Numerator Relationship Matrix Research Articles

Extension of the concept of kinship, relationship, and inbreeding to account for linked epistatic complexes

Abstract Although epistatic effects are well defined and, in principle, can be exploited in quantitative-genetic selection theory, they often are ignored or even treated as nuisance parameters in practical applications. Traditionally, epistasis is considered as an interaction between genes at unspecified loci. Inspired by the observation that functional genes are often organised in physical clusters, we developed a model to combine additive effects and additive × additive interactions in linked gene clusters of defined length. Malecot's kinship concept is extended to identity by descent probabilities for chromosome segments of a given length in Morgan units, called epistatic kinship. Using the analogy of Malecot's kinship and Wright's relationship and inbreeding coefficients, epistatic relationship coefficients and epistatic inbreeding coefficients are defined. Simple rules are given to set up the epistatic numerator relationship matrix and its inverse directly from a pedigree list. The well-known single locus parameters and algorithms to set up the additive numerator relationship matrix and its inverse are a special case of the suggested methodology for a chromosome segment length of null Morgan. A proof of concept of the suggested method is given with a small simulation study. Assuming additive, linked epistatic and residual variance components, 100 replicated data sets for 1000 individuals are generated. From these data, residual maximum likelihood estimates of the variance components and of the chromosome segment size are obtained. Potential applications of the methodology are discussed. Given that a substantial variance component is attributed to this effect, the expected genetic gain can be increased on the short term if selection is on additive and epistatic effects, the latter comprising additive × additive interaction effect of loci in linkage disequilibrium. This extra benefit, however, will diminish through crossing over in subsequent generations. Despite some practical problems yet to be solved, the suggested model and algorithms open new perspectives to use a higher proportion of genetic variability in selection and breeding.

Estimation of (co)variance components for reproductive traits in Angus beef cattle divergently selected for blood serum IGF-I concentration.

The objective of this study was to obtain estimates of (co)variance components for reproductive traits and insulin-like growth factor-I (IGF-I) concentration. Data were from a divergent selection experiment for blood serum IGF-I concentration in Angus beef cattle. Numbers of observations for mean IGF-I concentration of three blood samples taken at d 28, 42, and 56 of the 140-d postweaning test, scrotal circumference (SC), percentage of motile sperm cells (PMSC), percentage of morphologically normal sperm cells (PNSC), age of heifers at first calving (AFC), and calving rate (CR) were 1,848, 825, 596, 765, 294, and 2,092, respectively. Total number of animals in the numerator relationship matrix, including base animals, was 2,864, of which 1,861 were inbred. Estimates of direct heritability for IGF-I concentration of three blood samples collected at d 28, 42, and 56 of the postweaning test and for mean IGF-I concentration were 0.43+/-0.08, 0.51+/-0.09, 0.41+/-0.08, and 0.50+/-0.08, respectively. Estimates of direct heritability for SC, PMSC, PNSC, AFC, and CR were 0.51+/-0.13, 0.08+/-0.12, 0.47+/-0.07, 0.26+/-0.28, and 0.11+/-0.05, respectively. With the exception of age at first calving, estimates of maternal heritability and proportion of phenotypic variance that were due to permanent environmental effects of the dams were smaller than 0.21. Observations for calving rate were entered as either 1 (if calved) or 100 (if not calved). Estimates of additive genetic correlations of mean IGF-I concentration with SC, PMSC, PNSC, AFC, and CR were 0.35+/-0.11, 0.43+/-0.32, 0.00+/-0.03, -0.14+/-0.33, and -0.41+/-0.16, respectively. Environmental and phenotypic correlations for all of the traits with IGF-I measurements were smaller than 0.23. These results suggest that selection for increased serum IGF-I concentration should result in increased scrotal circumference, percent motile sperm cells, and calving rate.

