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.     

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.     

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.     

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.     

Variational bayesian method of estimating variance components

We developed a Bayesian analysis approach by using a variational inference method, a so‐called variational Bayesian method, to determine the posterior distributions of variance components. This variational Bayesian method and an alternative Bayesian method using Gibbs sampling were compared in estimating genetic and residual variance components from both simulated data and publically available real pig data. In the simulated data set, we observed strong bias toward overestimation of genetic variance for the variational Bayesian method in the case of low heritability and low population size, and less bias was detected with larger population sizes in both methods examined. The differences in the estimates of variance components between the variational Bayesian and the Gibbs sampling were not found in the real pig data. However, the posterior distributions of the variance components obtained with the variational Bayesian method had shorter tails than those obtained with the Gibbs sampling. Consequently, the posterior standard deviations of the genetic and residual variances of the variational Bayesian method were lower than those of the method using Gibbs sampling. The computing time required was much shorter with the variational Bayesian method than with the method using Gibbs sampling.     

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.     

Reliable computing in estimation of variance components

The purpose of this study is to present guidelines in selection of statistical and computing algorithms for variance components estimation when computing involves software packages. For this purpose two major methods are to be considered: residual maximal likelihood (REML) and Bayesian via Gibbs sampling. Expectation‐Maximization (EM) REML is regarded as a very stable algorithm that is able to converge when covariance matrices are close to singular, however it is slow. However, convergence problems can occur with random regression models, especially if the starting values are much lower than those at convergence. Average Information (AI) REML is much faster for common problems but it relies on heuristics for convergence, and it may be very slow or even diverge for complex models. REML algorithms for general models become unstable with larger number of traits. REML by canonical transformation is stable in such cases but can support only a limited class of models. In general, REML algorithms are difficult to program. Bayesian methods via Gibbs sampling are much easier to program than REML, especially for complex models, and they can support much larger datasets; however, the termination criterion can be hard to determine, and the quality of estimates depends on a number of details. Computing speed varies with computing optimizations, with which some large data sets and complex models can be supported in a reasonable time; however, optimizations increase complexity of programming and restrict the types of models applicable. Several examples from past research are discussed to illustrate the fact that different problems required different methods.     

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 breeding values from large‐sized routine carcass data in Japanese Black cattle using Bayesian analysis

Volumes of official data sets have been increasing rapidly in the genetic evaluation using the Japanese Black routine carcass field data. Therefore, an alternative approach with smaller memory requirement to the current one using the restricted maximum likelihood (REML) and the empirical best linear unbiased prediction (EBLUP) is desired. This study applied a Bayesian analysis using Gibbs sampling (GS) to a large data set of the routine carcass field data and practically verified its validity in the estimation of breeding values. A Bayesian analysis like REML‐EBLUP was implemented, and the posterior means were calculated using every 10th sample from 90 000 of samples after 10 000 samples discarded. Moment and rank correlations between breeding values estimated by GS and REML‐EBLUP were very close to one, and the linear regression coefficients and the intercepts of the GS on the REML‐EBLUP estimates were substantially one and zero, respectively, showing a very good agreement between breeding value estimation by the current GS and the REML‐EBLUP. The current GS required only one‐sixth of the memory space with REML‐EBLUP. It is confirmed that the current GS approach with relatively small memory requirement is valid as a genetic evaluation procedure using large routine carcass data.     

Estimation of direct, maternal, and grandmaternal genetic effects for weaning weight in several breeds of beef cattle.

