Genetic parameters for marbling and body score in Anglonubian goats using Bayesian inference via threshold and linear models

Objective The aim of this study was to estimate (co) variance components and genetic parameters for categorical carcass traits using Bayesian inference via mixed linear and threshold animal models in Anglonubian goats. Methods Data were obtained from Anglonubian goats reared in the Brazilian Mid-North region. The traits in study were body condition score, marbling in the rib eye, ribeye area, fat thickness of the sternum, hip height, leg perimeter, and body weight. The numerator relationship matrix contained information from 793 animals. The single- and two-trait analyses were performed to estimate (co) variance components and genetic parameters via linear and threshold animal models. For estimation of genetic parameters, chains with 2 and 4 million cycles were tested. An 1,000,000-cycle initial burn-in was considered with values taken every 250 cycles, in a total of 4,000 samples. Convergence was monitored by Geweke criteria and Monte Carlo error chain. Results Threshold model best fits categorical data since it is more efficient to detect genetic variability. In two-trait analysis the contribution of the increase in information and the correlations between traits contributed to increase the estimated values for (co) variance components and heritability, in comparison to single-trait analysis. Heritability estimates for the study traits were from low to moderate magnitude. Conclusion Direct selection of the continuous distribution of traits such as thickness sternal fat and hip height allows obtaining the indirect selection for marbling of ribeye.


INTRODUCTION
In animal breeding programs several traits are evaluated to identify genotypes with higher average according to the proposed objectives. The analyzed data for this purpose may be continuous, as an example of measurement performed in the loin eye area, which is assumed to have normal distribution, or discontinuous such as the case of categorical traits as carcass marbling. However, to obtain estimates of (co) variance components and genetic parameters for categorical data, some approaches that consider the discrete data distribution should be used to ensure greater accuracy of the estimates.
Within the categorical traits in meat goat production, the score assigned in the evaluation of body condition of animals is a criterion that has been used for carcass in vivo evaluation, whose advantage is to obtain information of the nutritional status of animals. In goats, this technique is widely applied, however the use of ultrasound to evaluate carcass in live animals has spread because it enables the identification of carcass marbling based on intramuscular fat. It is a type of data with discrete distribution in which the longissimus dorsi is the main muscle for evaluation in the living animal.
The estimation method with establishment of a model that correctly describes the nature of categorical data is an impor tant factor to obtain genetic parameters [1]. The methodology of Mixed Linear Models is the most common means of estima ting (co) variance components for traits of economic interest since this methodology has easy application in animal model and requires less time in data processing. However, it is nec essary that the analyzed traits have normal distribution, which is noticed only in continuous data. Thus, the use of important characteristics for animal breeding which assume discrete distribution is limited. Genetic analyses would not be appro priate using linear models [2,3]. Therefore, the use of threshold models is recommended, because they are more efficient in detecting genetic variability, in comparison to linear models [3].
With the use of threshold model it is considered there is an unobservable variable that takes the normal distribution un derlying the discrete variable. The variable connection observed with the underlying continuous scale is made by a set of fixed thresholds. Thus, the underlying variable, defined as the sum of genetic and environmental effects which affect the suscep tibility of a trait [2], is described by a linear model, but the relationship of this variable to the observed scale is nonlinear [4,5].
By comparison, the genetic gain estimated from genetic analysis of categorical data obtained by threshold models are higher due to the achievement of higher heritability in the un derlying scale [6], thus resulting in better identification of higher value genotypes.
With the advent of increasingly powerful processors, the Bayesian methodology reemerged as statistical tool to estimate components of (co) variance and genetic parameters in animal breeding. Bayesian inference allows the use of prior informa tion of the studied trait being included in the analysis through information of a prior distribution of the parameters to be analyzed along with its uncertainty before the observation data. It also considers the different distributions of the studied data, increasing the accuracy of estimates and predictions.
The aim of this study was to estimate (co) variance com ponents and genetic parameters for categorical carcass traits using Bayesian inference via mixed linear and threshold ani mal models in Anglonubian goats reared in the MidNorth region of Brazil.

