Investigation of Gibbs sampling conditions to estimate variance components from Japanese Black carcass field data

The genetic evaluation using the carcass field data in Japanese Black cattle has been carried out employing an animal model, implementing the restricted maximum likelihood (REML) estimation of additive genetic and residual variances. Because of rapidly increasing volumes of the official data sets and therefore larger memory spaces required, an alternative approach like the REML estimation could be useful. The purpose of this study was to investigate Gibbs sampling conditions for the single-trait variance component estimation using the carcass field data. As prior distributions, uniform and normal distributions and independent scaled inverted chi-square distributions were used for macro-environmental effects, breeding values, and the variance components, respectively. Using the data sets of different sizes, the influences of Gibbs chain length and thinning interval were investigated, after the burn-in period was determined using the coupling method. As would be expected, the chain lengths had obviously larger effects on the posterior means than those of thinning intervals. The posterior means calculated using every 10th sample from 90 000 of samples after 10 000 samples discarded as burn-in period were all considered to be reasonably comparable to the corresponding estimates by REML.     

Comparison of methods for handling censored records in beef fertility data: simulation study

A simulation study was conducted to compare methods for handling censored records for days to calving in beef cattle data. Days to calving was defined as the time, in days, between when a bull is turned out in the pasture and the subsequent parturition. Simulated data were generated to have data structure and genetic relationships similar to an available field data set. Records were simulated for 33,176 daughters of 4,238 sires. Data were simulated using a mixed linear model that included the fixed effects of contemporary group and sex of calf, linear and quadratic covariates for age at mating, and random effects of animal and residual error. Two methods for handling censored records were evaluated, and two censoring rates of 12 and 20% were applied to assess the influence of higher censoring rates on inferences. Censored records were assigned penalty values on a within-contemporary group basis under the first method (DCPEN). Under the second method (DCSIM), censored records were drawn from their respective predictive distributions. A Bayesian approach via Gibbs sampling was used to estimate variance components and predict breeding values. Posterior means (PM) and standard deviations (SD) of additive genetic variance for DCPEN at 12 and 20% censoring were 23.2 (3.7) and 21.0 (3.6), respectively, whereas the same estimates for DCSIM at 12 and 20% censoring were 23.7(3.3) and 21.9 (3.4), respectively. In all cases, the true value of the genetic variance was within the high posterior density (HPD) interval (95%). The PM (SD) of residual variance for DCPEN at 12 and 20% censoring were 415.7 (4.7) and 440.0 (4.8) respectively, whereas the same estimates for DCSIM at 12 and 20% censoring were 371.0 (4.3) and 365.4 (4.4), respectively. The true value of the residual variance was within the HPD (95%) for DCSIM, but it was outside this interval for DCPEN at both censoring rates, indicating a systematic bias for this parameter. Bayes Factor and Deviance Information Criteria were used for model comparisons, and both criteria indicated the superiority of the DCSIM method. However, little difference was observed between the two methods for correlations between true breeding values and posterior means of animal effects for sires, indicating that no major reranking of sires would be expected. This finding suggests that either censored data handling technique can be successfully used in a genetic evaluation for days to calving.     

The empirical bias of estimates by restricted maximum likelihood, Bayesian method, and method R under selection for additive, maternal, and dominance models.

Bayesian analysis via Gibbs sampling, restricted maximum likelihood (REML), and Method R were used to estimate variance components for several models of simulated data. Four simulated data sets that included direct genetic effects and different combinations of maternal, permanent environmental, and dominance effects were used. Parents were selected randomly, on phenotype across or within contemporary groups, or on BLUP of genetic value. Estimates by Bayesian analysis and REML were always empirically unbiased in large data sets. Estimates by Method R were biased only with phenotypic selection across contemporary groups; estimates of the additive variance were biased upward, and all the other estimates were biased downward. No empirical bias was observed for Method R under selection within contemporary groups or in data without contemporary group effects. The bias of Method R estimates in small data sets was evaluated using a simple direct additive model. Method R gave biased estimates in small data sets in all types of selection except BLUP. In populations where the selection is based on BLUP of genetic value or where phenotypic selection is practiced mostly within contemporary groups, estimates by Method R are likely to be unbiased. In this case, Method R is an alternative to single-trait REML and Bayesian analysis for analyses of large data sets when the other methods are too expensive to apply.     