Genetic evaluation of beef cattle accounting for uncertain paternity

A hierarchical Bayes (HIER) model for the quantitative genetic analysis of performance data was compared to a model based on the use of Henderson's average numerator relationship matrix (ANRM) when some pedigrees are characterized by uncertain paternity. A simulation study consisted of ten datasets characterized by 30% of animals having uncertain paternity for each of two traits: one having moderate heritabilities for direct and maternal genetic effects on weaning weight (WWT) and another having a high heritability for direct genetic effects on postweaning gain (PWG). Posterior inference on the variance components was very similar between the two models. In an application to WWT and PWG data from Brazilian Herefords, posterior inference on variance components was also very similar between ANRM and HIER. Furthermore, rank correlations on posterior means for genetic effects between the two models exceeded 0.90. Nevertheless, large differences in posterior means between these two models were observed for some animals. Furthermore, animals with uncertain paternity had generally larger posterior standard deviations of genetic effects using the HIER model compared to the ANRM model, likely because the HIER model infers upon sire assignment probabilities. Bayesian model choice criteria consistently favored the HIER model over the ANRM model in both simulation and Hereford data analysis studies.

THRESHOLD SIRE MODELS FOR ESTIMATING GENETIC PARAMETERS FOR STAYABILITY IN BEEF COWS

Stayability is the ability of a beef cow to remain in production to a specified age. In this study, the interest was in determining the genetic relationship between stayability to an early age with stayability to a later age. A nested threshold sire model for stayability was used here to estimate the genetic relationship between stayability to different ages. Genetic correlations were estimated among six different stayability traits using records from 1,868 Hereford cows. The model included period and year of birth as fixed factors and sire as a random factor. The numerator relationship matrix accounted for all known relationships among sires. Penalized quasilikelihood estimates were obtained using a probit link function. Estimates of heritability on the original scale were small and ranged from 0.09 to 0.17. Estimates of genetic correlations were low to moderate and variable in sign. Results indicate that selection for stayability to an early age would have a limited impact on stayability to later ages.

Bayesian inference strategies for the prediction of genetic merit using threshold models with an application to calving ease scores in Italian Piemontese cattle

SummaryFirst parity calving difficulty scores from Italian Piemontese cattle were analysed using a threshold mixed effects model. The model included the fixed effects of age of dam and sex of calf and their interaction and the random effects of sire, maternal grandsire, and herd‐year‐season. Covariances between sire and maternal grandsire effects were modelled using a numerator relationship matrix based on male ancestors. Field data consisted of 23 953 records collected between 1989 and 1998 from 4741 herd‐year‐seasons. Variance and covariance components were estimated using two alternative approximate marginal maximum likelihood (MML) methods, one based on expectation‐maximization (EM) and the other based on Laplacian integration. Inferences were compared to those based on three separate runs or sequences of Markov Chain Monte Carlo (MCMC) sampling in order to assess the validity of approximate MML estimates derived from data with similar size and design structure. Point estimates of direct heritability were 0.24, 0.25 and 0.26 for EM, Laplacian and MCMC (posterior mean), respectively, whereas corresponding maternal heritability estimates were 0.10, 0.11 and 0.12, respectively. The covariance between additive direct and maternal effects was found to be not different from zero based on MCMC‐derived confidence sets. The conventional joint modal estimates of sire effects and associated standard errors based on MML estimates of variance and covariance components differed little from the respective posterior means and standard deviations derived from MCMC. Therefore, there may be little need to pursue computation‐intensive MCMC methods for inference on genetic parameters and genetic merits using conventional threshold sire and maternal grandsire models for large datasets on calving ease.

An indirect approach to the extensive calculation of relationship coefficients

A method was described for calculating population statistics on relationship coefficients without using corresponding individual data. It relied on the structure of the inverse of the numerator relationship matrix between individuals under investigation and ancestors. Computation times were observed on simulated populations and were compared to those incurred with a conventional direct approach. The indirect approach turned out to be very efficient for multiplying the relationship matrix corresponding to planned matings (full design) by any vector. Efficiency was generally still good or very good for calculating statistics on these simulated populations. An extreme implementation of the method is the calculation of inbreeding coefficients themselves. Relative performances of the indirect method were good except when many full-sibs during many generations existed in the population.