MATERIALS AND METHODS
The experimental procedure was approved by the Institutional Animal Care and Use Committee at Federal University of Piauí, Brazil (009/14).
The database used in this research consisted of genea logical and production information measured in registered Anglonubian goats with genealogical records reared in the states of Piauí and Maranhão (MidNorth region of Brazil). The field data collection was carried out from the years 2012 to 2014.
The traits evaluated in this study were: body condition score (BCS), marbling in the rib eye (MRE), ribeye area (REA), fat thickness of the sternum (FTS), hip height (HH), leg perimeter (LP), and body weight. These traits were measured at the same time in each animal. Goats of both sexes, aged from seven months and healthy were eligible to participate.
For standardization during the farms data collection the following steps were adopted: after weighing on a scale with a capacity of 200 kg and precision of 0.10 kg, BCS of each animal was measured based on palpation and visualization of the lower back region, mimicking up motion grippers ap plying constant pressure around and between the transverse and spinal apophyses, also in the sternal region, assessing the amount of skin, muscle and fat density in two anatomi cal regions, according to methodology cited by [7]. The score assigned for each individual was based on the perception of the fat and muscle deposition in the evaluated areas, taking as a base values from one (lowest fat deposition) to five (ex cessive deposition of fat).
Thereafter, the following morphometric measurements were taken using hipometer and measure tape while the animal was restrained in a comfortable standing position: HH, dis tance between the sacral tuberosity of the ilium and the soil; and LP, measured on the median part of the leg above the femoraltibialrotulian articulation (in centimeters).
In vivo evaluation of carcass was performed by means of ultrasound images captured using the apparatus Chison 600M equipped with linear transducer (13 cm) using a set of fre quency 5.0 MHz.
The REA (in cm²) was measured through ultrasonographic crosssectional images of the longissimus dorsi muscle in the intercostal space between the 12th and 13th ribs. With the same image the MRE was evaluated with by assigning grades from 0 (absence of intramuscular fat) to 6 points (abundance of intramuscular fat). This visual grading scale was an adap tation of that used by [8].
The subcutaneous FTS, given in mm, proposed by [9] to indicate carcass fat thickness was measured from ultrasound images of mediastinal sternal region (3rd bone of the sternum). During ultrasound readings animals were restrained, in order to keep their comfort for better quality images.
In the statistical analyses it was assumed that the BCS and MRE data follow discrete distribution while REA, FTS, HH, and LP follow continuous distribution, so that these traits were considered anchors in twotrait analyses with the two cate gorical traits. Information of months in which data were collected was grouped in two collection seasons (CS): rainy season, from January to May; and dry season, from June to December. The birth month of animal was also distributed in two birth sea sons, namely rainy and dry seasons, similar to the format of CS. The age of the animal at the time of collection was grouped into age classes (AC), as follows: AC = 1, animals >six months and <two years; AC = 2, animals >two years but <four years; AC = 3, animals >four years. Finally, the evaluated goats were grouped into animal category (CAT) as follows: pregnant, kidding, and dry breeding does. In the formation of the con temporary group (CG), animals born on the same farm, in the same year, and same sex were taken into consideration.
The data were edited and formatted with the statistical program SAS (SAS Inst. Inc., Cary, NC, USA). After the con sistency analyses a file containing animal information, parent, CG, year of collection (YC), age group (AC), animal category (CAT), and observations relating to the analyzed traits, total ing 385 animals with observations was edited. At the end, the numerator relationship matrix of Wright contained informa tion from 793 different animals.
The components of (co) variance and genetic parameters were estimated via linear and threshold animal model by Bayesian inference in single and twotrait analyses. Twotrait analyses were performed by combining the two categorical traits (two by two) with the continuous ones, totaling 11 anal yses. Thus, twotrait analyses were performed in animal model, assuming two different distributions in the same analysis.
Estimates of genetic parameters via Bayesian Inference were calculated with the aid of GIBBS1F90 and THRGIBBS1F90 applications, used for linear and threshold models, respec tively [10], by testing chains with 2 and 4 million cycles. The length of the chain used to compare models and generate the posterior distribution of the (co) variance components and genetic parameters of the BCS and MRE was 2 million, since both generated similar estimates. This similarity was consid ered as the criterion of convergence.
After burnin of the first 1,000,000 samples samples were taken apart at every 250 cycles (sampling interval), resulting in a posteriori distribution with 4,000 samples in which in ferences were performed. Values for burnin and sampling interval were defined based on preliminary analyses in which the convergence and distribution of samples were evaluated through the POSTGIBBS1F90 program [10], which uses the Geweke diagnostic test [11] based on the Z test of average equality of the conditional distribution data logarithm.
The animal model in matrix notation was: y = Xβ+Zα+ε, in which: y = vector of observations of the studied traits (under lying scales for categorical traits); X = matrix n×f of incidence (n = total of observations, and f the number of fixed effects of classes), which relates the findings to the systemic effects; β = vector of systemic effects of CG (formed by animals raised on the same farm, born in the same year and season, and eval uated in the same season), YC, AC, and CAT; Z = the matrix n×N of incidence, which lists the observations to genetic addi tive direct effects, where n is the total number of observations and N number of individuals; α = vector of random effects representing the direct genetic additive values for each animal (animal model); and e = vector of residual random errors as sociated to the observations.
In Bayesian analysis the systemic and random effects in cluded in the model are considered as random variables.
The accepted assumptions, with a focus on Bayesian meth odology, about the information (y) and data (β, α, and