Impact of variation in length of individual testing periods on estimation of (co)variance components of a random regression model for feed intake of growing pigs

A simulation study was conducted to assess the influence of differences in the length of individual testing periods on estimates of (co)variance components of a random regression model for daily feed intake of growing pigs performance tested between 30 and 100 kg live weight. A quadratic polynomial in days on test with fixed regressions for sex, random regressions for additive genetic and permanent environmental effects and a constant residual variance was used for a bivariate simulation of feed intake and daily gain. (Co)variance components were estimated for feed intake only by means of a Bayesian analysis using Gibbs sampling and restricted maximum likelihood (REML). A single trait random regression model analogous to the one used for data simulation was used to analyse two versions of the data: full data sets with 18 weekly means of feed intake per animal and reduced data sets with the individual length of testing periods determined when tested animals reached 100 kg live weight. Only one significant difference between estimates from full and reduced data (REML estimate of genetic covariance between linear and quadratic regression parameters) and two significant differences from expected values (Gibbs estimates of permanent environmental variance of quadratic regression parameters) occurred. These differences are believed to be negligible, as the number lies within the expected range of type I error when testing at the 5% level. The course of test day variances calculated from estimates of additive genetic and permanent environmental covariance matrices also supports the conclusion that no bias in estimates of (co)variance components occurs due to the individual length of testing periods of performance‐tested growing pigs. A lower number of records per tested animal only results in more variation among estimates of (co)variance components from reduced compared with full data sets. Compared with the full data, the effective sample size of Gibbs samples from the reduced data decreased to 18% for residual variance and increased up to five times for other (co)variances. The data structure seems to influence the mixing of Gibbs chains.     

Efficiency of alternative MCMC strategies illustrated using the reaction norm model.

The Markov chain Monte Carlo (MCMC) strategy provides remarkable flexibility for fitting complex hierarchical models. However, when parameters are highly correlated in their posterior distributions and their number is large, a particular MCMC algorithm may perform poorly and the resulting inferences may be affected. The objective of this study was to compare the efficiency (in terms of the asymptotic variance of features of posterior distributions of chosen parameters, and in terms of computing cost) of six MCMC strategies to sample parameters using simulated data generated with a reaction norm model with unknown covariates as an example. The six strategies are single-site Gibbs updates (SG), single-site Gibbs sampler for updating transformed (a priori independent) additive genetic values (TSG), pairwise Gibbs updates (PG), blocked (all location parameters are updated jointly) Gibbs updates (BG), Langevin-Hastings (LH) proposals, and finally Langevin-Hastings proposals for updating transformed additive genetic values (TLH). The ranking of the methods in terms of asymptotic variance is affected by the degree of the correlation structure of the data and by the true values of the parameters, and no method comes out as an overall winner across all parameters. TSG and BG show very good performance in terms of asymptotic variance especially when the posterior correlation between genetic effects is high. In terms of computing cost, TSG performs best except for dispersion parameters in the low correlation scenario where SG was the best strategy. The two LH proposals could not compete with any of the Gibbs sampling algorithms. In this study it was not possible to find an MCMC strategy that performs optimally across the range of target distributions and across all possible values of parameters. However, when the posterior correlation between parameters is high, TSG, BG and even PG show better mixing than SG.     

