Implementation issues for Markov Chain Monte Carlo methods in random regression test-day models

Markov Chain Monte Carlo methods made possible estimation of parameters for complex random regression test‐day models. Models evolved from single‐trait with one set of random regressions to multiple‐trait applications with several random effects described by regressions. Gibbs sampling has been used for models with linear (with respect to coefficients) regressions and normality assumptions for random effects. Difficulties associated with implementations of Markov Chain Monte Carlo schemes include lack of good practical methods to assess convergence, slow mixing caused by high posterior correlations of parameters and long running time to generate enough posterior samples. Those problems are illustrated through comparison of Gibbs sampling schemes for single‐trait random regression test‐day models with different model parameterizations, different functions used for regressions and posterior chains of different sizes. Orthogonal polynomials showed better convergence and mixing properties in comparison with ‘lactation curve’ functions of the same number of parameters. Increasing the order of polynomials resulted in smaller number of independent samples for covariance components. Gibbs sampling under hierarchical model parameterization had a lower level of autocorrelation and required less time for computation. Posterior means and standard deviations of genetic parameters were very similar for chains of different size (from 20 000 to 1 000 000) after convergence. Single‐trait random regression models with large data sets can be analysed by Markov Chain Monte Carlo methods in relatively short time. Multiple‐trait (lactation) models are computationally more demanding and better algorithms are required.     

A comparison of restricted selection procedures to control genetic gains

Using Monte Carlo simulation, two schemes of restricted selection were compared under various combinations of genetic parameters and constraints on the genetic gains. The first selection scheme is the combination of best linear unbiased prediction (BLUP) evaluation and linear programming technique (BLUP + LP), and the second one is based on the restricted BLUP selection (RBLUP). Selection for two traits was supposed, in which animals were selected to maximize the genetic gain in trait 2 (Δg₂) under a proportional restriction on the genetic gain in trait 1 (Δg₁) to satisfy the intended ratio (Δg₁:Δg₂). Little differences were found between the two selection schemes with respect to the genetic gains averaged over replicates. However, in all the cases studied, the variance of genetic gains among replicates under BLUP + LP selection was smaller and less sensitive to the genetic parameters and the intended restriction than RBLUP selection. Under the situations of antagonistic selection, the difference tended to be larger. When the heritabilities of the two traits were different, RBLUP selection remarkably increased the variance of genetic gain in a trait with a higher heritability. These results suggest that BLUP + LP selection should always be preferable to RBLUP selection because of the smaller risk of selection. This choice is especially important for the situation where the genetic parameters act as limiting factors for the achievement of intended genetic gains.     

性别比例和性状遗传力对闭锁群体BLUP选择效果的影响

采用MonteCarlo方法模拟研究了性别比例和性状遗传力对闭锁群体动物模型BLUP选择效果的影响 ,选育过程中世代不重叠 ,共进行了 1 5个世代的选择。结果表明性别比例对群体育种值和近交系数的变化都有明显的影响。在育种值达到最大值以前 ,群体平均育种值提高的速度随着公畜比例的增加而有所减慢 ,但会使育种值达到最大值的时间后移 ,在育种值达到最大值后 ,其下降的速度则随着公畜比例的降低而加快。随着公畜比例的增加 ,群体近交系数的上升速度会明显变慢。高遗传力性状的选择效果要优于低遗传力性状     

Comparison of genetic improvement for litter size at birth by direct and indirect selection in swine herd

Responses to selection for number of piglets born alive (NBA) by the total number of piglets born (TNB), the NBA, and the NBA plus number of piglets born dead (NBD) were compared using the accuracy of selection and expected genetic gain calculated from the selection index with family information and the real response to selection, using data generated by Monte Carlo computer simulation. The accuracy of selection for NBA selected by TNB was higher than that by NBA only if the genetic correlation between TNB and NBA was close to 1.0, or the value of heritability for the TNB was much larger than that for the NBA. The accuracy of selection for the NBA selected by the combination of the TNB and the NBA was generally highest in the three selection methods in each family structure. Selection by the TNB resulted in the greatest expected genetic gain for the TNB among the selection methods. In the best linear unbiased prediction (BLUP) selection, the genetic gain for the NBA accumulated by the NBA tended to be similar to that accumulated by the combination of the NBA and the NBD, and both genetic gains at generation 10 were significantly larger than that by the TNB (P < 0.001). The accumulated responses selected by the two‐trait animal model BLUP estimated from genetic parameters with errors were similar to those estimated from the true parameters, and there was no significant difference between them. These results indicate that selection by the NBA or by the NBA and the NBD gives more genetic improvement in the NBA than that by the TNB.     