171
), which assumes multivariate normal distribution, is represent ed as: The animal model in matrix notation was: y = Xβ+Zα+ε, in which: y = vector of observations of the studied 152 traits (underlying scales for categorical traits); X = matrix n×f of incidence (n = total of observations, and f the

168
(fixed effects in frequentist analysis) and order s with a (random genetic additive direct effect); W is a matrix 169 incidence of order r by s; I is an identity matrix of order r by r; and 2 is the residual variance.

170
The conditional probability that that falls into the category j

161
The accepted assumptions, w 162 and 2 ), which assumes multiv     The conditional probability that that falls into the category j (1, 2, 3, 4, 5: BCS; and 1, 2, 171 thus: The animal model in matrix notation was: y = Xβ+Zα+ε, in which: y = vector of observatio 152 traits (underlying scales for categorical traits); X = matrix n×f of incidence (n = total of obser 153 number of fixed effects of classes), which relates the findings to the systemic effects; β = v 154 effects of CG (formed by animals raised on the same farm, born in the same year and season,     The conditional probability that that falls into the category j (1, 2, 3, 4, 5: BCS; and 1, 2,

and
The animal model in matrix notation was: y = Xβ+Zα+ε, in which: y = vector of observati 152 traits (underlying scales for categorical traits); X = matrix n×f of incidence (n = total of obser    The conditional probability that that falls into the category j (1, 2, 3, 4, 5: BCS; and 1, 2

171
, components of additive direct genetic and residual (co) variance, respectively; A, numerators matrix of Wright's inbreeding coefficient; and I, identity matrix of equal order to the number of animals with observations.
In the threshold model the underlying scale assumes nor mal distribution represented as: The animal model in matrix notation was: y = Xβ+Zα+ε, in which: y = vector of observations of the studied 152 traits (underlying scales for categorical traits); X = matrix n×f of incidence (n = total of observations, and f the

168
(fixed effects in frequentist analysis) and order s with a (random genetic additive direct effect); W is a matrix 169 incidence of order r by s; I is an identity matrix of order r by r; and 2 is the residual variance.