Comparison of (co)variance component estimates in control populations of red flour beetle (Tribolium castaneum) using restricted maximum likelihood and Gibbs sampling

Mixed model (co)variance component estimates by REML and Gibbs sampling for two traits were compared for base populations and control lines of Red Flour Beetle (Tribolium castaneum). Two base populations (1296 records in the first replication, 1292 in the second) were sampled from laboratory stock. Control lines were derived from corresponding base populations with random selection and mating for 16 generations. The REML estimate of each (co)variance component for both pupa weight and family size was compared with the mean and 95% central interval of the particular (co)variance estimated by Gibbs sampling with three different weights on the given priors: ‘flat’, smallest, and 3.7% degrees of belief. Results from Gibbs sampling showed that flat priors gave a wider and more skewed marginal posterior distribution than the other two weights on priors for all parameters. In contrast, the 3.7% degree of belief on priors provided reasonably narrow and symmetric marginal posterior distributions. Estimation by REML does not have the flexibility of changing the weight on prior information as does the Bayesian analysis implemented by Gibbs sampling. In general, the 95% central intervals from the three different weights on priors in the base populations were similar to those in control lines. Most REML estimates in base populations differed from REML estimates in control lines. Insufficient information from the data, and confounding of random effects contributed to the variability of REML estimates in base populations. Evidence is presented showing that some (co)variance components were estimated with less precision than others. Results also support the hypothesis that REML estimates were equivalent to the joint mode of posterior distribution obtained from a Bayesian analysis with flat priors, but only when there was sufficient information from data, and no confounding among random effects.     

Effects of nonrandom parental selection on estimation of variance components

Bayesian estimation via Gibbs sampling, REML, and Method R were compared for their empirical sampling properties in estimating genetic parameters from data subject to parental selection using an infinitesimal animal model. Models with and without contemporary groups, random or nonrandom parental selection, two levels of heritability, and none or 15% randomly missing pedigree information were considered. Nonrandom parental selection caused similar effects on estimates of variance components from all three methods. When pedigree information was complete, REML and Bayesian estimation were not biased by nonrandom parental selection for models with or without contemporary groups. Method R estimates, however, were strongly biased by nonrandom parental selection when contemporary groups were in the model. The bias was empirically shown to be a consequence of not fully accounting for gametic phase disequilibrium in the subsamples. The joint effects of nonrandom parental selection and missing pedigree information caused estimates from all methods to be highly biased. Missing pedigree information did not cause biased estimates in random mating populations. Method R estimates usually had greater mean square errors than did REML and Bayesian estimates.     

Comparison of methods for handling censored records in beef fertility data: field data

The purpose of this study was to compare methods for handling censored days to calving records in beef cattle data, and verify results of an earlier simulation study. Data were records from natural service matings of 33,176 first-calf females in Australian Angus herds. Three methods for handling censored records were evaluated. Censored records (records on noncalving females) were assigned penalty values on a within-contemporary group basis under the first method (DCPEN). Under the second method (DCSIM), censored records were drawn from their respective predictive truncated normal distributions, whereas censored records were deleted under the third method (DCMISS). Data were analyzed using a mixed linear model that included the fixed effects of contemporary group and sex of calf, linear and quadratic covariates for age at mating, and random effects of animal and residual error. A Bayesian approach via Gibbs sampling was used to estimate variance components and predict breeding values. Posterior means (PM) (SD) of additive genetic variance for DCPEN, DCSIM, and DCMISS were 22.6d2 (4.2d2), 26.1d2 (3.6d2), and 13.5d2 (2.9d2), respectively. The PM (SD) of residual variance for DCPEN, DCSIM, and DCMISS were 431.4d2 (5.0d2), 371.4d2 (4.5d2), and 262.2d2 (3.4d2), respectively. The PM (SD) of heritability for DCPEN, DCSIM, and DCMISS were 0.05 (0.01), 0.07 (0.01), and 0.05 (0.01), respectively. Simulating trait records for noncalving females resulted in similar heritability to the penalty method but lower residual variance. Pearson correlations between posterior means of animal effects for sires with more than 20 daughters with records were 0.99 between DCPEN and DCSIM, 0.77 between DCPEN and DCMISS, and 0.81 between DCSIM and DCMISS. Of the 424 sires ranked in the top 10% and bottom 10% of sires in DCPEN, 91% and 89%, respectively, were also ranked in the top 10% and bottom 10% in DCSIM. Little difference was observed between DCPEN and DCSIM for correlations between posterior means of animal effects for sires, indicating that no major reranking of sires would be expected. This finding suggests little difference between these two censored data handling techniques for use in genetic evaluation of days to calving.     