Optimal selection method for establishing an inbred strain of laboratory animals with high performance for litter size at weaning

The optimal selection method was investigated for establishing an inbred strain of laboratory animals with high performance for litter size at weaning (LSW). A Monte Carlo computer simulation was used to assess the effects of our selection methods on the genetic change of LSW under the continuous use of full‐sib mating for 20 generations. Smaller number of growing animals of each sex per litter and genetic evaluation for selection using a BLUP animal model increased LSW. Use of information on another trait genetically related to LSW, larger population size, and greater number of generations for random selection before starting full‐sib mating were useful for establishing an inbred strain of laboratory animals with high performance for LSW. It was concluded that LSW can be increased by directional selection when establishing inbred strains.     

Selective genotyping with a main trait and a correlated trait

In some instances of quantitative trait loci (QTL) mapping, a correlated trait may be available for which measurement is less costly or more convenient than direct measurement of the trait of main interest. In this paper we consider ways of combining phenotyping for a main trait and a correlated trait to maximize power within the overall framework of a selective genotyping design. Four schemes are compared: (1) Selection for the main trait only (2) Selection for the correlated trait only (3) Two-stage selection (4) Three-stage selection. Optimum proportion selected for the two traits were obtained for the various schemes, and power parameters compared, according to the correlation r, between the two traits, and the proportion of the population, Q, phenotyped for the main trait. The schemes were compared under two sets of assumptions: (1) Total population size and proportion that can be phenotyped for the main trait are fixed, and (2) Total costs are fixed, with relative cost, c', of rearing and phenotyping for the correlated trait as compared to the main trait. Under (1), the multi-stage schemes are very advantageous when r is high, and Q is low. Under (2), the single-stage schemes were always superior to the multi-stage schemes; Scheme 1 being preferred when r²/c' < 1.0, and Scheme 2 when r²/c' > 1.0.     

Efficiency of selection strategies for halothane-negative gene

Use of a method to estimate the frequency of the halothane-negative allele in boars is illustrated for different sampling schemes for boar testing programs and for testing within closed breeding populations. This method uses information not only on the individual, but also on all mates and relatives including parents, siblings and offspring. Accuracy of the estimates of the allelic frequency in boars was measured through use of Monte Carlo simulation. The selection differential in real frequency of the halothane-negative allele when boars were selected on estimated allelic frequency was used as the criterion for accuracy. In the progeny testing situation, phenotypes of base boars and one generation of offspring were available. The average selection differentials with 90% selection (i.e., culling 10% of boars on estimated allelic frequency) when 2 and 10 litters of two boars each were tested were .017 and .044 in base boars and .013 and .025 in the offspring. The value of the boar's own phenotype was small. Higher selection differentials were found in the closed herd situation, where data on two generations were available. The selection differential in base boars when 10 litters were tested increased from .046 to .066 when the proportion of boars selected decreased from 90% to 50%. No improvement in selection differential with proportion selected was found in the progeny testing situation. Intense selection is most effective when the number of litters per boar is large and data over several generations are used. The estimation procedure for allelic frequencies in boars should improve current screening and selection programs to reduce halothane sensitivity in pigs.     

The value of prior information for detection of QTL affecting longitudinal traits: an example using Von Bertalanffy growth function

A Bayesian procedure is presented for detecting quantitative trait loci (QTL) affecting longitudinal traits. The statistical model assumes a QTL affecting the prior distribution of the parameters of a given production function, under a hierarchical Bayesian scheme. Marginal posterior distributions for the effects associated with the QTL are calculated using Markov chain Monte Carlo methods. Furthermore, the Bayesian analysis allows the use of some available relevant information that can improve the detection of the QTL substantially. To illustrate the procedure, an example of QTL detection using the Von Bertalanffy growth function is presented with a F₂ pig population bred from Iberian boars and Landrace sows. Animals of the F₂ population were genotyped for seven markers in chromosome 2 (SSC2). Two prior distributions for the mean effect of the parameters related with birth and adult weight were compared. On the one hand, vague prior distributions were used, and, on the other, there were assumed univariate Gaussian distributions that ensure biologically meaningful adult and birth weights on the posterior growth curves. Results from the second prior distribution supported the presence of QTL, by showing that individuals with both alleles of Iberian origin had lower rates of maturation. On the contrary, when vague priors were used, the procedure was not able to detect QTL.     