171
in which U is a vector of scale base of origin r; θ = (b, a) is the location vector of order parameters s with b (fixed effects in frequentist analysis) and order s with a (random genetic additive direct effect); W is a matrix incidence of order r by s; I is an identity matrix of order r by r; and The animal model in matrix notation was: y = Xβ+Zα+ε, in which: y = vector of observations of the studied 152 traits (underlying scales for categorical traits); X = matrix n×f of incidence (n = total of observations, and f the

168
(fixed effects in frequentist analysis) and order s with a (random genetic additive direct effect); W is a matrix 169 incidence of order r by s; I is an identity matrix of order r by r; and 2 is the residual variance.

183
Where n is the number of observations for each trait.

184
The categories (scores) of the BCS and MRE traits for each i animals are defined by Ui, in the underlying scale, and BCS: with the given vectors β,α and t (t = tmin, t1, ..., tj-1, tmáx) and represented as: Where n is the number of observations for each trait.

184
As the residual variance is not estimable in threshold models, the paramet 185 order to obtain identifiability in the likelihood function [12]. This assumption 186 with the given vectors β,α and t (t = tmin, t1, ..., tj-1, tmáx) and represented as:

184
As the residual variance is not estimable in threshold models, the paramet 185 order to obtain identifiability in the likelihood function [12]. This assumption Where n is the number of observations for each trait. As the residual variance is not estimable in threshold mod els, the parameter was assigned a value of 1 in order to obtain identifiability in the likelihood function [12]. This assump tion is standard in categorical data analysis.
Flat was defined (uninformative a priori distribution) for all initial variance, in other words, did not reflect the knowl edge about the parameters to be estimated.
As the best model selection criteria for the traits in study it was used: deviance information criterion (DIC), which is obtained based on the posterior distribution of the deviation statistic [13]; and bayes factor (BF), which is the reason of mar ginal likelihoods between two models [14]. DIC is a model comparison criterion following the propo sition of [15], who suggests that comparisons among models are based on a posteriori distributions of the deviance of each model, being defined by: DIC = Where n is the number of observations for each trait.

184
As the residual variance is not estimable in threshold models, the parameter was assigned a value of 1 in 185 order to obtain identifiability in the likelihood function [12]. This assumption is standard in categorical data 186 analysis.

187
Flat was defined (uninformative a priori distribution) for all initial variance, in other words, did not reflect 188 the knowledge about the parameters to be estimated.

189
As the best model selection criteria for the traits in study it was used: deviance information criterion (DIC),

190
which is obtained based on the posterior distribution of the deviation statistic [13]; and bayes factor (BF), which 191 is the reason of marginal likelihoods between two models [14].
192 DIC is a model comparison criterion following the proposition of [15], who suggests that comparisons in which: Where n is the number of observations for each trait.

184
As the residual variance is not estimable in threshold models, the parameter was assigned a value of 1 in 185 order to obtain identifiability in the likelihood function [12]. This assumption is standard in categorical data 186 analysis.

187
Flat was defined (uninformative a priori distribution) for all initial variance, in other words, did not reflect 188 the knowledge about the parameters to be estimated.

189
As the best model selection criteria for the traits in study it was used: deviance information criterion (DIC),

190
which is obtained based on the posterior distribution of the deviation statistic [13]; and bayes factor (BF), which 191 is the reason of marginal likelihoods between two models [14].

192
DIC is a model comparison criterion following the proposition of [15], who suggests that comparisons Thus, the smaller the value of DIC the better the fit of the eval uated model [13]. As for FB, the marginal likelihood of a given model M is given by: . interpretation of this ratio was given according to [14], ergence was monitored by Geweke criteria [11] and error of Monte Carlo chain, which was obtained by ng the variance of samples for each component divided by the number of samples. Thus, the square root alue refers to the approach of the error standard deviation associated with the size of Gibbs chain [16].