Maternal animal model with correlation between maternal environmental effects of related dams

A procedure to take into account the nongenetic relationship between maternal effects in adjacent generations is presented. It considers a correlation between maternal environments provided by a dam and its daughters (lambda). The dispersion structure of the maternal animal model was modified to include a correlation matrix (E) that relates the maternal permanent environmental effects. The structures of the E matrix and its inverse (E(-1)) are described. Both matrices are completely defined by the correlation coefficient lambda. An algorithm to compute these matrices from pedigree information was also developed. Furthermore, a Bayesian analysis of this model including the lambda parameter was developed using Gibbs sampling, with Metropolis steps for the nonstandard conditional distributions. With simulated data, the proposed model reduced the bias in all estimates of dispersion parameters when an antagonism between the maternal effects received by a daughter and its future maternal environment existed. This model also provides an estimate of the environmental relationship between the maternal effects of dams and daughters by the lambda parameter. The same Bayesian analysis was also carried out with weaning weight data of the Bruna dels Pirineus breed. The posterior means (standard deviation) of (co)variance ratios were .214 (.081) for direct heritability (h2d), .107 (.033) for maternal heritability (h2m), .047 (.020) for the proportion of variance due to maternal environmental effects (c2m), and -.034 (.043) for the genetic correlation between direct and maternal effects (r(dm)). The posterior mean of lambda parameter was -.190, and 76% of its marginal posterior distribution took negative values. As occurred with simulated data, considering the maternal environmental correlation in the analysis implied higher h2m estimates, lower c2m and h2d estimates, and less negative values for the marginal posterior distribution of r(dm). These results were considered as evidence of the environmental antagonism between maternal effects provided by a dam and its daughters to weaning weight of their progeny in the Bruna dels Pirineus breed.     

Bayesian conjugate analysis using a generalized inverted Wishart distribution accounts for differential uncertainty among the genetic parameters--an application to the maternal animal model

Consider the estimation of genetic (co)variance components from a maternal animal model (MAM) using a conjugated Bayesian approach. Usually, more uncertainty is expected a priori on the value of the maternal additive variance than on the value of the direct additive variance. However, it is not possible to model such differential uncertainty when assuming an inverted Wishart (IW) distribution for the genetic covariance matrix. Instead, consider the use of a generalized inverted Wishart (GIW) distribution. The GIW is essentially an extension of the IW distribution with a larger set of distinct parameters. In this study, the GIW distribution in its full generality is introduced and theoretical results regarding its use as the prior distribution for the genetic covariance matrix of the MAM are derived. In particular, we prove that the conditional conjugacy property holds so that parameter estimation can be accomplished via the Gibbs sampler. A sampling algorithm is also sketched. Furthermore, we describe how to specify the hyperparameters to account for differential prior opinion on the (co)variance components. A recursive strategy to elicit these parameters is then presented and tested using field records and simulated data. The procedure returned accurate estimates and reduced standard errors when compared with non-informative prior settings while improving the convergence rates. In general, faster convergence was always observed when a stronger weight was placed on the prior distributions. However, analyses based on the IW distribution have also produced biased estimates when the prior means were set to over-dispersed values.     

The relative effect of genomic information on efficiency of Bayesian analysis of the mixed linear model with unknown variance

This work focuses on the effects of variable amount of genomic information in the Bayesian estimation of unknown variance components associated with single‐step genomic prediction. We propose a quantitative criterion for the amount of genomic information included in the model and use it to study the relative effect of genomic data on efficiency of sampling from the posterior distribution of parameters of the single‐step model when conducting a Bayesian analysis with estimating unknown variances. The rate of change of estimated variances was dependent on the amount of genomic information involved in the analysis, but did not depend on the Gibbs updating schemes applied for sampling realizations of the posterior distribution. Simulation revealed a gradual deterioration of convergence rates for the locations parameters when new genomic data were gradually added into the analysis. In contrast, the convergence of variance components showed continuous improvement under the same conditions. The sampling efficiency increased proportionally to the amount of genomic information. In addition, an optimal amount of genomic information in variance–covariance matrix that guaranty the most (computationally) efficient analysis was found to correspond a proportion of animals genotyped ***0.8. The proposed criterion yield a characterization of expected performance of the Gibbs sampler if the analysis is subject to adjustment of the amount of genomic data and can be used to guide researchers on how large a proportion of animals should be genotyped in order to attain an efficient analysis.     