Hamiltonian Monte Carlo method for estimating variance components

A Hamiltonian Monte Carlo algorithm is a Markov chain Monte Carlo method, and the method has a potential to improve estimating parameters effectively. Hamiltonian Monte Carlo is based on Hamiltonian dynamics, and it follows Hamilton's equations, which are expressed as two differential equations. In the sampling process of Hamiltonian Monte Carlo, a numerical integration method called leapfrog integration is used to approximately solve Hamilton's equations, and the integration is required to set the number of discrete time steps and the integration stepsize. These two parameters require some amount of tuning and calibration for effective sampling. In this study, we applied the Hamiltonian Monte Carlo method to animal breeding data and identified the optimal tunings of leapfrog integration for normal and inverse chi-square distributions. Then, using real pig data, we revealed the properties of the Hamiltonian Monte Carlo method with the optimal tuning by applying models including variance explained by pedigree information or genomic information. Compared with the Gibbs sampling method, the Hamiltonian Monte Carlo method had superior performance in both models. We have provided the source codes of this method written in the Fortran language at https://github.com/A-ARAKAWA/HMC .     

Plenary contribution to International Conference on Boar Semen Preservation 2011. Genetic selection for freezability and its controversy with selection for performance

Little data are available in the literature regarding freezability of boar sperm or its relationship with other traits. Existing data suggest the trait would respond favourably to selection, and information is available from other species suggesting components that might have changed. Genetic parameters are estimated for boar sperm freezability including heritability and correlations with other production traits. Sperm freezability is an ideal candidate for marker assisted-selection or selection for favourable alleles.     

Bayesian reanalysis of a quantitative trait locus accounting for multiple environments by scaling in broilers

A Bayesian method was developed to handle QTL analyses of multiple experimental data of outbred populations with heterogeneity of variance between sexes for all random effects. The method employed a scaled reduced animal model with random polygenic and QTL allelic effects. A parsimonious model specification was applied by choosing assumptions regarding the covariance structure to limit the number of parameters to estimate. Markov chain Monte Carlo algorithms were applied to obtain marginal posterior densities. Simulation demonstrated that joint analysis of multiple environments is more powerful than separate single trait analyses of each environment. Measurements on broiler BW obtained from 2 experiments concerning growth efficiency and carcass traits were used to illustrate the method. The population consisted of 10 full-sib families from a cross between 2 broiler lines. Microsatellite genotypes were determined on generations 1 and 2, and phenotypes were collected on groups of generation 3 animals. The model included a polygenic correlation, which had a posterior mean of 0.70 in the analyses. The reanalysis agreed on the presence of a QTL in marker bracket MCW0058-LEI0071 accounting for 34% of the genetic variation in males and 24% in females in the growth efficiency experiment. In the carcass experiment, this QTL accounted for 19% of the genetic variation in males and 6% in females.     

Correlated responses to selection for large body size in oMt1a-oGH mice: growth, feed efficiency and body composition

Correlated responses were determined for growth, feed consumption, feed efficiency and body composition following short‐term selection for large 8‐week body weight in transgenic and nontransgenic mice. Replicate lines which either carried or did not carry the sheep metallothionein 1a‐sheep growth hormone transgene (oMt1a‐oGH) were established. The lines carrying the transgene at an initial frequency of 0.5 came from a high‐growth (TM) and a randomly selected (TC) background. The respective nontransgenic lines were identified as NM and NC. Control replicates (CC) came from the randomly selected background. During the selection process the transgene was activated by adding 25 mm ZnSO₄ to the drinking water of all mice. Correlated responses were measured with (Z) and without (C) the addition of zinc. After seven and eight generations of selection, the frequency of transgenic mice in line TM had fallen sharply, whereas transgene frequency had risen moderately in TC. The reduced frequency of oMt1a‐oGH in the high‐growth genetic background may have been caused by a lower additive effect compared with the randomly selected background combined with a fitness disadvantage of the transgene. Therefore, the utility of a transgene in improving a quantitative trait may depend in part on genetic background. Correlated responses for most traits in NC were similar for Z and C. In contrast, correlated responses in TC showed marked differences in C compared with Z. For example, daily gain and feed efficiency showed no significantly correlated responses under C and positive responses (p < 0.001) under Z, and the reverse was found for indicators of body fat percentage. These line by environment (Z versus C) interactions may represent a genetic correlation of less than one between a trait expressed in two distinct environments. Thus, in developing lines with a transgene that can be regulated, a critical question is whether selection for quantitative trait(s) should be conducted when the transgene is activated or not activated.     