TS
shows the descriptive statistics of MRE and BCS traits. For the other characteristics considered as In analysis with two models (M i and M j ), the BF was defined as the reason for the marginal likelihoods of these two models:

8
. Thus, the smaller the value of DIC the better the fit of the evaluated model [13]. As for FB, the marginal od of a given model M is given by: . interpretation of this ratio was given according to [14], vergence was monitored by Geweke criteria [11] and error of Monte Carlo chain, which was obtained by ting the variance of samples for each component divided by the number of samples. Thus, the square root value refers to the approach of the error standard deviation associated with the size of Gibbs chain [16].

TS
shows the descriptive statistics of MRE and BCS traits. For the other characteristics considered as rs" in multi-trait analysis descriptive statistics were presented in [17]. Because they are categorical The interpretation of this ratio was given according to [14], where: FB i,j >1 is the indication that the numerator of the model (M i ) is the most plausible if FB i,j <1, the denominator model (M j ), is preferred, and if FB i,j = 1, the quality of the two models is the same (M i = M j ). The threshold model was the FB de nominator.
In twotrait analyses it was assumed normal distribution traits for REA, FTS, HH, LP, and body weight, whereas for MRE and BCS these traits were considered as categorical discrete dis tribution. The same animal model of single trait was used in the analyses with the following presupposition: . is ratio was given according to [14], where: FBi,j>1 is the indication that the is the most plausible if FBi,j<1, the denominator model (Mj), is preferred, and if FBi,j dels is the same (Mi = Mj). The threshold model was the FB denominator.    .

201
The interpretation of this ratio was given according to [14], where: FBi,j>1 is the indication that the     .

201
The interpretation of this ratio was given according to [14], where: FBi,j>1 is the indication that the

217
, thus: A, additive genetic relationship matrix among animals; G 0 , additive genetic (co) variance matrix among traits; I, iden tity matrix; and R 0 , residual (co) variance matrix among traits. Convergence was monitored by Geweke criteria [11] and error of Monte Carlo chain, which was obtained by calculat ing the variance of samples for each component divided by the number of samples. Thus, the square root of this value refers to the approach of the error standard deviation associ ated with the size of Gibbs chain [16]. Table 1 shows the descriptive statistics of MRE and BCS traits. For the other characteristics considered as "anchors" in multi trait analysis descriptive statistics were presented in [17]. Because they are categorical qualitative data, MRE and BCS were represented more efficiently by the central tendency statistics, namely Median and Mode, which showed value 3.

RESULTS
In the examination of chain convergence, the Geweke cri terion was significant (p<0.05) ( Table 2), indicating that the chains of the converged parameters and the number of iter ations were appropriate, thus validating the estimates of a posteriori distribution parameters, according to [11].
Comparing the linear model with the threshold applied to categorical data, using the DIC criteria, and the BF, it was ob served that the threshold DIC model showed values equal to 421.00 and 706.74 (Table 2), whereas the linear model pre sented 956.44 and 1,176.10 for BCS and MRE, respectively. Thus, as the values of the threshold model of DIC were lower than those for linear model, this indicates that it was the best model to fit the data in study, consequently the most recom mended model for obtaining estimates of genetic parameters for BCS and MRE in goats.
BF, according to the used criterion, showed values lower  Estimates of genetic parameters via Bayesian Inference were calculated with the aid of GIBBS1F90 and THRGIBBS1F90 applications, used for linear and threshold models, respectively [10], by testing chains with 2 and 4 million cycles. The length of the chain used to compare models and generate the posterior distribution of the (co) variance components and genetic parameters of the BCS and MRE was 2 million, since both generated similar estimates. This similarity was considered as the criterion of convergence.
After burn-in of the first 1,000,000 samples samples were taken apart at every 250 cycles (sampling interval), resulting in a posteriori distribution with 4,000 samples in which inferences were performed. Values for burn-in and sampling interval were defined based on preliminary analyses in which the convergence and distribution of samples were evaluated through the POSTGIBBS1F90 program [10], which uses the Geweke diagnostic test [11] based on the Z test of average equality of the conditional distribution data logarithm.
The animal model in (fixed effects in frequentist analysis) and order s with a (random genetic additive direct effect); W is a matrix incidence of order r by s; I is an identity matrix of order r by r; and 2 is the residual variance.   than 1, since the denominator model was the threshold, then confirming that it is the most recommended for the inference of parameters, as interpretation proposed by [14].
Estimates of heritability for BCS and MRE traits (0.11 and 0.03, respectively) obtained by the threshold model were lower than the values estimated by linear model. BCS equals to 0.29 and MRE equals to 0.12. The confidence intervals of herita bility estimates showed variation of low magnitude when estimated by the threshold model. This amplitude permits inference with a greater reliability of the quality of the esti mates.
In twotrait analyses, where the marbling and BCS were paired with other traits with continuous distribution, it was used the threshold model for the categories that were the best adjustment for the ones used in singletrait analysis, while for continuous distribution the model was linear ( Table 2).
The estimated heritability for continuous traits ranged from moderate (0.24) to high magnitude (0.54) ( Table 3). Table 4 provides estimates of genetic correlations of the categorical traits BCS and MRE with those considered "anchor" in two trait analyses (REA, FTS, HH, LP, and body weight), in which values were positive and had low to moderate magnitude with the exception of the association between BCS and HH equals zero.