Technical Note: Software for Calculation of the Inverse Numerator Relationship Matrix

Software was developed to compute the non-zero elements of the inverse of the numerator relationship matrix used in estimation of (co)variances and breeding values. The program was written to be flexible with regard to format of the input pedigree file and integration with existing software packages for genetic evaluation. The program was further designed to be highly portable, with a minimum of compiler dependence, and to use dynamic rather than static memory allocation. Real time required to read and sort input pedigrees containing 5,000 to 100,000 animals with varying levels of inbreeding and to compute the non-zero elements of the matrix was < 9.5min and increased as the numbers of animals in the pedigree file increased beyond 20,000 animals. Increased numbers of inbred animals in large (≥20,000 animals) pedigrees increased the time required to complete computations. Although a combination of alphanumeric animal identification codes were allowed, the time required to initially read pedigrees and, therefore, total run time was significantly decreased (P < 0.001) for large pedigrees when identification codes were strictly numeric for all animals.

Dynamic selection for maximizing response with constrained inbreeding in schemes with overlapping generations

AbstractA dynamic selection algorithm for maximizing annual genetic response while constraining the rate of inbreeding per generation in populations with overlapping generations is presented. The procedure gives the optimum number of individuals to be selected and the progeny they each produce. The solution to the problem was obtained by using BLUP estimated breeding values, the augmented numerator relationship matrix and lifetime breeding profiles. The procedure was able to constrain the rate of inbreeding per generation to a predefined level across generations of selection by considering all gene flow pathways. The optimization procedure represents an improvement on standard truncation BLUP selection, as it yielded substantially more genetic response (up to 35%) at the same rate of inbreeding.

Detecting QTLs for uni- and bipolar disorder using a variance component method.

The objective of this study was to use a robust variance component method to analyse unipolar and bipolar disorder in a large Scottish extended family (n = 168) in which linkage between markers and disease has been previously reported on the short arm of chromosome 4. Data consisted of diagnosed clinical uni- or bipolar disorder on 143 individuals, with microsatellite marker information on 109 of these individuals. The incidence of unipolar and bipolar disorder in the family was 17/143, and 11/143, respectively. Eleven linked markers on chromosome 4, spanning a region of approximately 26 cM, were used in the analysis. The statistical analysis was performed in two steps. First, pairwise identify-by-descent (IBD) coefficients for all individuals in the pedigree were calculated at 1 cM intervals, using all marker data simultaneously, with a Monte Carlo Markov Chain algorithm. Second, the variance in the trait of interest was partitioned using residual maximum likelihood (REML). Three components of variance were estimated: (i) a genetic component associated with the average relationship between individuals using the numerator relationship matrix, (ii) a genetic component associated with a chromosome location using the estimated IBD coefficients, and (iii) a residual component. The test statistic (LOD score) was calculated from the maximum likelihood of the full model, fitting all three variance components, and the maximum likelihood value from the reduced model, fitting a polygenic and residual component. The largest LOD scores (maximum LOD = 5.9), were found in a region spanning about 10 cM, when the trait was defined as the occurrence of either uni- or bipolar disorder. The putative QTL explained about 25% of the total variation in the trait.

The Additive Genetic Gamma Frailty Model

In this paper the additive genetic gamma frailty model is defined. Individual frailties are correlated as a result of an additive genetic model. An algorithm to construct additive genetic gamma frailties for any pedigree is given so that the variance–covariance structure among individual frailties equals the numerator relationship matrix times a variance. The EM algorithm can be used to estimate the parameters in the model. Calculations are similar using the EM algorithm in the shared frailty model, however the E step is not correspondingly simple. This is illustrated re‐analysing data, analysed by the shared frailty model in Nielsen et al. (1992), from the Danish adoptive register. Goodness of fit of the additive genetic gamma frailty model can be tested after analysing data with the correlated frailty model. Doing so, a “defect” in the often used and otherwise well behaving likelihood was found

Calculation of breed direct and maternal genetic fractions and breed specific direct and maternal heterozygosity for crossbreeding data

