Comparison between estimation of breeding values and fixed effects using Bayesian and empirical BLUP estimation under selection on parents and missing pedigree information

Bayesian (via Gibbs sampling) and empirical BLUP (EBLUP) estimation of fixed effects and breeding values were compared by simulation. Combinations of two simulation models (with or without effect of contemporary group (CG)), three selection schemes (random, phenotypic and BLUP selection), two levels of heritability (0.20 and 0.50) and two levels of pedigree information (0% and 15% randomly missing) were considered. Populations consisted of 450 animals spread over six discrete generations. An infinitesimal additive genetic animal model was assumed while simulating data. EBLUP and Bayesian estimates of CG effects and breeding values were, in all situations, essentially the same with respect to Spearman''s rank correlation between true and estimated values. Bias and mean square error (MSE) of EBLUP and Bayesian estimates of CG effects and breeding values showed the same pattern over the range of simulated scenarios. Methods were not biased by phenotypic and BLUP selection when pedigree information was complete, albeit MSE of estimated breeding values increased for situations where CG effects were present. Estimation of breeding values by Bayesian and EBLUP was similarly affected by joint effect of phenotypic or BLUP selection and randomly missing pedigree information. For both methods, bias and MSE of estimated breeding values and CG effects substantially increased across generations.     

Best Linear Unbiased Prediction (BLUP) for regional yield trials: a comparison to additive main effects and multiplicative interaction (AMMI) analysis

Multilocation trials are often used to analyse the adaptability of genotypes in different environments and to find for each environment the genotype that is best adapted; i.e. that is highest yielding in that environment. For this purpose, it is of interest to obtain a reliable estimate of the mean yield of a cultivar in a given environment. This article compares two different statistical estimation procedures for this task: the Additive Main Effects and Multiplicative Interaction (AMMI) analysis and Best Linear Unbiased Prediction (BLUP). A modification of a cross validation procedure commonly used with AMMI is suggested for trials that are laid out as a randomized complete block design. The use of these procedure is exemplified using five faba bean datasets from German registration trails. BLUP was found to outperform AMMI in four of five faba bean datasets.     

R法遗传参数估值置信区间的计算方法和重复估计次数的研究

探讨R法遗传参数估值置信区间的计算方法和重复估计次数(NORE)对参数估值的影响,利用4种模型通过模拟产生数据集。基础群中公、母畜数分别为200和2000头,BLUP育种值选择5个世代。利用多变量乘法迭代(MMI)法,结合先决条件的共扼梯度(PCG)法求解混合模型方程组估计方差组分。用经典方法、Box-Cox变换后的经典方法和自助法计算参数估值的均数、标准误和置信区间。结果表明,重复估计次数较多时,3种方法均可;重复估计次数较少时,建议使用自助法。简单模型下需要较少的重复估计,但对于复杂模型则需要较多的重复估计。随模型中随机效应数的增加,直接遗传力高估。随着PCG和MMI轮次的增大,参数估值表现出低估的趋势。     

Best linear prediction of breeding values in a forest tree improvement program

Summary Best Linear Prediction (BLP) was used to predict breeding values for 1,396 parents from progeny test data in an operational slash pine breeding program. BLP rankings of parents were compared to rankings of averaged standard scores, a common approach in forestry. Using BLP rankings, selection of higher ranking parents tends to choose parents in a larger number of more precise progeny tests. The trend is the opposite with standard scores; higher ranking parents tend to be those in fewer, less precise tests. BLP and a related methodology, Best Linear Unbiased Prediction (BLUP), were developed by dairy cattle breeders and have not been used widely outside of animal breeding for predicting breeding values from messy progeny test data. Application of either of these techniques usually requires simplifying assumptions to keep the problem computationally tractable. The more appropriate technique for a given application depends upon which set of assumptions are better for the given problem. An assumption of homogeneous genetic and error variances and covariances, generally made by animal breeders when applying BLUP, was inappropriate for our data. We employed an approach that treated fixed effects as known and treated the same trait measured in different environments as different traits with heterogeneous variance structures. As tree improvement programs become more complex, the ease with which BLP and BLUP handle messy data and incorporate diverse sources of information should make these techniques appealing to forest tree breeders.     