Bayesian inference for genetic parameter estimation on growth traits for Nelore cattle in Brazil,using the Gibbs sampler

This data set consisted of over 29 245 field records from 24 herds of registered Nelore cattle born between 1980 and 1993, with calves sires by 657 sires and 12 151 dams. The records were collected in south‐eastern and midwestern Brazil and animals were raised on pasture in a tropical climate. Three growth traits were included in these analyses: 205‐ (W205), 365‐ (W365) and 550‐day (W550) weight. The linear model included fixed effects for contemporary groups (herd‐year‐season‐sex) and age of dam at calving. The model also included random effects for direct genetic, maternal genetic and maternal permanent environmental (MPE) contributions to observations. The analyses were conducted using single‐trait and multiple‐trait animal models. Variance and covariance components were estimated by restricted maximum likelihood (REML) using a derivative‐free algorithm (DFREML) for multiple traits (MTDFREML). Bayesian inference was obtained by a multiple trait Gibbs sampling algorithm (GS) for (co)variance component inference in animal models (MTGSAM). Three different sets of prior distributions for the (co)variance components were used: flat, symmetric, and sharp. The shape parameters (ν) were 0, 5 and 9, respectively. The results suggested that the shape of the prior distributions did not affect the estimates of (co)variance components. From the REML analyses, for all traits, direct heritabilities obtained from single trait analyses were smaller than those obtained from bivariate analyses and by the GS method. Estimates of genetic correlations between direct and maternal effects obtained using REML were positive but very low, indicating that genetic selection programs should consider both components jointly. GS produced similar but slightly higher estimates of genetic parameters than REML, however, the greater robustness of GS makes it the method of choice for many applications.     

Bayesian inference in a piecewise Weibull proportional hazards model with unknown change points

The main difference between parametric and non-parametric survival analyses relies on model flexibility. Parametric models have been suggested as preferable because of their lower programming needs although they generally suffer from a reduced flexibility to fit field data. In this sense, parametric survival functions can be redefined as piecewise survival functions whose slopes change at given points. It substantially increases the flexibility of the parametric survival model. Unfortunately, we lack accurate methods to establish a required number of change points and their position within the time space. In this study, a Weibull survival model with a piecewise baseline hazard function was developed, with change points included as unknown parameters in the model. Concretely, a Weibull log-normal animal frailty model was assumed, and it was solved with a Bayesian approach. The required fully conditional posterior distributions were derived. During the sampling process, all the parameters in the model were updated using a Metropolis-Hastings step, with the exception of the genetic variance that was updated with a standard Gibbs sampler. This methodology was tested with simulated data sets, each one analysed through several models with different number of change points. The models were compared with the Deviance Information Criterion, with appealing results. Simulation results showed that the estimated marginal posterior distributions covered well and placed high density to the true parameter values used in the simulation data. Moreover, results showed that the piecewise baseline hazard function could appropriately fit survival data, as well as other smooth distributions, with a reduced number of change points.     