Teaching, research, and herd breeding applications may require calculation of breed additive contributions for direct and maternal genetic effects and fractions of heterozygosity associated with breed specific direct and maternal heterosis effects. These coefficients can be obtained from the first NB rows of a pseudo numerator relationship matrix where the first NB rows represent fractional contributions by breed to each animal or group representing a specific breed cross. The table begins with an NB x NB identity matrix representing pure breeds. Initial animals or representative crosses must be purebreds or two-breed crosses. Parents of initial purebreds are represented by the corresponding column and initial two-breed cross progeny by the two corresponding columns of the identity matrix. After that, usual rules are used to calculate the NB column entries corresponding to breeds for each animal. The NB entries are fractions of genes expected to be contributed by each of the pure breeds and correspond to the breed additive direct fractions. Entries in the column corresponding to the dam represent breed additive maternal fractions. Breed specific direct heterozygosity coefficients are entries of an NB x NB matrix formed by the outer product of the two NB by 1 columns associated with sire and dam of the animal. One minus sum of the diagonals represents total direct heterozygosity. Similarly, the NB x NB matrix formed by the outer product of columns associated with sire of dam and dam of dam contains breed specific maternal heterozygosity coefficients. These steps can be programmed to create covariates to merge with data. If X represents these coefficients for all unique breed crosses, then the reduced row echelon form function of MATLAB or SAS can be used on X to determine estimable functions of additive breed direct and maternal effects and breed specific direct and maternal heterosis effects

Breeding value and variance component estimation from data containing inbred individuals: application to gynogenetic families in common carp (Cyprinus carpio L.).

Under gynogenetic reproduction, offspring receive genes only from their dams and completely homozygous offspring are produced within one generation. When gynogenetic reproduction is applied to fully inbred individuals, homozygous clone lines are produced. A mixed model method was developed for breeding value and variance component estimation in gynogenetic families, which requires the inverse of the numerator relationship matrix. A general method for creating the inverse for a population with unusual relationships between animals is presented, which reduces to simple rules as is illustrated for gynogenetic populations. The presence of clones in gynogenetic populations causes singularity of the numerator relationship matrix. However, clones can be regarded as repeated observations of the same genotype, which can be accommodated by modifying the incidence matrix, and by considering only unique genotypes in the estimation procedure. Optimum gynogenetic sib family sizes for estimating heritabilities and estimates of their accuracy were derived and compared to those for conventional full-sib designs. This was done by means of a deterministic derivation and by stochastic simulation using Gibbs sampling. Optimum family sizes were smallest for gynogenetic families. Only for low heritabilities, there was a small advantage in accuracy under the gynogenetic design.

Maximising response with a predefined rate of inbreeding for overlapping generation structure

Wray and Thompson (1990) described the relationship between genetic contributions and the rate of inbreeding and this work was extended by Woolliams and Thompson (1994) to interpret genetic response in terms of genetic contributions. By using these relationships Meuwissen (1997) developed an algorithm for obtaining optimum selection decisions at a given (discrete) generation and level of inbreeding by maximising a function of accumulated gain corrected for squared contributions. Here we extend this method for optimising populations with overlapping generation structure and maximising response at a given rate rather than level of inbreeding. Three modifications to the afore mentioned model are presented: an augmented numerator relationship matrix which has not been corrected for reduced Mendelian variance with inbreeding, the consideration of committed future contributions beyond the next cohort derived from the lifetime breeding profile of the population, and a constraint to achieve the desired rate of inbreeding.

Alternatives for genetic evaluation with uncertain parentage

Alternative methods of genetic evaluation which consider uncertain paternity were compared theoretically and through Monte Carlo simulation. Records were simulated for 300 base generation animals and 10 subsequent generations of 100 animals each. Probabilities of paternal uncertainty were either 20 or 50%, heritability was 0.05, 0.25 or 0.50, mating was random with a female-to-male ratio of 5, and selection of breeding animals was either random or by truncation on phenotype. Simulations were replicated 10 times. Differences in expected selection response, for the genetic evaluation methods studied, were largest when heritability was low and degree of paternal uncertainty high. Expected response to selection was maximized in all cases by using uncertain paternity methods instead of genetic grouping or other methods. Advantages over genetic grouping ranged from 0 to 16%. The exclusion of performance records of animals with paternal uncertainty reduced expected selection response by as much as 37%, in addition to the reduction in expected response caused by excluding from selection animals with uncertain parents. All methods that accounted for true or probable paternity, either directly or through genetic groups, yielded unbiased estimates of genetic trends. Key words: Uncertain parentage, numerator relationship matrix, genetic grouping, genetic evaluation, genetic progress, Monte Carlo Simulation

Use of multiple genetic markers in prediction of breeding values.