Best Linear Unbiased Prediction of Genomic Breeding Values Using a Trait-Specific Marker-Derived Relationship Matrix

Background

With the availability of high density whole-genome single nucleotide polymorphism chips, genomic selection has become a promising method to estimate genetic merit with potentially high accuracy for animal, plant and aquaculture species of economic importance. With markers covering the entire genome, genetic merit of genotyped individuals can be predicted directly within the framework of mixed model equations, by using a matrix of relationships among individuals that is derived from the markers. Here we extend that approach by deriving a marker-based relationship matrix specifically for the trait of interest.

Methodology/Principal Findings

In the framework of mixed model equations, a new best linear unbiased prediction (BLUP) method including a trait-specific relationship matrix (TA) was presented and termed TABLUP. The TA matrix was constructed on the basis of marker genotypes and their weights in relation to the trait of interest. A simulation study with 1,000 individuals as the training population and five successive generations as candidate population was carried out to validate the proposed method. The proposed TABLUP method outperformed the ridge regression BLUP (RRBLUP) and BLUP with realized relationship matrix (GBLUP). It performed slightly worse than BayesB with an accuracy of 0.79 in the standard scenario.

Conclusions/Significance

The proposed TABLUP method is an improvement of the RRBLUP and GBLUP method. It might be equivalent to the BayesB method but it has additional benefits like the calculation of accuracies for individual breeding values. The results also showed that the TA-matrix performs better in predicting ability than the classical numerator relationship matrix and the realized relationship matrix which are derived solely from pedigree or markers without regard to the trait. This is because the TA-matrix not only accounts for the Mendelian sampling term, but also puts the greater emphasis on those markers that explain more of the genetic variance in the trait.     

On selection criteria and estimation of parameters when the variance is heterogeneous

Summary Procedures for ranking candidates for selection and for estimating genetic and environmental parameters when variances are heterogeneous are discussed. The best linear unbiased predictor (BLUP) accounts automatically for heterogeneous variance provided that the covariance structure is known and that the assumptions of the model hold. Under multivariate normality BLUP allowing for heterogeneous variance maximizes expected genetic progress. Examples of application of BLUP to selection when residual or genetic variances are heterogeneous are given. Restricted maximum likelihood estimation of heterogeneous variances and covariances via the expectation-maximization algorithm is presented.     

Multiple-trait BLUP: a suitable strategy for genetic selection of <Emphasis Type="Italic">Eucalyptus</Emphasis>

Usually, genetic selection is carried out based on several traits, which can be genetically correlated. In this case, selection bias may occur if these traits are analyzed individually. Thus, the present work aimed to evaluate the applicability and efficiency of multiple-trait best linear unbiased prediction (BLUP) in the genetic selection of Eucalyptus. The data used in this work refer to the evaluation of a partial diallel of Eucalyptus spp. in relation to height, diameter at breast height (DBH), and volume. Variance components and genetic and non-genetic parameters were estimated via residual maximum likelihood (REML). Multiple-trait BLUP led to estimates of mean additive genetic variance higher than the estimates obtained via single-trait BLUP and, consequently, led to higher estimates of narrow-sense individual interpopulational heritabilities and mean accuracies. Partial genetic correlations obtained via multiple-trait BLUP allowed a real understanding of the association between traits, differently from those obtained via single-trait BLUP. Multiple-trait BLUP led to higher gains predicted with the selection for height, DBH, and volume and can be efficiently applied in the genetic selection of Eucalyptus.     

