REML (restricted maximum likelihood) has become the standard method of variance component estimation in animal breeding. Inference in Bayesian animal models is typically based upon Markov chain Monte Carlo (MCMC) methods, which are generally flexible but time-consuming. Recently, a new Bayesian computational method, integrated nested Laplace approximation (INLA), has been introduced for making fast non-sampling-based Bayesian inference for hierarchical latent Gaussian models. This paper is concerned with the comparison of estimates provided by three representative programs (ASReml, WinBUGS and the R package AnimalINLA) of the corresponding methods (REML, MCMC and INLA), with a view to their applicability for the typical animal breeder. Gaussian and binary as well as simulated data were used to assess the relative efficiency of the methods. Analysis of 2319 records of body weight at 35 days of age from a broiler line suggested a purely additive animal model, in which the heritability estimates ranged from 0.31 to 0.34 for the Gaussian trait and from 0.19 to 0.36 for the binary trait, depending on the estimation method. Although in need of further development, AnimalINLA seems a fast program for Bayesian modeling, particularly suitable for the inference of Gaussian traits, while WinBUGS appeared to successfully accommodate a complicated structure between the random effects. However, ASReml remains the best practical choice for the serious animal breeder.

The restricted maximum likelihood (REML) method (Patterson and Thompson, 1971) for unbalanced mixed models has been extensively used in animal breeding and has become the standard method for the estimation of variance components. The Bayesian Markov chain Monte Carlo (MCMC) methods were introduced in quantitative genetics in the early 1990s (Wang et al., 1993; Sorensen et al., 1994), facilitated by the development of the Gibbs sampling procedure (Geman and Geman, 1984; Gelfand and Smith, 1990). The Gibbs sampler successively samples from conditional distributions of all parameters in a model in order to generate a random sample of the marginal posterior distribution, which is the target for Bayesian inference. MCMC methods represent the standard inference procedure for Bayesian animal models (Sorensen and Gianola, 2002), and through the years they have become an attractive alternative to REML. Recently, a non-sampling-based alternative to MCMC, the integrated nested Laplace approximations (INLAs), has been introduced (Rue et al., 2009). Using INLA, marginal posteriors for all parameters and random effects can be calculated. Because INLA is based on direct numerical integration instead of simulations, it is much faster than MCMC (Rue et al., 2009). Furthermore, Holand et al. (2013) have developed an R package (AnimalINLA) making Bayesian animal models more accessible to animal breeders.

Several programs are available for MCMC methods, but very few provide a
flexible environment. WinBUGS (Lunn et al., 2000) is the most well-developed
and general-purpose Bayesian software available to date. It has an
interactive environment that enables the user to specify models that need to
be compiled before starting the Gibbs sampling. Convergence diagnostics,
model comparisons, e.g., via DIC (deviance information criterion), and other useful plots and diagnostics are
available. Several distributions can be used for modeling the observations
as well as priors, while full conditional distributions are automatically
constructed and the appropriate MCMC algorithm for sampling is chosen (Lunn
et al., 2000). In WinBUGS and in the context of animal breeding, an important
issue is the importation of the animals' genetic relationship matrix.
Methods proposed so far (Damgaard, 2007; Waldmann, 2009) either require prior
transformation of the data using complex code or do not provide a generic
procedure independent of the data structure. A good solution here is the use
of the inverse of the numerator relationship matrix

The primary goals of the present study were to apply and investigate the relative merits of three methods (REML, Gibbs sampling and INLA) in the context of animal breeding, using representative programs such as ASReml 3.0 (Gilmour et al., 2009), WinBUGS and AnimalINLA. For this purpose, both a Gaussian and a binary trait were explored and variance components and the genetic parameters along with breeding values across the three methods were estimated and compared.