Analysis of beef cattle longitudinal data applying a nonlinear model

The objective of this work was to evaluate the Nelore beef cattle, growth curve parameters using the Von Bertalanffy function in a nested Bayesian procedure that allowed estimation of the joint posterior distribution of growth curve parameters, their (co)variance components, and the environmental and additive genetic components affecting them. A hierarchical model was applied; each individual had a growth trajectory described by the nonlinear function, and each parameter of this function was considered to be affected by genetic and environmental effects that were described by an animal model. Random samples of the posterior distributions were drawn using Gibbs sampling and Metropolis-Hastings algorithms. The data set consisted of a total of 145,961 BW recorded from 15,386 animals. Even though the curve parameters were estimated for animals with few records, given that the information from related animals and the structure of systematic effects were considered in the curve fitting, all mature BW predicted were suitable. A large additive genetic variance for mature BW was observed. The parameter a of growth curves, which represents asymptotic adult BW, could be used as a selection criterion to control increases in adult BW when selecting for growth rate. The effect of maternal environment on growth was carried through to maturity and should be considered when evaluating adult BW. Other growth curve parameters showed small additive genetic and maternal effects. Mature BW and parameter k, related to the slope of the curve, presented a large, positive genetic correlation. The results indicated that selection for growth rate would increase adult BW without substantially changing the shape of the growth curve. Selection to change the slope of the growth curve without modifying adult BW would be inefficient because their genetic correlation is large. However, adult BW could be considered in a selection index with its corresponding economic weight to improve the overall efficiency of beef cattle production.     

Estimation of variances for gametic effects on litter size in Yorkshire and Landrace swine

The objective of this study was to test for effects of gametic imprinting on litter size in swine by estimating variances for parent-specific gametic effects. Data were 64,047 and 137,009 multiparous records of number born alive for the U.S. Landrace and Yorkshire breeds, respectively. The statistical model included fixed effects of parity number and herd, and random effects of herd-year-season, mate, permanent environment, animal (additive genetic), and either maternal or paternal gametes. A Bayesian approach that used Gibbs sampling to obtain posterior distributions was employed. To aid in the interpretation of results, the Landrace data structure was used to simulate data with and without effects of imprinting. Analyses of the simulated records indicated that the model applied was capable of detecting effects of imprinting when such effects were present. Small, but non-zero, estimates of gametic variances were obtained when no imprinting was simulated. Estimates of the proportion of total variance accounted for by paternally transmitted gametes were 0.8 and 0.9% for Landrace and Yorkshires, respectively. These estimates were different from zero, but were similar to the results observed for data simulated without an imprinting effect. Corresponding results for maternally transmitted gametes were 1.6% for Landrace and 0.8% for Yorkshires. The estimate for Landrace was significantly greater than that observed for Yorkshires and for the simulations without a true effect and suggested the presence of a non-Mendelian genetic influence on litter size. Paternally imprinted genes are a plausible reason for the observed results. Assuming that the effect observed was due to paternal imprinting at a single biallelic locus, the substitution effect of the superior allele could be greater than 0.7 piglets per litter. Identification of a genetic marker for such an allele would be useful in marker-assisted selection of females. Other possible explanations exist for the increased gametic variance in the Landrace breed, but these explanations (such as maternal or cytoplasmic effects) may be less likely than paternal imprinting.     

Genetic parameters estimated with multitrait and linear spline-random regression models using Gelbvieh early growth data

Estimates of direct and maternal genetic parameters in beef cattle were obtained with a random regression model with a linear spline function (SFM) and were compared with those obtained by a multitrait model (MTM). Weight data of 18,900 Gelbvieh calves were used, of which 100, 75, and 17% had birth (BWT), weaning (WWT), and yearling (YWT) weights, respectively. The MTM analysis was conducted with a three-trait maternal animal model. The MTM included an overall linear partial fixed regression on age at recording for WWT and YWT, and direct-maternal genetic and maternal permanent environmental effects. The SFM included the same effects as MTM, plus a direct permanent environmental effect and heterogeneous residual variance. Three knots, or breakpoints, were set to 1, 205, and 365 d. (Co)variance components in both models were estimated with a Bayesian implementation via Gibbs sampling using flat priors. Because BWT had no variability of age at recording, there was good agreement between corresponding components of variance estimated from both models. For WWT and YWT, with the exception of the sum of direct permanent environmental and residual variances, there was a general tendency for SFM estimates of variances to be lower than MTM estimates. Direct and maternal heritability estimates with SFM tended to be lower than those estimated with MTM. For example, the direct heritability for YWT was 0.59 with MTM, and 0.48 with SFM. Estimated genetic correlations for direct and maternal effects with SFM were less negative than those with MTM. For example, the direct-maternal correlation for WWT was -0.43 with MTM and -0.33 with SFM. Estimates with SFM may be superior to MTM due to better modeling of age in both fixed and random effects.     

