Prediction of rates of inbreeding in selected populations

A method is presented for the prediction of rate of inbreeding for populations with discrete generations. The matrix of Wright's numerator relationships is partitioned into 'contribution' matrices which describe the contribution of the Mendelian sampling of genes of ancestors in a given generation to the relationship between individuals in later generations. These contributions stabilize with time and the value to which they stabilize is shown to be related to the asymptotic rate of inbreeding and therefore also the effective population size, Ne approximately 2N/(mu 2r + sigma 2r), where N is the number of individuals per generation and mu r and sigma 2r are the mean and variance of long-term relationships or long-term contributions. These stabilized values are then predicted using a recursive equation via the concept of selective advantage for populations with hierarchical mating structures undergoing mass selection. Account is taken of the change in genetic parameters as a consequence of selection and also the increasing 'competitiveness' of contemporaries as selection proceeds. Examples are given and predicted rates of inbreeding are compared to those calculated in simulations. For populations of 20 males and 20, 40, 100 or 200 females the rate of inbreeding was found to increase by as much as 75% over the rate of inbreeding in an unselected population depending on mating ratio, selection intensity and heritability of the selected trait. The prediction presented here estimated the rate of inbreeding usually within 5% of that calculated from simulation.     

Asymptotic rates of response from forest tree breeding strategies using best linear unbiased prediction

 Genetic gain equations are developed for selection on multiple traits using either multi- or univariate best linear unbiased predictors (BLUP) and for selection under controlled and open pollination and polymix mating schemes. The equations assume an infinite population and account for the effects of selection. A comparison with simulated populations under the same mating schemes show that the gain equations predict selection response well, with the predictions having some upward bias. The gain equations are used to compare across mating schemes, to compare univariate to multivariate analyses, and to measure the reduction in the rate of genetic gain due to selection disequilibrium. Results show controlled pollination schemes can offer as much as a 56% advantage in genetic gain relative to open pollination. The reduction in the rate of genetic gain due to selection disequilibrium is approximately 27% under controlled pollination for the breeding goals studied. The results show a limited benefit in using multivariate analyses for predicting breeding values. Received: 20 April 1997 / Accepted: 8 October 1997     

Methods for predicting rates of inbreeding in selected populations

Summary In selected populations, families superior for the selected trait are likely to contribute more offspring to the next generation than inferior families and, as a consequence, the rate of inbreeding is likely to be higher in selected populations than in randomly mated populations of the same structure. Methods to predict rates of inbreeding in selected populations are discussed. The method of Burrows based on probabilities of coselection is reappraised in conjunction with the transition matrix method of Woolliams. The method of Latter based on variances and covariances of family size is also examined. These methods are one-generation approaches in the sense that they only account for selective advantage over a single generation, from parents to offspring. Two-generation methods are developed that account for selective advantage over two generations, from grandparent to grandoffspring as well as from parent to offspring. Predictions are compared to results from simulation. The best one-generation method was found to underpredict rates of inbreeding by 10–25%, and the two-generation methods were found to underpredict rates of inbreeding by 9–18%.     

Evaluation of the best linear unbiased prediction method for breeding values of fruit-quality traits in citrus

Fruit-quality trait improvement is an important objective in citrus breeding; however, fruit breeding programs often accumulate highly unbalanced phenotypic records, which are a serious obstacle in comparing and selecting genotypes. The best linear unbiased prediction (BLUP) method can be used to overcome these difficulties, but few fruit breeding programs have adopted the method, and to our knowledge, the method has not yet been used to predict breeding values of traits based on pedigree information and genetic correlations between traits in citrus. Accordingly, we used the BLUP method to predict the breeding values of nine fruit-quality traits (fruit weight, fruit skin color, fruit surface texture, peelability, flesh color, pulp firmness, segment firmness, sugar content, and acid content) utilizing phenotypic records collected over several years as part of the citrus breeding program conducted at the Kuchinotsu branch of the National Institute of Fruit Tree Science in Japan. Although the accumulated phenotypic records were highly unbalanced, the BLUP method was able to predict the breeding values of all 2122 genotypes (111 parental cultivars and 2011 F₁ offspring from 126 pair-cross families), as well as estimates of several genetic parameters, including narrow-sense heritability and phenotypic and genotypic correlations. These findings demonstrate the utility of the BLUP method in citrus crossbreeding and provide predicted breeding values, which can be used to select superior genotypes in the Kuchinotsu Citrus Breeding Program and related genetic selection endeavors.     