Genotypes at a marker locus give information on transmission of genes from parents to offspring and that information can be used in predicting the individuals' additive genetic value at a linked quantitative trait locus (MQTL). In this paper a recursive method is presented to build the gametic relationship matrix for an autosomal MQTL which requires knowledge on recombination rate between the marker locus and the MQTL linked to it. A method is also presented to obtain the inverse of the gametic relationship matrix. This information can be used in a mixed linear model for simultaneous evaluation of fixed effects, gametic effects at the MQTL and additive genetic effects due to quantitative trait loci unlinked to the marker locus (polygenes). An equivalent model can be written at the animal level using the numerator relationship matrix for the MQTL and a method for obtaining the inverse of this matrix is presented. Information on several unlinked marker loci, each of them linked to a different locus affecting the trait of interest, can be used by including an effect for each MQTL. The number of equations per animal in this case is 2m + 1 where m is the number of MQTL. A method is presented to reduce the number of equations per animal to one by combining information on all MQTL and polygenes into one numerator relationship matrix. It is illustrated how the method can accommodate individuals with partial or no marker information. Numerical examples are given to illustrate the methods presented. Opportunities to use the presented model in constructing genetic maps are discussed.

PREDICTION OF BREEDING VALUES FOR UNMEASURED TRAITS FROM MEASURED TRAITS

Henderson (1977, 1984) described a method for prediction of breeding values for traits not in the model for records. This method may be practical for animal or sire models for the case when no measurements can be obtained on any animals for some traits to be evaluated. The least squares equations are augmented with A-1⊗GN-1 rather than with A-1⊗G0-1 where A is the numerator relationship and G0and GN are the genetic covariance matrices for measured and for all traits. This method can be used for each unmeasured trait or simultaneously for measured and all unmeasured traits. An option in the MTDFRUN module of the Multiple Trait Derivative Free REML (MTDFREML) program of Boldman et al. (1993) is to obtain solutions for breeding values and their prediction error variances. However, the preparation program (MTDFPREP) must be tricked to set-up equation numbers for breeding values of the unmeasured traits. Adding dummy records for the unmeasured traits but with missing records for the measured traits for dummy animals to the data file of animals with measured traits but with missing unmeasurable traits will result in the needed equations. At least two dummy records are needed to avoid a divide by zero error in calculating the sample standard deviation. The dummy records need to be associated with a level of at least one fixed factor. The dummy animals also must be added to the pedigree file with unknown sires and dams before running the program to obtain the inverse of the numerator relationship matrix (MTDFNRM). In the program to obtain solutions to the multiple trait mixed model equations (MTDFRUN), the full genetic (co)vaiiance matrix, GN, for both measured and unmeasured traits is needed. The residual (co)variance matrix must have zero covariances between pairs of measured and unmeasured traits but the variance of the unmeasured trait must not be zero. This procedure provides direct solutions for breeding values of unmeasured traits based on mixed model predictions of breeding values of the measured traits and also allows calculation of standard errors of prediction for the solutions directly from elements of the inverse of the ugmented coefficient matrix. For example, this procedure can be used to predict breeding values of bulls (which have tenderness measurements) for the correlated trait of tenderness as a steer or heifer which cannot be measured on the bull.

Effects of multiple-sire matings on genetic evaluations, selection response and rates of inbreeding

Multiple-sire mating is evaluated for a range of conditions, for its impact on regular genetic evaluation analyses in performance recording, and on selection response and inbreeding in breeding programs as a whole. Substitution of the average for the complete numerator relationship matrix (NRM) in the mixed model equations produced little bias in genetic trend estimates but substantial bias of inbreeding estimates. By recording sire group membership and dam identification under multiple-sire mating and using these and the average NRM in genetic evaluation analyses, it is shown that: 26–63% more genetic gain is achievable compared with selection on individual performance alone (no genetic analysis); and realised genetic gain is reduced by less than 6–24% compared with single-sire mating and use of complete BLUP analysis.