Testcross additive and dominance effects in best linear unbiased prediction of maize single-cross performance

Best linear unbiased prediction (BLUP) has been found to be useful in maize (Zea mays L.) breeding. The advantage of including both testcross additive and dominance effects (Intralocus Model) in BLUP, rather than only testcross additive effects (Additive Model), has not been clearly demonstrated. The objective of this study was to compare the usefulness of Intralocus and Additive Models for BLUP of maize single-cross performance. Multilocation data from 1990 to 1995 were obtained from the hybrid testing program of Limagrain Genetics. Grain yield, moisture, stalk lodging, and root lodging of untested single crosses were predicted from (1) the performance of tested single crosses and (2) known genetic relationships among the parental inbreds. Correlations between predicted and observed performance were obtained with a delete-one cross-validation procedure. For the Intralocus Model, the correlations ranged from 0.50 to 0.66 for yield, 0.88 to 0.94 for moisture, 0.47 to 0.69 for stalk lodging, and 0.31 to 0.45 for root lodging. The BLUP procedure was consistently more effective with the Intralocus Model than with the Additive Model. When the Additive Model was used instead of the Intralocus Model, the reductions in the correlation were largest for root lodging (0.06–0.35), smallest for moisture (0.00–0.02), and intermediate for yield (0.02–0.06) and stalk lodging (0.02–0.08). The ratio of dominance variance (v _D) to total genetic variance (v _G) was highest for root lodging (0.47) and lowest for moisture (0.10). The Additive Model may be used if prior information indicates that V_D for a given trait has little contribution to V_G. Otherwise, the continued use of the Intralocus Model for BLUP of single-cross performance is recommended.     

A Function Accounting for Training Set Size and Marker Density to Model the Average Accuracy of Genomic Prediction

Prediction of genomic breeding values is of major practical relevance in dairy cattle breeding. Deterministic equations have been suggested to predict the accuracy of genomic breeding values in a given design which are based on training set size, reliability of phenotypes, and the number of independent chromosome segments (). The aim of our study was to find a general deterministic equation for the average accuracy of genomic breeding values that also accounts for marker density and can be fitted empirically. Two data sets of 5′698 Holstein Friesian bulls genotyped with 50 K SNPs and 1′332 Brown Swiss bulls genotyped with 50 K SNPs and imputed to ∼600 K SNPs were available. Different k-fold (k = 2–10, 15, 20) cross-validation scenarios (50 replicates, random assignment) were performed using a genomic BLUP approach. A maximum likelihood approach was used to estimate the parameters of different prediction equations. The highest likelihood was obtained when using a modified form of the deterministic equation of Daetwyler et al. (2010), augmented by a weighting factor (w) based on the assumption that the maximum achievable accuracy is . The proportion of genetic variance captured by the complete SNP sets () was 0.76 to 0.82 for Holstein Friesian and 0.72 to 0.75 for Brown Swiss. When modifying the number of SNPs, w was found to be proportional to the log of the marker density up to a limit which is population and trait specific and was found to be reached with ∼20′000 SNPs in the Brown Swiss population studied.     

Linear Mixed Effects Models under Inequality Constraints with Applications

Constraints arise naturally in many scientific experiments/studies such as in, epidemiology, biology, toxicology, etc. and often researchers ignore such information when analyzing their data and use standard methods such as the analysis of variance (ANOVA). Such methods may not only result in a loss of power and efficiency in costs of experimentation but also may result poor interpretation of the data. In this paper we discuss constrained statistical inference in the context of linear mixed effects models that arise naturally in many applications, such as in repeated measurements designs, familial studies and others. We introduce a novel methodology that is broadly applicable for a variety of constraints on the parameters. Since in many applications sample sizes are small and/or the data are not necessarily normally distributed and furthermore error variances need not be homoscedastic (i.e. heterogeneity in the data) we use an empirical best linear unbiased predictor (EBLUP) type residual based bootstrap methodology for deriving critical values of the proposed test. Our simulation studies suggest that the proposed procedure maintains the desired nominal Type I error while competing well with other tests in terms of power. We illustrate the proposed methodology by re-analyzing a clinical trial data on blood mercury level. The methodology introduced in this paper can be easily extended to other settings such as nonlinear and generalized regression models.     

