replications. Variance components were estimated by analysis of variance (Henderson's Method 1) and by maximum likelihood (ML). For the design and model used in this study, bias in heritability based on Method 1 and ML estimates of variance components was negligible. Effect of imbalance on variance of heritability was smaller for ML than for Method 1 estimation, and was smaller for heritability based on estimates of sire-plus-dam variance components than for heritability based on estimates of sire or dam variance components. Mean square error for heritability based on estimates of sire-plus-dam variance components appears to be less sensitive to data imbalance than heritability based on estimates of sire or dam variance components, especially when using Method 1 estimation. Estimation of heritability from sire-plus-dam components was insensitive to differences in data imbalance, especially for the larger sample size.

https://link.springer.com/content/pdf/10.1007/BF00251098.pdf

Effects of data imbalance on estimation of heritability

The tabular method to compute coancestry between two individuals is based on the principle that coancestry may be computed as the average coancestry between one individual and the parents of the other, on the condition that the former individual is not a direct descendent of the latter. It follows that coancestry also may be computed as the average of the four coancestries between the parents of the two individuals, on the condition that each individual is not a direct descendent of the other. The requirement for these conditions is explained.

M. Grossman

Papers

Covariance between relatives in multibreed populations: additive model.

Genetic evaluation with autosomal and X-chromosomal inheritance.

Theory for modelling means and covariances in a two-breed population with dominance inheritance.

Effects of data imbalance on estimation of heritability

On the principle underlying the tabular method to compute coancestry