Livestock Research for Rural Development 26 (6) 2014  Guide for preparation of papers  LRRD Newsletter  Citation of this paper 
A Bayesian analysis via Gibbs sampling was applied to estimate the genetic parameters and investigate the mode of inheritance of litter size (LS) in Iranian Moghani native sheep breed. The data used in this study were collected at the breeding station of Moghani sheep (Jafar Abad Moghan, Iran) during 19952009. LS was defined as a binary trait (single or multiple births), and estimations of (co) variance components were done by three different thresholdlinear models in Bayesian framework and by using Thrf90 software. Bayesian Segregation analysis was done with a mixed inheritance model by using iBay software. To apply mixed inheritance model, the effect of a major gene was added to the polygenic model. The major gene was modeled as an autosomal biallelic locus with Mendelian transmission probabilities. In both of the analyses, Gibbs sampling and Monte Carlo Markov`s Chain algorithm were used.
The posterior means of direct heritability and repeatability obtained by the best model (model 2) were 0.14 and 0.24 respectively. Fractions of variance due to maternal permanent environmental effects on phenotypic variance were 0.11 for this trait. The results of fitting mixed inheritance model with Bayesian complex segregation analysis did not show the evidence of a major gene affecting LS in Moghani sheep.
Key words: Bayesian analysis, genetic parameters, major gene, Moghani sheep
Moghani sheep breed is one of the most important meat breeds among Iranian native sheep. This is a fattailed breed, categorized among meatwool breeds of sheep (Vatankhah et al 2005), numbering about 5.5 million heads, and is well known for its large body size, tolerance, and capability in producing heavy lambs (Shodja et al 2006). The main place for rearing Moghani breed is Moghan plain of Ardebil province in Iran. The area has a semitropical weather with hot and almost dry summers. Moghani breed is raised through a traditional migratory system, spending summers in the mountainous areas and winters in even lands and pastures. Breeding station of Moghani sheep was established in Jafar Abad of Moghan in order to improve genetic performance of this breed. There is a controlled mating system there, so that the sires and dams of each lamb should be identified. Ewe lambs and ram lambs are bred at 18 months of age. The rams are used only for 1 year, while the ewes may be used for up to 8 years. Male and female lambs are kept in separate flocks from the 6^{th} months of age. The breeding season begins from August till October. The breeding animals are selected primarily based on the general appearance and coat color of animals. The best rams are identified and distributed among the commercial flocks.
Litter size (LS. number of lamb born per ewe lambing) is economically the most important reproductive trait in all systems of sheep production and in all environments (Gallivan 1996; Matika 2003; Vatankhah 2005). Analla et al (1997) showed that energy requirements (metabolism energy) for the production of 1 kg of lamb was about 30% lower for ewes with twin lambs as compared to ewes with singles. More offspring also result in more candidates to select from, and thus, in more selection pressure for any other trait. These indirect effects make LS an important trait for genetic analysis. Efficient selection programs rely on accurate estimation of genetic parameters of traits. But, because of discrete or categorical expression of LS trait, it is difficult to improve this trait. Generally, there are two groups of procedures for analyzing categorical traits: linear model and threshold model. Previous researches have indicated the usefulness of threshold model in analyzing LS trait in sheep, which results in higher heritability estimation, compared to that of linear model (Uriosit and Danell 1987; Olesen et al 1995; Matos et al 1997; Gernande et al 2008). In fact, threshold model was able to explain a large proportion of the variation and, increased the accuracy of these predictions as compared to linear model (Janssens et al 2004). Few studies have been conducted on the estimation of genetic parameter for LS trait of Moghani sheep. Therefore, the main objective of this study was to estimate genetic parameters of LS and investigate whether a segregating major gene affecting LS exists in Moghani native sheep breed.
The data set (for descriptive statistics see Table 1) and pedigree information used in this study were collected at the breeding station of Moghani sheep (Jafar Abad Moghan, Iran) from 1995 to 2009. LS were defined as the total number of lambs born per ewe at every lambing. Prior to analysis, data were checked for discovery of incomplete or aberrant records. Records with missing information were removed. At last 5308 records of 1703 ewe of Moghani sheep breed were used in this study.
Table 1. Descriptive statistics for litter size in Moghani sheep. 