Data on body weight (BW) at 35 days of age from a broiler line were made
available by Aviagen Ltd. Given that, in the Windows version of AnimalINLA
1.1, limitations in the size of the data set exist, a small data set was
randomly selected, consisting of 2319 records. This comprised 1171 males
and 1148 females in 40 hatch weeks, while the pedigree included a total of
2456 animals. All sires (

A binary response trait was also constructed, using the original BW values
and a threshold at the highest 20 % phenotypic values. Thus, the new
variable

Three animal models were considered for BW. Model M

From a Bayesian perspective, the data

Gelman (2006) investigated the statistical properties of different priors on
variance components and found that a uniform prior on the standard deviation
is a reasonable choice in a number of situations. Therefore, vague uniform
priors were utilized for the standard deviation of the additive genetic
effects

For measuring the mixing and efficiency of the MCMC samples, the effective
sample size (ESS) was used. The ESS of the posterior samples of each
parameter corresponds to the number of independent samples having the same
estimation accuracy as the dependent MCMC samples and is given by
Waagepetersen et al. (2008): ESS

Initially, a simple animal model was fitted via REML, considering

In order to investigate the relative merits of the three approaches, data for both the Gaussian and the binomial case were simulated and models were applied accordingly.

The initial analysis of data revealed a marginal importance of the

In total, 30 samples from each scenario were generated. These samples were
then analyzed via models M

According to the method applied, the model comparison was based on four
evaluation criteria: the Akaike information criterion (AIC; Akaike, 1973),
the Bayesian information criterion (BIC; Schwarz 1978), the conditional
Akaike information criterion (cAIC; Vaida and Blanchard, 2005) and the
DIC (Spiegelhalter et al., 2002). All
criteria are based upon the computation of the deviance (

Table 1 summarizes the estimated variance components and genetic parameters
of BW, along with likelihoods,

Under model M

This could be effectively modeled only via the WinBUGS software. Under model
M

To further quantify the implications of model and method evaluation on selection decisions, Pearson as well as rank correlations of animals' EBVs and the percentage of common animals selected were calculated across the models and methods applied (results not shown). The correlations in question were extremely high (0.97–0.99) when the focus was on the whole population and/or a proportion of the best 20 % of animals. During this phase, an additional advantage of the WinBUGS software was its ability to estimate (via the rank tool) the uncertainty associated with the ranking of the individuals from the posterior distributions of the EBVs. Figure 1 presents 12 selected examples from the posterior distribution of the EBV ranks, with four animals each from the top, middle and low end of the spectrum. These ranks were based upon the whole posterior density and properly accounted for characteristics such as the variance and skewness of the posterior. Both, a 95 % rank interval as well as the median rank are provided, thus presenting an easy and flexible way of animal selection. The large uncertainty associated with selecting among similar animals is also illustrated. Here, rank correlations were remarkably high, ranging from 0.96 to 0.99 among all methods and models considered. Furthermore, standard errors of the EBVs and solutions for the fixed effects were comparable among the methods, with no statistically significant differences. All models and methods suggested the same animals, resulting in correlations between the estimated breeding values that ranged from 0.96 to 0.99.

Estimates of variance components and genetic parameters for body weight (BW) at 35 days of age.

Estimates of variance components and genetic parameters for the binary transformed BW.

The estimated variance components and genetic parameters of

Distribution of ranking for 12 representative animals, based
on the EBVs estimated by WinBUGS. Four animals each were taken from the top, middle and low end of the spectrum.

As in the case of the Gaussian trait, rank correlations across the three methods remained high, ranging from 0.92 to 0.99 (results not shown). In addition, the proportion of common animals selected among the three methods exceeded 93 %, suggesting minor implications of method usage on selection decisions.

Descriptive statistics of the simulated data and the estimators across
models and methods are given in Table 3. Average values of the simulated
data were equal to the true ones (

True values and descriptive statistics of the estimators under two levels of additive genetic maternal environmental correlation.

Estimates under model M

The MSEs across models and methods are presented in Table 4. Irrespectively
of the method and/or model, MSEs were lower in the low- vs. the high-correlation scenario. Furthermore, better estimates (in terms of MSEs) were
attained in ASReml using M

Mean squared errors of the variance components and the genetic parameters under two levels of additive genetic maternal environmental correlation.

Actual coverage of nominal 95 % intervals of estimated variance components and genetic parameters.