Genomic BLUP Decoded: A Look into the Black Box of Genomic Prediction

Genomic best linear unbiased prediction (BLUP) is a statistical method that uses relationships between individuals calculated from single-nucleotide polymorphisms (SNPs) to capture relationships at quantitative trait loci (QTL). We show that genomic BLUP exploits not only linkage disequilibrium (LD) and additive-genetic relationships, but also cosegregation to capture relationships at QTL. Simulations were used to study the contributions of those types of information to accuracy of genomic estimated breeding values (GEBVs), their persistence over generations without retraining, and their effect on the correlation of GEBVs within families. We show that accuracy of GEBVs based on additive-genetic relationships can decline with increasing training data size and speculate that modeling polygenic effects via pedigree relationships jointly with genomic breeding values using Bayesian methods may prevent that decline. Cosegregation information from half sibs contributes little to accuracy of GEBVs in current dairy cattle breeding schemes but from full sibs it contributes considerably to accuracy within family in corn breeding. Cosegregation information also declines with increasing training data size, and its persistence over generations is lower than that of LD, suggesting the need to model LD and cosegregation explicitly. The correlation between GEBVs within families depends largely on additive-genetic relationship information, which is determined by the effective number of SNPs and training data size. As genomic BLUP cannot capture short-range LD information well, we recommend Bayesian methods with t-distributed priors.     

Model comparison for genetic evaluation of milk yield in crossbred Holsteins in the tropics

The objective of this study was to compare models for appropriate genetic parameter estimation for milk yield (305-day) in crossbred Holsteins in the tropics, where only records from crossbred cows were available. Eleven models with different effects of contemporary group (CG) at calving (herd-year-season or herd-year-month as fixed, and herd-year-month as random), age at calving (as linear or quadratic covariates, age-class, and age-class x lactation), and dominance were considered. On-farm records from small herds (n < 50) were included or excluded to validate the parameter estimates. Average Information Restricted Maximum Likelihood (AIREML) and Best Linear Unbiased Prediction (BLUP) were used to estimate variance components and breeding values. R-square (R2) and standard error of heritability (h2) were used to determine the appropriate model. The estimates of heritability from most models ranged from 0.18 to 0.22. CG formation of herd-year-month as a random effect slightly lowered the additive genetic variance but considerably decreased the permanent environmental variance. The model with age-class x lactation gave better R2 than other age adjustments. The models including records from smallholders gave similar estimates of heritability and a lower standard error than the models excluding them. The estimate of dominance variance as a proportion of total variance was close to zero. The low ratio of dominance to additive genetic variance suggested that the inclusion of dominance effects in the model was unjustified. In conclusion, the model including the effects of herd-year-month, age-class x lactation, as well as additive genetic, permanent environmental and residual effects, was the most appropriate for genetic evaluation in crossbred Holsteins, where records from smallholders could be included.     

Impact of genotyping strategy on the accuracy of genomic prediction in simulated populations of purebred swine

Single-step genomic BLUP (ssGBLUP) has been widely used in genomic evaluation due to relatively higher prediction accuracy and simplicity of use. The prediction accuracy from ssGBLUP depends on the amount of information available concerning both genotype and phenotype. This study investigated how information on genotype and phenotype that had been acquired from previous generations influences the prediction accuracy of ssGBLUP, and thus we sought an optimal balance about genotypic and phenotypic information to achieve a cost-effective and computationally efficient genomic evaluation. We generated two genetically correlated traits (h² = 0.35 for trait A, h² = 0.10 for trait B and genetic correlation 0.20) as well as two distinct populations mimicking purebred swine. Phenotypic and genotypic information in different numbers of previous generations and different genotyping rates for each litter were set to generate different datasets. Prediction accuracy was evaluated by correlating genomic estimated breeding values with true breeding values for genotyped animals in the last generation. The results revealed a negligible impact of previous generations that lacked genotyped animals on the prediction accuracy. Phenotypic and genotypic data, including the most recent three to four generations with a genotyping rate of 40% or 50% for each litter, could lead to asymptotic maximum prediction accuracy for genotyped animals in the last generation. Single-step genomic best linear unbiased prediction yielded an optimal balance about genotypic and phenotypic information to ensure a cost-effective and computationally efficient genomic evaluation of populations of polytocous animals such as purebred pigs.     