Restricted best linear unbiased prediction and a selection model

Summary An equivalence between restricted best linear unbiased prediction (and thus restricted selection index) and a particular example of a selection model is presented. Specifically, the equivalence is between restricted selection and a model of selection on the residuals of the general mixed linear model. This result illustrates that restricted selection acts by nonrandomly sampling those genes that act pleiotropically in multiple trait genetic models. An expression for a mixed linear model which includes restrictions is also presented.     

Prediction of maize double-cross hybrids using the best linear unbiased prediction with microsatellite marker information

We examined the usefulness of the best linear unbiased prediction associated with molecular markers for prediction of untested maize double-cross hybrids. Ten single-cross hybrids from different commercial backgrounds were crossed using a complete diallel design. These 10 single-cross hybrids were genotyped with 20 microsatellite markers. The best linear unbiased prediction associated with microsatellite information gave relatively good prediction ability of the double-cross hybrid performance, with correlations between observed phenotypic values and genotypic prediction values varying from 0.27 to 0.54. Taking into account the predictions of specific combing ability, the correlation between observed and predicted specific combining ability varied from 0.50 to 0.88. Based on these results, we infer that it is feasible to predict maize double-cross hybrids with different unbalance degrees without including any prior information about parental inbreed lines or single-cross hybrid performance.     

Prediction of rates of inbreeding in populations undergoing index selection

For populations undergoing mass selection, previous studies have shown that the rate of inbreeding is directly related to the mean and variance of long-term contributions from ancestors to descendants, and thus prediction of the rate of inbreeding can be achieved via the prediction of long-term contributions. In this paper, it is shown that the same relationship between the rate of inbreeding and long-term contributions is found when selection is based on an index of individual and sib records (index selection) and where sib records may be influenced by a common environment. In these situations, rates of inbreeding may be considerably higher than under mass selection. An expression for the rate of inbreeding is derived for populations undergoing index selection based on variances of (one-generation) family size and incorporating the concept of long-term selective advantage. When the mating structure is hierarchical, and when half-sib records are included in the index, the correlation between parental breeding values and the index values of their offspring is higher for male parents than female parents. This introduces an important asymmetry between the contributions of male and female ancestors to the evolution of inbreeding which is not present when selection is based on individual and/or full-sib records alone. The prediction equation for index selection accounts for this asymmetry. The prediction is compared to rates of inbreeding calculated from simulation. The prediction is good when family size is small relative to the number selected. The reasons for overprediction in other situations are discussed.     

Empirical best linear unbiased prediction in cultivar trials using factor-analytic variance-covariance structures

 Results of multi-environment trials to evaluate new plant cultivars may be displayed in a two-way table of genotypes by environments. Different estimators are available to fill the cells of such tables. It has been shown previously that the predictive accuracy of the simple genotype by environment mean is often lower than that of other estimators, e.g. least-squares estimators based on multiplicative models, such as the additive main effects multiplicative interaction (AMMI) model, or empirical best-linear unbiased predictors (BLUPs) based on a two-way analysis-of-variance (ANOVA) model. This paper proposes a method to obtain BLUPs based on models with multiplicative terms. It is shown by cross-validation using five real data sets (oilseed rape, Brassica napus L.) that the predictive accuracy of BLUPs based on models with multiplicative terms may be better than that of least-squares estimators based on the same models and also better than BLUPs based on ANOVA models. Received: 18 October 1997 / Accepted: 31 March 1998     

Genomic best linear unbiased prediction method including imprinting effects for genomic evaluation

Background

Genomic best linear unbiased prediction (GBLUP) is a statistical method used to predict breeding values using single nucleotide polymorphisms for selection in animal and plant breeding. Genetic effects are often modeled as additively acting marker allele effects. However, the actual mode of biological action can differ from this assumption. Many livestock traits exhibit genomic imprinting, which may substantially contribute to the total genetic variation of quantitative traits. Here, we present two statistical models of GBLUP including imprinting effects (GBLUP-I) on the basis of genotypic values (GBLUP-I1) and gametic values (GBLUP-I2). The performance of these models for the estimation of variance components and prediction of genetic values across a range of genetic variations was evaluated in simulations.