The coverage of interval estimates for the three models and the respective
methods of analysis are shown in Table 5. To construct Bayesian 95 %
credible intervals, the quantiles of the relevant posterior distributions
(as estimated by MCMC and INLA) were used. ASReml's intervals were
constructed based on asymptotic normality of the maximum likelihood using

The theoretical aspects and advantages of REML and MCMC methods for fitting hierarchical multilevel models, such as the animal model, have been extensively explored elsewhere, either with a statistical focus (Browne and Draper, 2006) or from an animal breeder's perspective (van Tassel et al., 1995; van Tassel and van Vleck, 1996). However, this is the first study applying REML and MCMC methods along with another Bayesian approach, i.e., INLA, within the context of poultry breeding. Our main concerns were the practical aspects of the applicability of three available typical software programs for the standard animal breeder. Given that both the size and the structure of data sets may have an impact on the performance of the analytical approach (Blasco, 2001), no general inference can be made based on the present results.

In the present study, an attempt to compare coverage intervals derived from Bayesian and REML approaches was pursued. However, there are two main differences between credible and confidence intervals. While a credible interval incorporates information from the prior distribution into the estimate, confidence intervals are based solely on the data, treating the parameter as fixed and the interval itself as random. Credible intervals are different from confidence intervals essentially because credible intervals are probability intervals; i.e., they say that the true value should be within the interval with a determined probability. Confidence intervals do not say that the true value is within the limits with a determined probability. In conceptual repetitions of an experiment, different confidence intervals can be obtained; 95 % of these intervals contain the true value. Thus, we treat the interval as containing it, knowing that, in the long run, we will be wrong 5 % of times. Although different in philosophy, the comparison between these types of intervals may be useful within the context of a study such as ours.

From a frequentist's point of view, the standard method entails the use of the REML and BLUP methods. In the present study, ASReml (Gilmour et al., 2009) software was employed. The software is stable and fast and can handle many different models, data structures and thousands of data records. In addition, the necessary files are not especially complicated to construct, while a valuable manual, containing a lot of information and numerous examples, is available for the animal breeder. For binary trait modeling, a variety of link functions (logit, probit, cloglog) can be chosen.

An obvious obstacle when using commercial programs is their
limited flexibility, i.e., the inability to model complex structures between
(random) effects. A good example here was the presence of negative
correlation between

Modeling the covariance in question was made possible only via WinBUGS. This is a very valuable feature when testing assumptions of the standard animal model with regard to possible correlation structures between the various random effects. This program allows for the application of a large group of competing models and Bayesian model evaluation criteria (Sorensen and Gianola, 2002). A further important attribute of WinBUGS is the rank tool, which can simultaneously incorporate the uncertainty associated with ranking the individuals, thus assisting in animal selection. In theory, REML and INLA would probably struggle if the likelihood was very flat, whereas MCMC methods should be able to cope (Blasco, 2001). Such scenarios could be important for practical breeding purposes and might be properly encountered by MCMC methods. Bayesian methods, such as MCMC implemented in WinBUGS, can be especially useful in complex situations at the cost of being computationally expensive and time-consuming. For our data, approximately 14 to 16 h were needed to achieve convergence, depending on modeling assumptions.

The AnimalINLA has proved to be a remarkably time-efficient experience. It took less than 10 s to produce the required posterior distributions, while providing comparable estimates with the other packages. Although computationally efficient, the current version of this R package (AnimalINLA 1.1) could not accommodate more than 4000 records in the animal model, probably due to compatibility problems with Windows. Although time-efficient, AnimalINLA has displayed certain problems in terms of bias and accuracy, particularly for a binary trait. The latter has also been confirmed by Holand et al. (2013) and is supported by a more detailed investigation of simulated data. Finally, it is not as flexible in modeling as the WinBUGS and the documentation is still under development.

In conclusion, WinBUGS can be of great assistance to the animal breeder because of its flexibility in modeling complex models while unraveling existent data structures that the usual REML-based packages neglect. Within the animal breeding context, its applicability remains rather limited since only small to moderate data sets or populations can be handled in a time-efficient manner. Furthermore, the choice of the priors should be made with caution, particularly when the posteriors may vary with priors. The AnimalINLA software appears to be a promising future perspective for the animal breeder dedicated to the Bayesian paradigm since it is remarkably fast. It seems, however, to be a package still under development. Our own experience on large data sets has shown that ASReml can effectively handle analyses for up to 200 000 records and related pedigree structures fast (< 1 h) and mostly independent of initial values (Maniatis et al., 2013). Furthermore, as the simulation results have shown, even when a large covariance between random effects is neglected, it may provide estimates of the parameters in question with relatively small bias and error. Given all the above, ASReml remains the best practical choice for the serious animal breeder among the software packages examined.