Estimation of the additive and dominance variances in SA Duroc pigs

The objective of this study was to estimate dominance variance for number born alive (NBA), 21-day litter weight (LWT21) and interval between parities (FI) in South African Duroc pigs. A total of 10,703 NBA, 6883 LWT 21 and 6881 FI records were analysed. Bayesian analysis via Gibbs sampling was used to estimate variance components and genetic parameters. Estimates of additive genetic variance were 0.554, 16.84 and 4.535 for NBA, FI and LWT21, respectively. Corresponding estimates of dominance variance were 0.246, 9.572 and 0.661 respectively. Dominance effects were statistically not significant for all traits studied. Further research utilizing a larger data set is necessary to make concrete conclusions on the importance of dominance genetic effects for the traits studied.     

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 .     

Genetic parameters and evaluation of the Tunisian dairy cattle population for milk yield by Bayesian and BLUP analyses

A Markov Chain Monte Carlo Bayesian method and BLUP analyses were used on Tunisian dairy cattle data. Data included 92,106 lactation records collected on 37,536 animals over 19 freshening years, from 1983 to 2001. Each record was partitioned into the fixed effects of herd-year, month of calving, and age-parity, a permanent environmental effect, an additive genetic effect, and a residual effect.

Posterior conditional distributions were determined for variance components and model effects. Solutions (BLUE) and posterior means for levels of herd-year, month of calving, and age-parity showed similar patterns. Posterior means of heritability and repeatability were 0.17 ± 18 × 10^− 5 and 0.39 ± 8 × 10^− 5, respectively. Posterior means of bull's breeding values were compared to bull's BLUP solutions. BLUP solutions were obtained using 0.17 and 0.39, estimated from the data, and 0.25 and 0.40 estimates for heritability and repeatability, respectively. Rank correlations between bull's posterior means and BLUP breeding values were 0.998 and 0.994 using genetic parameters estimated from the data and from the literature, respectively. This correlation coefficient was 0.995 between bull's BLUP solutions using either of the two sets of genetic parameters.     

Bayesian threshold analysis of direct and maternal genetic parameters for piglet mortality at farrowing in Large White, Landrace, and Pietrain populations

A Bayesian threshold model was fitted to analyze the genetic parameters for farrowing mortality at the piglet level in Large White, Landrace, and Pietrain populations. Field data were collected between 1999 and 2006. They were provided by 3 pig selection nucleus farms of a commercial breeding company registered in the Spanish Pig Data Bank (BDporc). Analyses were performed on 3 data sets of Large White (60,535 piglets born from 4,551 litters), Landrace (57,987 piglets from 5,008 litters), and Pietrain (42,707 piglets from 4,328 litters) populations. In the analysis, farrowing mortality was considered as a binary trait at the piglet level and scored as 1 (alive piglet) or 0 (dead piglet) at farrowing or within the first 12 h of life. Each breed was analyzed separately, and operational models included systematic effects (year-season, sex, litter size, and order of parity), direct and maternal additive genetic effects, and common litter effects. Analyses were performed by Bayesian methods using Gibbs sampling. The posterior means of direct heritability were 0.02, 0.06, and 0.10, and the posterior means of maternal heritability were 0.05, 0.13, and 0.06 for Large White, Landrace, and Pietrain populations, respectively. The posterior means of genetic correlation between the direct and maternal genetic effects for Landrace and Pietrain populations were -0.56 and -0.53, and the highest posterior intervals at 95% did not include zero. In contrast, the posterior mean of the genetic correlation between direct and maternal effects was 0.15 in the Large White population, with the null correlation included in the highest posterior interval at 95%. These results suggest that the genetic model of evaluation for the Landrace and Pietrain populations should include direct and maternal genetic effects, whereas farrowing mortality could be considered as a sow trait in the Large White population.