No. of records 
No. of Single birth 
No. of Twin birth 
No. of Triplet birth 
No. of ewes 
mean 
S.D. 
C.V. (%) 
5308 
4076 
1232 
18 
1703 
1.24 
0.43 
35.3 
Preliminary least squares analyses were performed, applying the general linear model (GLM) procedure of SAS (SAS Institute 2004) software to determine significant fixed effects included in final model. The level of significance for including the effects in the model of the analyses was declared at P < 0.02. The final model included the fixed effects of birth year of ewe, lambing year and parity number, the linear covariate effect of dam age (in days), additive direct genetic of ewe, maternal genetic of the dam, and maternal permanent environmental effects.
We considered the threshold Wright (1934) model for LS trait, based on an underlying Gussian random variable. Only 0.003 of all births was triplets. Hence, for statistical analysis, twins and triplets were put in one single class (0.08 of all births); then, we assumed that there is an unobservable random variable, associated with two categories of LS (single and multiple) with only one threshold (t = 1). The following threshold model was used for LS trait.
Where t represents the threshold that defines the two categories of response, nd is the total numbers of data points and I _{ls} is the underlying distribution of LS. The distribution of t was assumed to be fixed. Prior distribution for variance and covariance was inverted Wishart distribution. The Prior distribution for threshold was uniform distribution with starting value calculated from the data.
Three different singletrait animal models were used to estimate co (variance) components and genetic parameters of LS trait:
y = Xβ + Z_{1}u + e (Model1)
y = Xβ + Z_{1}u + Z_{2}pe + e (Model 2)
y = Xβ + Z_{1u}+ Z_{2}pe+ Z_{3}m+e Cov (u, m) = 0 (Model 3)
where y is the vector of observations, β is the vector of fixed effects, X is a design matrix relating fixed effects to y, u is a vector of additive genetic effects of animals, m is a vector of maternal genetic effects, pe is a vector of permanent environmental effects corresponding to the ewes, with incidence matrices Z_{1}, Z_{2}, and Z_{3 }relating the effects to y, and e is a vector of random residual effects. Nonadditive genetic effects were assumed not to exist.
Janss et al (1995) presented a Bayesian approach for segregation analysis in livestock. For segregation analysis, a mixed inheritance model that included nongenetic fixed and random effects, random polygenic effects and fixed effects of major gene was used. The major gene was modeled as an autosomal biallelic locus with Mendelian transmission probability and contained two alleles A_{1} and A_{2} with frequencies of q and 1q, respectively. Three genotypes A_{1}A_{1}, A_{1}A_{2}, and A_{2}A_{2} were assumed to occur in HardyWeinberg frequencies, q^{2}, 2q (1q) and (1q^{2}) in the base population, with effect on LS a, d and, a, respectively. The equation for the mixed inheritance model is written as follows:
y = Xβ + Zu + ZWm + e
Where y is a vector of observation, β is a vector of fixed effects of birth years, parity years and parity number, u is a vector of random additive polygenic effects, e is a vector of random errors, and X and Z are incidence matrices of relating fixed and genetic effects to observations. Vector m contains unknown additive (a) and dominance (d) genotypic values related to genotypes, m` {a, d, a}. W is unknown three column matrix that represents genotype configuration. Three columns correspond to possible genotypes at a major locus, and each row contains two 0 and 1 to indicate the genotype of an individual. Hence Wm is a vector of single gene effects.
Prior distributions for the frequency of positive allele (q) and additive and dominance genotypic values were uniform and defined as [0, 1], [0, ∞>] and <−∞, ∞>, respectively. The distributional assumption for U was U ~ N (0, Aσ^{2}_{u}) where A is the numerator relationship matrix and σ^{2}_{u}is the polygenic variance. The relationship matrix of full pedigree was used in the analyses. Distributional assumptions for e were specified as e ~ N (0, Iσ^{2}e), where σ^{2}e is an error variance component with uniform prior distribution on <0, ∞>. The complete set of parameters for the specified model was θ1 = (b, u, σ^{2}u, W, q, a, d, σ^{2}e. The variance explained by a single gene (σ^{2}m) is defined as (Falconer et al 1996):
σ^{2}m = 2q (1 − q) [a + d (1 – 2q)]^{2} + [2q (1 − q) d] ^{2}.