Results

Estimates of total genetic variances and residual variances with GBLUP-I1 and GBLUP-I2 were close to the true values and the regression coefficients of total genetic values on their estimates were close to 1. Accuracies of estimated total genetic values in both GBLUP-I methods increased with increasing degree of imprinting and broad-sense heritability. When the imprinting variances were equal to 1.4% to 6.0% of the phenotypic variances, the accuracies of estimated total genetic values with GBLUP-I1 exceeded those with GBLUP by 1.4% to 7.8%. In comparison with GBLUP-I1, the superiority of GBLUP-I2 over GBLUP depended strongly on degree of imprinting and difference in genetic values between paternal and maternal alleles. When paternal and maternal alleles were predicted (phasing accuracy was equal to 0.979), accuracies of the estimated total genetic values in GBLUP-I1 and GBLUP-I2 were 1.7% and 1.2% lower than when paternal and maternal alleles were known.

Conclusions

This simulation study shows that GBLUP-I1 and GBLUP-I2 can accurately estimate total genetic variance and perform well for the prediction of total genetic values. GBLUP-I1 is preferred for genomic evaluation, while GBLUP-I2 is preferred when the imprinting effects are large, and the genetic effects differ substantially between sexes.     

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.     

Prediction of additive and dominance effects in selected or unselected populations with inbreeding

Summary A genetic model with either 64 or 1,600 unlinked biallelic loci and complete dominance was used to study prediction of additive and dominance effects in selected or unselected populations with inbreeding. For each locus the initial frequency of the favourable allele was 0.2, 0.5, or 0.8 in different alternatives, while the initial narrow-sense heritability was fixed at 0.30. A population of size 40 (20 males and 20 females) was simulated 1,000 times for five generations. In each generation 5 males and 10 or 20 females were mated, with each mating producing four or two offspring, respectively. Breeding individuals were selected randomly, on own phenotypic performance or such yielding increased inbreeding levels in subsequent generations. A statistical model containing individual additive and dominance effects but ignoring changes in mean and genetic covariances associated with dominance due to inbreeding resulted in significantly biased predictions of both effects in generations with inbreeding. Bias, assessed as the average difference between predicted and simulated genetic effects in each generation, increased almost linearly with the inbreeding coefficient. In a second statistical model the average effect of inbreeding on the mean was accounted for by a regression of phenotypic value on the inbreeding coefficient. The total dominance effect of an individual in that case was the sum of the average effect of inbreeding and an individual effect of dominance. Despite a high mean inbreeding coefficient (up to 0.35), predictions of additive and dominance effects obtained with this model were empirically unbiased for each initial frequency in the absence of selection and 64 unlinked loci. With phenotypic selection of 5 males and only 10 females in each generation and 64 loci, however, predictions of additive and dominance effects were significantly biased. Observed biases disappeared with 1,600 loci for allelic frequencies at 0.2 and 0.5. Bias was due to a considerable change in allelic frequency with phenotypic selection. Ignoring both the covariance between additive and dominance effects with inbreeding and the change in dominance variance due to inbreeding did not significantly bias prediction of additive and dominance effects in selected or unselected populations with inbreeding.     

Predicting genetic values: a kernel-based best linear unbiased prediction with genomic data

Genomic data provide a valuable source of information for modeling covariance structures, allowing a more accurate prediction of total genetic values (GVs). We apply the kriging concept, originally developed in the geostatistical context for predictions in the low-dimensional space, to the high-dimensional space spanned by genomic single nucleotide polymorphism (SNP) vectors and study its properties in different gene-action scenarios. Two different kriging methods ["universal kriging" (UK) and "simple kriging" (SK)] are presented. As a novelty, we suggest use of the family of Matérn covariance functions to model the covariance structure of SNP vectors. A genomic best linear unbiased prediction (GBLUP) is applied as a reference method. The three approaches are compared in a whole-genome simulation study considering additive, additive-dominance, and epistatic gene-action models. Predictive performance is measured in terms of correlation between true and predicted GVs and average true GVs of the individuals ranked best by prediction. We show that UK outperforms GBLUP in the presence of dominance and epistatic effects. In a limiting case, it is shown that the genomic covariance structure proposed by VanRaden (2008) can be considered as a covariance function with corresponding quadratic variogram. We also prove theoretically that if a specific linear relationship exists between covariance matrices for two linear mixed models, the GVs resulting from BLUP are linked by a scaling factor. Finally, the relation of kriging to other models is discussed and further options for modeling the covariance structure, which might be more appropriate in the genomic context, are suggested.     