Prediction of F3 row performance from F2 individual plant data in oats

Summary Parameters estimated from a Gardner-Eberhart analysis of the F₂ generation of a six-parent diallel in oats (Avena sativa L.) were used to compare methods for predicting the performance of F₃ row plots. The prediction methods were: (1) individual F₂ plant performance (F2I), (2) parent average plus F₂ plot deviations (PF2), (3) parent average plus weighted F₂ plot deviations (PF2P), (4) best linear unbiased prediction (BLUP) of parent average plus F₂ plot deviations (BPF2), and (5) BLUP plus weighted F₂ deviations (BF2). The F₂ single-plant traits used for prediction were biological yield to predict F₃ biological yield, whole plant and primary tiller grain yield for prediction of F₃ grain yield, and whole plant and primary tiller harvest index (HI) to predict F₃ HI. Prediction methods were evaluated by correlations between predicted and observed F₃ performance. Prediction methods and traits for which correlations were greater than for F2I included: BF2 for biological yield, PF2, PF2P and BF2 for whole plant grain yield, PF2, BPF2, and BF2 for primary tiller grain yield. None had a correlation significantly greater than F2I for either measure of HI, where heritability was large. PF2 is the recommended method for traits with low heritability because of its simplicity and because it had the largest or nearly the largest correlation for each of the yield traits. F2I is the recommended method for traits with larger heritability.Contribution No. 8821 of the U.S. Regional Pasture Research LaboratoryDeceased     

Determination of the parameters of the single 2-allele autosomal locus model with incomplete penetrance

A method for estimating the parameters of the single autosomal two-allele locus model with incomplete penetrance (the gene frequency and penetrances of the three genotypes) is presented. The method requires the presence of (1) the population frequency of the trait; (2) any set of incidences in relatives of probands that would allow to obtain the estimations of additive and dominant genetic variances and (3) segregation frequencies determined by the generalized Weinberg's method. The method proposed yields the uniquel estimations of the model parameters for traits which have additive genetic variance differed from zero. Some aspects of practical use of the method are discussed.     

Genetic evaluation methods for populations with dominance and inbreeding

Summary The effect of inbreeding on mean and genetic covariance matrix for a quantitative trait in a population with additive and dominance effects is shown. This genetic covariance matrix is a function of five relationship matrices and five genetic parameters describing the population. Elements of the relationship matrices are functions of Gillois (1964) identity coefficients for the four genes at a locus in two individuals. The equivalence of the path coefficient method (Jacquard 1966) and the tabular method (Smith and Mäki-Tanila 1990) to compute the covariance matrix of additive and dominance effects in a population with inbreeding is shown. The tabular method is modified to compute relationship matrices rather than the covariance matrix, which is trait dependent. Finally, approximate and exact Best Linear Unbiased Predictions (BLUP) of additive and dominance effects are compared using simulated data with inbreeding but no directional selection. The trait simulated was affected by 64 unlinked biallelic loci with equal effect and complete dominance. Simulated average inbreeding levels ranged from zero in generation one to 0.35 in generation five. The approximate method only accounted for the effect of inbreeding on mean and additive genetic covariance matrix, whereas the exact accounted for all of the changes in mean and genetic covariance matrix due to inbreeding. Approximate BLUP, which is computable for large populations where exact BLUP is not feasible, yielded unbiased predictions of additive and dominance effects in each generation with only slightly reduced accuracies relative to exact BLUP.     

A Comprehensive Genetic Approach for Improving Prediction of Skin Cancer Risk in Humans

Prediction of genetic risk for disease is needed for preventive and personalized medicine. Genome-wide association studies have found unprecedented numbers of variants associated with complex human traits and diseases. However, these variants explain only a small proportion of genetic risk. Mounting evidence suggests that many traits, relevant to public health, are affected by large numbers of small-effect genes and that prediction of genetic risk to those traits and diseases could be improved by incorporating large numbers of markers into whole-genome prediction (WGP) models. We developed a WGP model incorporating thousands of markers for prediction of skin cancer risk in humans. We also considered other ways of incorporating genetic information into prediction models, such as family history or ancestry (using principal components, PCs, of informative markers). Prediction accuracy was evaluated using the area under the receiver operating characteristic curve (AUC) estimated in a cross-validation. Incorporation of genetic information (i.e., familial relationships, PCs, or WGP) yielded a significant increase in prediction accuracy: from an AUC of 0.53 for a baseline model that accounted for nongenetic covariates to AUCs of 0.58 (pedigree), 0.62 (PCs), and 0.64 (WGP). In summary, prediction of skin cancer risk could be improved by considering genetic information and using a large number of single-nucleotide polymorphisms (SNPs) in a WGP model, which allows for the detection of patterns of genetic risk that are above and beyond those that can be captured using family history. We discuss avenues for improving prediction accuracy and speculate on the possible use of WGP to prospectively identify individuals at high risk.     