"Quasi-REML" correlation estimates between production and health traits in the presence of selection and confounding: a simulation study

Performance of the "quasi-REML" method for estimating correlations between a continuous trait and a categorical trait, and between two categorical traits, was studied with Monte Carlo simulations. Three continuous, correlated traits were simulated for identical populations and three scenarios with either no selection, selection for one moderately heritable trait (Trait 1, h2 = .25), and selection for the same trait plus confounding between sires and management groups. The "true" environmental correlations between Traits 2 (h2 = .10) and 3 (h2 = .05) were always of the same absolute size (.20), but further data scenarios were generated by setting the sign of environmental correlation to either positive or negative. Observations for Traits 2 and 3 were then reassigned to binomial categories to simulate health or reproductive traits with incidences of 15 and 5%, respectively. Genetic correlations (r(g12), r(g13), and r(g23) and environmental correlations (r(e12), r(e13), and r(e23)) were estimated for the underlying continuous scale (REML) and the visible categorical scales ("quasi-REML") with linear multiple-trait sire and animal models. Contrary to theory, practically all "quasi-REML" genetic correlations were underestimated to some extent with the sire and animal models. Selection inflated this negative bias for sire model estimates, and the sign of r(e23) noticeably affected r(g23) estimates for the animal model, with greater bias and SD for estimates when the "true" r(e23) was positive. Transformed "quasi-REML" environmental correlations between a continuous and a categorical trait were estimated with good efficiency and little bias, and corresponding correlations between two categorical traits were systematically overestimated. Confounding between sires and contemporary groups negatively affected all correlation estimates on the underlying and the visible scales, especially for sire model "quasi-REML" estimates of genetic correlation. Selection, data structure, and the (co)variance structure influences how well correlations involving categorical traits are estimated with "quasi-REML" methods.     

Comparison of two models to estimate genetic parameters for number of born alive in pigs

The performance of the two‐trait animal model that regards the first parity and later parities as two different traits in estimating genetic parameters for number of born alive (NBA) was examined using real and simulated data. Genetic parameters for NBA were estimated in purebred Landrace and Large White pigs using a single‐trait repeatability model (Model 1) that regards all parities as the same trait and a two‐trait animal model (Model 2) that regards the first and the later parities as different traits. For Model 2, the permanent environmental effect was fitted to only the records of the later parities. Heritability for NBA estimated using Model 1 was 0.12 for Landrace and 0.11 for Large White. Estimated heritability for NBA of the first parity and the later parities was 0.21 and 0.16, respectively, for Landrace; 0.18 and 0.16, respectively, for Large White obtained using Model 2, and higher than those in both breeds obtained using Model 1. Further results based on data simulated using the Monte Carlo method suggest that estimated additive genetic variance could be more biased using Model 2 than Model 1.     

A multiple-phenotype imputation procedure as a method for prediction of cheese-making efficiency in Spanish Assaf sheep

Sheep milk is mainly intended to manufacture a wide variety of high-quality cheeses. The ovine cheese industry would benefit from an improvement, through genetic selection, of traits related to the milk coagulation properties (MCPs) and cheese yield-related traits, broadly denoted as “cheese-making traits.” Considering that routine measurements of these traits needed for genetic selection are expensive and time-consuming, this study aimed to evaluate the accuracy of a cheese-making phenotype imputation method based on the information from official milk control records combined with the pH of the milk. For this study, we analyzed records of milk production traits, milk composition traits, and measurements of cheese-making traits available from a total of 1,145 dairy ewes of the Spanish Assaf sheep breed. Cheese-making traits included five related to the MCPs and two cheese yield-related traits. The milk and cheese-making phenotypes were adjusted for significant effects based on a general linear model. The adjusted phenotypes were used to define a multiple-phenotype imputation procedure for the cheese-making traits based on multivariate normality and Markov chain Monte Carlo sampling. Five of the seven cheese-making traits considered in this study achieved a prediction accuracy of 0.60 computed as the correlation between the adjusted phenotypes and the imputed phenotypes. Particularly the logarithm of curd-firming time since rennet addition (logK₂₀) (0.68), which has been previously suggested as a potential candidate trait to improve the cheese ability in this breed, and the logarithm of the ratio between the rennet clotting time and the curd firmness at 60 min (logRCT/A60) (0.65), which has been defined by other studies as an indicator trait of milk coagulation efficiency. This study represents a first step toward the possible use of the phenotype imputation of cheese-making traits to develop a practical methodology for the dairy sheep industry to impute cheese-making traits only based on the analysis of a milk sample without the need of pedigree information. This information could be also used in future planning of specific breeding programs considering the importance of the cheese-making efficiency in dairy sheep and highlights the potential of phenotype imputation to leverage sample size on expensive, hard-to-measure phenotypes.     