Predicting rates of inbreeding in populations undergoing selection

Tractable forms of predicting rates of inbreeding (DeltaF) in selected populations with general indices, nonrandom mating, and overlapping generations were developed, with the principal results assuming a period of equilibrium in the selection process. An existing theorem concerning the relationship between squared long-term genetic contributions and rates of inbreeding was extended to nonrandom mating and to overlapping generations. DeltaF was shown to be approximately (1)/(4)(1 - omega) times the expected sum of squared lifetime contributions, where omega is the deviation from Hardy-Weinberg proportions. This relationship cannot be used for prediction since it is based upon observed quantities. Therefore, the relationship was further developed to express DeltaF in terms of expected long-term contributions that are conditional on a set of selective advantages that relate the selection processes in two consecutive generations and are predictable quantities. With random mating, if selected family sizes are assumed to be independent Poisson variables then the expected long-term contribution could be substituted for the observed, providing (1)/(4) (since omega = 0) was increased to (1)/(2). Established theory was used to provide a correction term to account for deviations from the Poisson assumptions. The equations were successfully applied, using simple linear models, to the problem of predicting DeltaF with sib indices in discrete generations since previously published solutions had proved complex.     

Best linear unbiased prediction and optimum allocation of test resources in maize breeding with doubled haploids

With best linear unbiased prediction (BLUP), information from genetically related candidates is combined to obtain more precise estimates of genotypic values of test candidates and thereby increase progress from selection. We developed and applied theory and Monte Carlo simulations implementing BLUP in 2 two-stage maize breeding schemes and various selection strategies. Our objectives were to (1) derive analytical solutions of the mixed model equations under two breeding schemes, (2) determine the optimum allocation of test resources with BLUP under different assumptions regarding the variance component ratios for grain yield in maize, (3) compare the progress from selection using BLUP and conventional phenotypic selection based on mean performance solely of the candidates, and (4) analyze the potential of BLUP for further improving the progress from selection. The breeding schemes involved selection for testcross performance either of DH lines at both stages (DHTC) or of S₁ families at the first stage and DH lines at the second stage (S₁TC-DHTC). Our analytical solutions allowed much faster calculations of the optimum allocations and superseded matrix inversions to solve the mixed model equations. Compared to conventional phenotypic selection, the progress from selection was slightly higher with BLUP for both optimization criteria, namely the selection gain and the probability to select the best genotypes. The optimum allocation of test resources in S₁TC-DHTC involved ≥10 test locations at both stages, a low number of crosses (≤6) each with 100–300 S₁ families at the first stage, and 500–1,000 DH lines at the second stage. In breeding scheme DHTC, the optimum number of test candidates at the first stage was 5–10 times larger, whereas the number of test locations at the first stage and the number of test candidates at the second stage were strongly reduced compared to S₁TC-DHTC.     

A comparison of two methods for prediction of response and rates of inbreeding in selected populations with the results obtained in two selection experiments

Selection programmes are mainly concerned with increasing genetic gain. However, short-term progress should not be obtained at the expense of the within-population genetic variability. Different prediction models for the evolution within a small population of the genetic mean of a selected trait, its genetic variance and its inbreeding have been developed but have mainly been validated through Monte Carlo simulation studies. The purpose of this study was to compare theoretical predictions to experimental results. Two deterministic methods were considered, both grounded on a polygenic additive model. Differences between theoretical predictions and experimental results arise from differences between the true and the assumed genetic model, and from mathematical simplifications applied in the prediction methods. Two sets of experimental lines of chickens were used in this study: the Dutch lines undergoing true truncation mass selection, the other lines (French) undergoing mass selection with a restriction on the representation of the different families. This study confirmed, on an experimental basis, that modelling is an efficient approach to make useful predictions of the evolution of selected populations although the basic assumptions considered in the models (polygenic additive model, normality of the distribution, base population at the equilibrium, etc.) are not met in reality. The two deterministic methods compared yielded results that were close to those observed in real data, especially when the selection scheme followed the rules of strict mass selection: for instance, both predictions overestimated the genetic gain in the French experiment, whereas both predictions were close to the observed values in the Dutch experiment.