DISCUSSION
The good availability of food for the evaluated animals in study is well portrayed by the amount of body condition presented, in which the average was 3.27, higher than the median and mode. Therefore, animals with good tissue composition in terms of muscle mass in the lower back were samped. It is noteworthy, however, that animals which participated in ex hibitions at agricultural fairs were also sampled, and this partly explains the relatively high value of the observed variation coefficient.
In addition, since BCS and MRE are evaluated subjective ly, these traits are greatly influenced by the experience of the appraiser and environmental factors such as food, physiologi cal status, and health, which in this study are embedded in the value of CV 45.71 and 34.11 for BCS and MRE, respec tively, enhancing the role of phenotypic variability that should be considered carefully if included in animal breeding pro grams.
For the carcass fat thickness based on the value of MRE, the average of 2.55 points, that median and mode corrected to 3.0, is greater than what was found by [18] in crossbred goats SRD×Boer (2.10) and SRD×Anglonubian (2.20). Despite these authors having assigned a slightly larger value directly to the carcass, which reinforces the evidence that intramuscular fat condition observed in this study has the potential to meet the market demands, considering that goats are seen as animal lean meat.
The superiority when comparing to the goats values to other genotypes, focusing on meat production, is explained by the case that the animals were intended for breeding were kept under more appropriate management. However, such superiority was also influenced by age, since animals aged 7 months were evaluated. After this age there is the possibility of increased deposition of fat in the carcass [19].
The good body condition and potential of carcass marbling of the animals in study are emphasized when compared to results obtained in sheep. For MRE, the average value was 2.55 points, close to that observed in crossbred undefined sheep breed (SRD)×Santa Inês (2.98), which did not differ statisti cally from crossbred lambs SRD×Texel [20]. Thus, due the lack of studies to assess these traits in goats, the observed values Table 3. Estimates of the (co) variance and heritability (h²) components of body condition score characteristics (BCS), the loin marbling (MRE), ribeye area (REA), fat thickness of the sternum (FTS), hip height (HH), the leg perimeter (LP) and body weight, obtained in two-trait analysis