Considerations on genetic connectedness between management units under an animal model.

Connectedness among management units (e.g., herds or regions) is of concern in genetic evaluation. When genetic evaluation is under an animal model, connections occur through A, the numerator relationship matrix. It is argued that the most appropriate measure of connectedness is the average prediction error variance (PEV) of differences in EBV between animals in different management units. It is shown that PEV of differences is influenced by average genetic relationship between and within management units, which in turn affects the variances of estimates of differences between management unit effects. When PEV of differences cannot be computed, use of one of three alternative measures is proposed; the gene-flow method that measures the exchange of genes between management units, measurement of genetic drift variance based on average relationships between and within management units, and measurement of the variance of estimated differences between management units effects. These were correlated with PEV of differences in a test simulation. The gene-flow method, which is simplest to compute, had the lowest correlation (.671). The drift variance and variance of management unit effects methods were highly correlated with PEV of differences (.924 and .995, respectively).

Selection for reproductive rate in Rambouillet sheep: Estimated genetic change in reproductive rate

Rambouillet sheep were selected for high or low reproductive rate based on an index of reproductive rate, reproductive index = dam's total lifetime lambs born/(age in years-1). Selected ewes were first born in 1969. A random bred control line was established from the remaining foundation ewes with the first ewes born in 1973. Genetic variances were estimated with REML procedures for reproductive index and number of lambs born or weaned either per ewe exposed for breeding or per ewe lambing. Heritability of reproductive index was 0.26. Heritability for number of lambs born were 0.11 and 0.12 and for number of lambs weaned were 0.04 and 0.04 per ewe exposed for breeding and per ewe lambing, respectively. Breeding values for the reproductive index were estimated using the full animal model (BLUP) with the complete numerator relationship matrix on reproductive index, with fixed effect of year of birth of ewe and breeding values for number of lambs born or weaned either per ewe exposed for breeding or per ewe lambing, estimated with the fixed effects of year of birth of ewe and age of ewe and the random effect of permanent environmental effect among repeated lambing records. Genetic change was estimated as the regression of breeding value for reproductive index and number of lambs born or weaned either per ewe exposed for breeding or per ewe lambing on year of birth of the ewe. High and low lines both responded to selection for reproductive index as compared to the control line ( P<0.01). The high line increased at a rate of b = 0.0134 ± 0.0006 reproductive index units and the low line decreased at a rate of b = − 0.0098 ± 0.0005 reproductive index units per year. Response to selection for reproductive index resulted in b = − 0.0074 ± 0.0007 and b = 0.0163 ± 0.0006 lambs per ewe exposed for breeding, and b = − 0.0041 ± 0.0002 and b = 0.0075 ± 0.0002 lambs weaned per ewe exposed for breeding in the low and high lines, respectively. There was a greater response to selection in the high line compared to the low line. These results indicated that the reproductive index did respond to selection.

Estimation of genetic parameters of preweaning performance in the French Limousin cattle breed

Summary - Direct and maternal genetic and environmental parameters of preweaning growth and conformation at weaning were estimated in the French Limousin beef cattle field recording program using the tilde-hat approach of Van Raden and Jung (1988) with a sire, maternal grandsire (MGS) and dam within MGS model. The numerator relationship matrix among bulls was included in the estimation. The data available after editing contained 169 391 calves with performance records, from 43 683 dams, 7 265 sires, 5 664 maternal grandsires and 1 605 herds, for the years 1972-1989. The traits involved were: birth, 120-d and 210-d weights, average daily gains from birth to 120-d, from 120-d to 210-d, from birth to 210-d, muscular development (MD) and skeletal development (SD) scores at weaning. Estimates ranged from 0.22 to 0.32 for additive direct heritabilities and from 0.06 to 0.16 for maternal heritabilities. Correlations between direct and maternal genetic effects for these traits were negative, ranging from -0.23 to -0.49. Maternal permanent environmental effects were small for all traits, accounting for 5-9% of the phenotypic variances for preweaning growth performance, and 3% and 4% for MD and SD, respectively. beef cattle / variance components / preweaning growth / conformation score / direct and maternal effects / field data