Weaning weights from nine parental breeds and three composites were analyzed to estimate variance due to grandmaternal genetic effects and to compare estimates for variance due to maternal genetic effects from two different models. Number of observations ranged from 794 to 3,465 per population. Number of animals in the pedigree file ranged from 1,244 to 4,326 per population. Two single-trait animal models were used to obtain estimates of covariance components by REML using an average information method. Model 1 included random direct and maternal genetic, permanent maternal environmental, and residual environmental effects as well as fixed sex x year and age of dam effects. Model 2 in addition included random grandmaternal genetic and permanent grandmaternal environmental effects to account for maternal effects of a cow on her daughter's maternal ability. Non-zero estimates of proportion of variance due to grandmaternal effects were obtained for 7 of the 12 populations and ranged from .03 to .06. Direct heritability estimates in these populations were similar with both models. Existence of variance due to grandmaternal effects did not affect the estimates of maternal heritability (m2) or the correlation between direct and maternal genetic effects (r(am)) for Angus and Gelbvieh. For the other five populations, magnitude of estimates increased for both m2 and r(am) when estimates of variance due to grandmaternal effects were not zero. Estimates of the correlation between maternal and grandmaternal genetic effects were large and negative. These results suggest that grand-maternal effects exist in some populations, that when such effects are ignored in analyses maternal heritability may be underestimated, and that the correlation between direct and maternal genetic effects may be biased downward if grandmaternal effects are not included in the model for weaning weight of beef cattle.     

Variance decomposition in the estimation of genetic variance with selected data.

Estimation of genetic variance in populations under selection involves assumptions on base animals. Base animals are often considered unselected and it also has been proposed to treat selected base animals as fixed. The consequences of assumptions on base animals in the estimation of genetic variance in selected populations are not fully understood. Variance decompositions are introduced for simple designs to quantify the differences between models that treat base animals in different ways. Independent contrasts were constructed and REML estimates of variance components were compared for different designs and selection rules. The method shows how selection is accounted for in a complete model and why estimation of variance components can become biased when base animals are treated as fixed.     

A Bayesian analysis of response to selection for uterine capacity in rabbits

A divergent, eight generation selection experiment on uterine capacity in rabbits was performed. Rabbit does were ovariectomized unilaterally before puberty, and selected for increased and decreased litter size by 'best linear unbiased prediction' using data from up to four parities. Two different analyses were performed to estimate the response to selection. The first was based on least squares analysis; the second was based on Bayesian methods using Gibbs sampling techniques. Three different priors were used for variance components, but these had little influence on the results. Posterior means of heritabilities for uterine capacity, varied from 0.09 to 0.12, and repeatabilities from 0.18 to 0.22. The response to eight generations of selection was symmetrical and led to a divergence of 0.16 young rabbits per generation, which amounts to about 2% of the average litter size of the base population per generation. The pattern of response however, was not linear: a high initial response was followed by a period where little further response was observed, and a final burst of response was obtained during the last two cycles of selection.     

Direct and maternal (co)variance components and heritability estimates for body weights in Chokla sheep

Estimates of (co)variance components were obtained for weights at birth, weaning and 6, 9 and 12 months of age in Chokla sheep maintained at the Central Sheep and Wool Research Institute, Avikanagar, Rajasthan, India, over a period of 21 years (1980–2000). Records of 2030 lambs descended from 150 rams and 616 ewes were used in the study. Analyses were carried out by restricted maximum likelihood (REML) fitting an animal model and ignoring or including maternal genetic or permanent environmental effects. Six different animal models were fitted for all traits. The best model was chosen after testing the improvement of the log-likelihood values. Direct heritability estimates were inflated substantially for all traits when maternal effects were ignored. Heritability estimates for weight at birth, weaning and 6, 9 and 12 months of age were 0.20, 0.18, 0.16, 0.22 and 0.23, respectively in the best models. Additive maternal and maternal permanent environmental effects were both significant at birth, accounting for 9% and 12% of phenotypic variance, respectively, but the source of maternal effects (additive versus permanent environmental) at later ages could not be clearly identified. The estimated repeatabilities across years of ewe effects on lamb body weights were 0.26, 0.14, 0.12, 0.13, and 0.15 at birth, weaning, 6, 9 and 12 months of age, respectively. These results indicate that modest rates of genetic progress are possible for all weights.     

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.     

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.     

Threshold-linear versus linear-linear analysis of birth weight and calving ease using an animal model: I. Variance component estimation.