6
Estimates of genetic parameters via Bayesian Inference were calculated with the aid of GIBBS1F90 and RGIBBS1F90 applications, used for linear and threshold models, respectively [10], by testing chains with 2 d 4 million cycles. The length of the chain used to compare models and generate the posterior distribution of (co) variance components and genetic parameters of the BCS and MRE was 2 million, since both generated ilar estimates. This similarity was considered as the criterion of convergence.
After burn-in of the first 1,000,000 samples samples were taken apart at every 250 cycles (sampling interval), ulting in a posteriori distribution with 4,000 samples in which inferences were performed. Values for burn-in d sampling interval were defined based on preliminary analyses in which the convergence and distribution of ples were evaluated through the POSTGIBBS1F90 program [10], which uses the Geweke diagnostic test ] based on the Z test of average equality of the conditional distribution data logarithm.
The animal model in matrix notation was: y = Xβ+Zα+ε, in which: y = vector of observations of the studied its (underlying scales for categorical traits); X = matrix n×f of incidence (n = total of observations, and f the The accepted assumptions, with a focus on Bayesian methodology, about the information (y) and data (β, α,

68
(fixed effects in frequentist analysis) and order s with a (random genetic additive direct effect); W is a matrix 69 incidence of order r by s; I is an identity matrix of order r by r; and 2 is the residual variance.