Use of Generalised Linear Mixed Models for the Analysis of Clustered Survival Data

Data from a litter matched tumorigenesis experiment are analysed using a generalised linear mixed model (GLMM) approach to the analysis of clustered survival data in which there is a dependence of failure time observations within the same litter. Maximum likelihood (ML) and residual maximum likelihood (REML) estimates of risk variable parameters, variance component parameters and the prediction of random effects are given. Estimation of treatment effect parameter (carcinogen effect) has good agreement with previous analyses obtained in the literature though the dependence structure within a litter is modelled in different ways. The variance component estimation provides the estimated dispersion of the random effects. The prediction of random effects, is useful, for instance, in identifying high risk litters and individuals. The present analysis illustrates its wider application to detecting increased risk of occurrence of disease in particular families of a study population.     

Accounting for genomic pre-selection in national BLUP evaluations in dairy cattle

Background

In future Best Linear Unbiased Prediction (BLUP) evaluations of dairy cattle, genomic selection of young sires will cause evaluation biases and loss of accuracy once the selected ones get progeny.

Methods

To avoid such bias in the estimation of breeding values, we propose to include information on all genotyped bulls, including the culled ones, in BLUP evaluations. Estimated breeding values based on genomic information were converted into genomic pseudo-performances and then analyzed simultaneously with actual performances. Using simulations based on actual data from the French Holstein population, bias and accuracy of BLUP evaluations were computed for young sires undergoing progeny testing or genomic pre-selection. For bulls pre-selected based on their genomic profile, three different types of information can be included in the BLUP evaluations: (1) data from pre-selected genotyped candidate bulls with actual performances on their daughters, (2) data from bulls with both actual and genomic pseudo-performances, or (3) data from all the genotyped candidates with genomic pseudo-performances. The effects of different levels of heritability, genomic pre-selection intensity and accuracy of genomic evaluation were considered.

Results

Including information from all the genotyped candidates, i.e. genomic pseudo-performances for both selected and culled candidates, removed bias from genetic evaluation and increased accuracy. This approach was effective regardless of the magnitude of the initial bias and as long as the accuracy of the genomic evaluations was sufficiently high.

Conclusions

The proposed method can be easily and quickly implemented in BLUP evaluations at the national level, although some improvement is necessary to more accurately propagate genomic information from genotyped to non-genotyped animals. In addition, it is a convenient method to combine direct genomic, phenotypic and pedigree-based information in a multiple-step procedure.     

Genetic steady-state under BLUP selection for an infinite and homogeneous population with discrete generations

A matrix derivation is proposed to analytically calculate the asymptotic genetic variance-covariance matrix under BLUP selection according to the initial genetic parameters in a large population with discrete generations. The asymptotic genetic evolution of a homogeneous population with discrete generations is calculated for a selection operating on an index including all information (pedigree and records) from a non-inbred and unselected base population (BLUP selection) or on an index restricted to records of a few ancestral generations. Under the first hypothesis, the prediction error variance of the selection index is independent of selection and is calculated from the genetic parameters of the base population. Under the second hypothesis, the prediction error variance depends on selection. Furthermore, records of several generations of ancestors of the candidates for selection must be used to maintain a constant prediction error variance over time. The number of ancestral generations needed depends on the population structure and on the occurrence of fixed effects. Without fixed effects to estimate, accounting for two generations of ancestors is sufficient to estimate the asymptotic prediction error variance. The amassing of information from an unselected base population proves to be important in order not to overestimate the asymptotic genetic gains and not to underestimate the asymptotic genetic variances.