Birth weight and calving difficulty were analyzed with Bayesian methodology using univariate linear models, a bivariate linear model, a threshold model for calving difficulty, and a joint threshold-linear model using a probit approach. Field data included 26,006 records of Gelbvieh cattle. Simulated populations were generated using parameters estimated from the field data. The Gibbs sampler was used to obtain estimates of the marginal posterior mean and standard deviation of the (co)variance components, heritabilities, and correlations. In the univariate analyses, the posterior mean of direct heritability for calving difficulty was .23 with the threshold model and .18 with the linear model. Maternal heritabilities were .10 and .08, respectively. In the bivariate analysis, posterior means of direct heritability for calving difficulty were .21 and .18 for the bivariate linear-threshold and linear-linear model, respectively. Maternal heritabilities were .09 and .06, respectively. Direct heritability for birth weight was .25 for the univariate model and .26 for bivariate models. Maternal heritability was .05 for the linear-threshold model and the univariate model and .06 for the bivariate linear model. Genetic correlation between direct genetic effects in both traits was .81 for the linear-threshold model and .79 for the bivariate linear. Residual correlation was .35 for the bivariate linear model and .50 for the bivariate linear-threshold. A simulation study confirmed that the posterior mean of the marginal distribution was suitable as a point estimate for univariate threshold and bivariate linear-threshold models.     

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.     

A Bayesian approach to the accuracy of clinical observations

The accuracy of clinical observations was estimated using Bayesian latent-class models with two or more independent tests. Four veterinarians carried out systematic independent clinical examinations on 155 pigs in three herds. Based on the results of binary recordings of clinical observations on dullness, poor body condition (PBC), skin lesions, lameness, respiratory disease, and diarrhea, a latent disease state for each clinical disease was estimated using Gibbs sampling.

The accuracy of the clinical observations differed for the four observers and for different clinical signs. Population parameters were estimated from a Bayesian hierarchical model, and the accuracy of a random observer was calculated. We concluded that the accuracy of the veterinarians in this study substantiated the need to pursue more-precise definitions of the clinical findings and that larger sample sizes would be needed to provide reasonable variance estimates. Finally, we concluded that the uncertainty in the clinical decision-making process (starting with the clinical examination) needs to be represented fully.     

Response to 13 generations of selection for increased 8-week body weight in lines of mice carrying a sheep growth hormone-based transgene

The purpose of this study was to evaluate selection in lines of transgenic mice. Two replicates of lines that either carried or did not carry the sheep metallothionein-1a sheep growth hormone transgene (oMt1a-oGH) were established. The host lines had been previously selected for rapid growth or selected randomly. Within-litter selection for increased 8-wk body weight was carried out for 13 generations. The frequency of oMt1a-oGH was monitored in all generations in the transgenic lines, but no genotypic information regarding the transgene was used as an aid to selection. The oMt1a-oGH was activated from weaning, at 3 wk, until 8 wk of age by adding ZnSO4 to the drinking water. Zinc stimulation of the transgene was not done during mating, gestation, or lactation. Data on body weights and weight gains were analyzed with a conventional mixed model and with an animal model. Genetic progress was achieved in all lines subjected to directional selection. In the control background, response to selection for 8-wk body weight was larger in the nontransgenic lines than in the transgenic lines, whereas no difference was found in the selected background. The frequency of the transgene was increased from the initial .5 to .62 in the randomly selected background but decreased to .04 in lines from a selected background. The REML estimates of variance components and genetic gain estimates varied greatly between the two methods. In general, there was better agreement between the realized heritability estimates and the heritability estimates obtained from the conventional mixed model analysis than between realized heritability estimates and results obtained using the animal model. Favorable correlated responses were obtained for 3- and 6-wk body weights and on 3- to 6- and 6- to 8-wk weight gains. Correlated responses to selection were larger in the selected than in the nonselected background but were not affected by the presence of the transgene. Results suggest that constructs similar to the oMt1a-oGH, which allow tight regulation, may be successfully incorporated into commercial livestock and should have larger effects in populations that have not been subject to selection.