70
The conditional probability that that falls into the category j (1, 2, 3   in this study are close to those observed in sheep [21]. Based on the Monte Carlo error (MCE), low values for both the analyses with linear and threshold models confirm the convergence, because according to [16], this occurs when the error value is added to the average estimate of distribution posteriori heritability coefficient, it does not change the value of this estimate, considering the second decimal place, as shown in Table 2.
Therefore, the convergence monitored by Geweke criterion and MCE indicated that the chain size used in this study was appropriate and the amount a posteriori were heritability valid estimates of categorical traits.
There is tendency of BF to be more sensitive to the choice of the prior distribution than the a posteriori interval proba bility [14]. Thus, the threshold model was presented as the greater one with ability to detect the genetic variability in MRE and BCS, when compared to the linear model, according to [3], related to the efficiency of this model for categorical data.
Even considering that in the literature there are few stud ies that compare linear and threshold models for analysis of categorical carcass traits in goats, our results agree with the consulted studies, mostly with cattle and sheep, indicating that the use of threshold model provides more accurate esti mates than the linear model [22,23]. However, studies have not verified differences among models [24].
The lower variability of the genetic components of cate gorical traits, when estimated by linear model, could be an evidence of the inadequacy of its use for traits with this dis tribution, which may result in estimates that cause wrong inferences [22]. Reflections of this are that estimates of heri tability for BCS and MRE traits may be overestimated using this model, leading to genetic progress estimation with low accuracy.
It was found that the use of the threshold model resulted in lower estimates for obtaining the studied heritability of cate gorical traits, because the residual variance estimates differed markedly between threshold and linear models. In addition, as the credibility regions do not overlap, this leads to rejection of statistical hypothesis of equality between the estimates generated by the two models [1].
As seen in the singletrait analysis, it was found that the Geweke convergence criterion presented significance level lower than 0.05, and MCE was low for all analyses (Table 3), implying that the Bayesian analysis with linear threshold model was adequate for obtaining a posteriori distribution estimates of the parameters in twotrait analyses.
It was found in twotrait analyses increased contribution of the considered information number and also influence of the correlations between the characteristics to increase the estimated values for the components of (co) variance and heritability values of the traits in study (Table 3), since there was recovery of part of the additive genetic variance that was incorporated into the residual variance in singletrait analysis (Table 2).
Adipose tissue is the component of the carcass which has the highest variability and influence of the environment [25], which might also be inherent in the MRE behavior that is in tramuscular fat located in the longissimus dorsi. Thus, it is considered that this fact is an explanation for the low herita bility estimates for MRE, and then much of the phenotypic variability of the trait is explained by environmental compo nent, so it is important to pay attention to the environmental factors that influence the phenotypic expression of marbling.
The consulted literature was incipient to the heritability estimates of BCS and carcass marbling in goats. However, the heritability values of the traits MRE and BCS equal to 0.13 and 0.09, respectively. These values higher than the univariate analysis with threshold model, are of low magnitude, then show little potential response to selection based on these char acteristics in the evaluated breed.
So, there is low possibility for response to direct selection or by including these traits in selection indexes. Therefore, considering the producer interest in higher meat production and carcass yield, the index should prioritize animal selection with higher genetic value in REA, if the interest is the finish ing precocity the FTS gains importance.
These categorical traits should not be considered solely in meat goat breeding programs, because besides the amount of meat, the carcass needs adipose protection for cooling, which is guaranteed by the presence of subcutaneous fat, addition ally it can be considered an early indicator of development [26]. Therefore, both REA and FTS should be considered in meat goats breeding programs and the heritability coefficient estimates of 0.28 and 0.24, respectively, both have potential to make beef goats selection indexes.
It is observed in Table 4 that MRE showed moderate genetic association with BCS, so there are common genes responsible for the expression of these traits. Therefore, there is potential for genetic gain by selecting one of them as a phenotypic marker, or both do not necessarily need to be in the same selection index. Thus, it is evident the importance of identi fying correlated response when selection is based on a trait and the intention is also to improve another one, which is defined by the genetic correlation between them, but also in fluences the heritability of traits involved [27].
Identical situation was observed in the genetic correlation between FTS and MRE (0.58). So, considering the ease of mea suring the thickness of subcutaneous fat in the sternal region, it is indication useful means of indirect selection for marbling. The heritability of FTS being 0.24 suggests the inclusion of this trait in the selection indexes for carcass production with good grades of marbling.
Thus, BCS and FTS would be more suitable to be included in the composition of selection indexes, since they have higher heritability, but with equal values to 0.13 and 0.24, respectively, which are considered low values for direct selection purposes (Table 3). However, as the genetic correlations between these traits and marbling were equal to 0.58 (Table 4), this indicates that there is an increasing trend in the carcass marbling.
However, BCS and FTS did not appear to be genetically correlated with each other, which was unexpected and fails to explain the presence of genetic correlation of moderate am plitude of both to the marbling. In other words due to existence of genes in common. However, it is relevant to consider that these traits are measured almost directly in the longissimus dorsi.
Considering that the estimated genetic correlation between MRE and HH was equal to 0.88 (Table 4) and the estimated heritability for HH was 0.32 (Table 3), it is a good indication that the selection based on HH can provide, indirectly, higher genetic progress in the carcass marbling. In this case the HH could be a potential alternative of a phenotypic marker for carcass marbling improvement. This has also the advantage of high heritability, combined with ease of measurement at low cost. However, it can be considered that taller animals grow faster and tend to have less fat [28].
The genetic association of MRE with REA was small (0.03), implying low potential response correlated by selection based on the other one. As REA showed heritability equals to 0.28, considered moderate magnitude and response potential to direct selection, which was also found in cattle. It is recom mended to compose selection indexes in the breed and it will not interfere in carcass marbling. The deposition profile of these tissues in the carcass may be related to this result, be cause as the muscle is earlier developed than the adipose tissue, which implies in low correlation between these characteristics [28].
It is also observed that the genetic correlation between body weight and loin marbling had low magnitude, showing that heavy animals and, consequently, with better body condition, do not necessarily present greater amount of fat in the carcass. This result can be seen as an indication that goats present the "lean meat animal" profile, which has the disadvantage of limiting cryopreservation. On the other hand, it would meet market demands of lowfat products.

IMPLICATIONS
Heritability estimates for categorical traits of BCS and car cass marbling using ultrasound were from low to moderate magnitude, so genetic progress is possible with animal se lection based on these characteristics. The twotrait analysis provided higher estimates of heritability for the traits in study. The thres hold model showed best adjustment for categorical characteristics, BCS, and carcass marbling using ultrasound, with overestimation trend of heritability when the data were submitted to analysis under the linear model. Direct selec tion of the continuous distribution of characteristics, such as thickness sternal fat and HH, allows obtaining the indirect selection for ribeye marbling.

CONFLICT OF INTEREST
We certify that there is no conflict of interest with any financial organization regarding the material discussed in the manu script.