Genetic analysis of 'breeding field test status' in Icelandic horses

Genetic evaluation of Icelandic horses is currently based on results from breeding field tests where riding ability and conformation of the horses are evaluated over the course of 1-2 days. Only a small part of registered horses attend these field tests, and it can be assumed that these are not a random sample of the population. In this study, the trait test status was introduced, describing whether a horse was assessed in a breeding field test. This trait was analysed to find out whether it has a genetic variation and how it correlates genetically to other traits in the breeding goal. Breeding field test data included 39,443 mares born in Iceland in 1990-2001, of which 7431 were assessed in the period 1994-2007. The trait was defined in relation to age, gender and stud of horses. Variance and covariance components were estimated using the Markov Chain Monte Carlo method by applying the Gibbs sampler procedure in the DMU program. Three multivariate analyses were performed where the test status trait was analysed with breeding field test traits. Animal models and sire models were applied. Based on estimated heritabilities (0.51-0.67) and genetic correlations (0.00-0.87), the test status trait showed significant genetic variation and was strongly correlated to some traits. The test status trait reflects preselection in the breeding field test traits and should be included in the genetic evaluation to enhance the procedure, reduce selection bias and increase accuracy of the estimation.     

Analysis of longitudinal data of beef cattle raised on pasture from northern Brazil using nonlinear models

This study was undertaken to aim of estimating the genetic parameters and trends for asymptotic weight (A) and maturity rate (k) of Nellore cattle from northern Brazil. The data set was made available by the Brazilian Association of Zebu Breeders and collected between the years of 1997 and 2007. The Von Bertalanffy, Brody, Gompertz, and logistic nonlinear models were fitted by the Gauss?CNewton method to weight?Cage data of 45,895 animals collected quarterly of the birth to 750?days old. The curve parameters were analyzed using the procedures GLM and CORR. The estimation of (co)variance components and genetic parameters was obtained using the MTDFREML software. The estimated heritability coefficients were 0.21?±?0.013 and 0.25?±?0.014 for asymptotic weight and maturity rate, respectively. This indicates that selection for any trait shall results in genetic progress in the herd. The genetic correlation between A and k was negative (?0.57?±?0.03) and indicated that animals selected for high maturity rate shall result in low asymptotic weight. The Von Bertalanffy function is adequate to establish the mean growth patterns and to predict the adult weight of Nellore cattle. This model is more accurate in predicting the birth weight of these animals and has better overall fit. The prediction of adult weight using nonlinear functions can be accurate when growth curve parameters and their (co)variance components are estimated jointly. The model used in this study can be applied to the prediction of mature weight in herds where a portion of the animals are culled before they reach the adult age.     

Testing genetic determinism in rate of hoof growth in pigs using Bayes Factors

Excessive growth and consequent deformity of hooves is a frequent disorder in some purebred pig populations. A test to detect possible genetic determinism related with this phenomenon was performed using the Bayes Factor (BF). Data were available for females from three purebred selection lines: Landrace (561 records), Pietrain (183) and Large White (225). Animals were scored in four categories, according to the overall growth rate of their hooves. A Bayesian analysis was performed separately for each line using a threshold model with a probit approach, and Bayes Factors between models with and without additive genetic effects were computed. Results from the three lines showed that models exhibiting genetic variability were much more probable than those that did not include a genetic component, with BF values of 312, 35 and 40 (and posterior probabilities of 0.99, 0.97 and 0.98), respectively, for the Landrace, Pietrain and Large White lines. Monte Carlo estimates of posterior means of heritabilities were medium to high (0.25, 0.41 and 0.38, respectively), and the highest posterior density region for heritability at 99% did not include zero in any of the three lines. These results allow us to conclude that genetic determinism has an important influence upon the rate of hoof growth in the pig. A potential genetic response can be achieved in the populations analysed, but further studies are needed to determine the genetic architecture of hoof growth disorders in pigs.     

A review of theoretical aspects in the estimation of breeding values for multi-trait selection

The natural links between best linear unbiased prediction and selection indices are reviewed. These links allow a simple interpretation of the estimation equations and allow results from selection index theory to be used in investigating the accuracy of predictors. The gain in accuracy in prediction for one trait from using other correlated traits is partitioned into a direct gain from measuring other traits and a gain because fixed effects are estimated more precisely. There will be a loss of accuracy in predictors and loss of selection response when variance parameters are not known exactly. A regression argument is used to explain this phenomenon and formulae to quantify the loss of selection response are interpreted using canonical variates. Often predictors and variance parameters are derived from selected populations. The relevance of procedures developed for unselected populations is discussed and unsolved problems indicated.