The Gibbs sampler was used to generate random samples from the marginal posterior distribution by sampling successively from the full conditional distribution of the parameters. To estimate the genetic parameters, a single chain of 50000 samples were generated by using Thrf90 software (Misztal et al 2002). The first 10000 samples were considered to belong to the "burnin" period and discarded. Samples were saved every100 iterations. Then, 400 samples were used for statistical inference of genetic parameters. Gibbs sampling algorithm with blocked genotypic sampling (Janss et al 1995) was used for inference in the mixed model and implemented, using iBay software (Janss et al 2008). To start the Gibbs chain, zero was given as the starting value for β and u, a reasonable guess based on the literature was used as initial values for σ^{2}u and σ^{2}e, genotypes of major gene initialized as heterozygotes and initial allele frequency was 0.5. A single chain of 500000 iterations and a burnin period of 40000 iterations were run for Bayesian complex segregation analyses. Samples were saved every 50 iterations. Therefore, densities were estimated from 9160 independent joint samples of the parameters, generated via Gibbs sampling.
For inference of genetic parameters, posterior mean and standard deviation of samples were computed. Posterior means were used as point estimates for parameters. For gene segregation analysis, Gibbs sampling algorithm was applied to sample directly marginal posterior densities of the following parameters: error variance(σ^{2}_{e}), variance of polygenic effect (σ^{2}_{u}),frequency of major gene allele (q), additive effect of major gene (a) and dominance effect of major gene (d). Statistical inference focused on genetic variance components (σ^{2}_{u}, σ^{2}_{m)} and in particular major gene variance (σ^{2}_{m}). According to Bayesian theory, the presence of zero in right or left side of 95% highest posterior density region (HPDR 95%) of σ^{2}_{m} means the lack of an evidence of major gene segregation.
Descriptive statistics of Moghani sheep has been represented in table 1. In this study the mean and standard deviation of LS trait in Moghani sheep during 1995 to 2009 years were 1.24 ± 0.43 lambs per ewe lambing. Mokhtari and et al (2010) reported the mean and standard deviation of 1.05 ± 0.21lambs per ewe lambing for LS trait of Kermani sheep breed of Iran. Also, Hanford et al (2003) reported a mean of 1.33 lambs per ewe lambing for LS trait of Rambouillet sheep. In Vatankhah and Talebi (2008) study, the mean and standard deviation of LS per ewe lambing trait was lower than this study (1.17 ± 0.38).
Marginal posterior distribution of the (co) variance components and genetic parameters are summarized in table 2, in which the mean, mode, median, standard deviation (SD) and the 95% highest posterior density region (HPDR95%) are shown. The mean for direct heritability, and repeatability obtained by the best model (model 2) were 0.14 and 0.24 respectively. Our criterion for choosing the best model was the residual variance. As the results show, model 2 has the least residual variance. Fractions of variance due to maternal permanent environmental effects on phenotypic variance were 0.11 for this trait. These results indicate that LS trait in Moghani sheep is influenced by maternal permanent invironmental effects (nutrition, management). According to results, the estimated error variance for each parameter is normal. A Histogram of the marginal posterior distributions of litter size heritability with three different models is presented in figure 1.
Table 2. Genetic parameter estimates from three models of analyses of litter size trait 

Parameters and Models 
Median 
Mode 
m(± S.E) 
HPD (95%) 
Model 1 

0.03 – 0.07 
0.05 
0.05 
0.05 ± 0.01 
σ^{2}a 
0.18 – 0.26 
0.22 
0.22 
0.22 ± 0.02 
σ^{2}e 
0.22 – 0.28 
0.25 
0.25 
0.27 ± 0.02 
σ^{2}p 
0.14 – 0.26 
0.20 
0.19 
0.20 ± 0.03 
h^{2}_{d} 
Model 2 

0.02 – 0.06 
0.03 
0.03 
0.04 ± 0.01 
σ^{2}a 
0.01 – 0.05 
0.031 
0.035 
0.03 ± 0.01 
σ^{2}pe 
0.17 – 0.25 
0.22 
0.22 
0.21 ± 0.02 
σ^{2}e 
0.11 – 0.13 
0.095 
0.10 
0.14 ± 0.05 
h^{2}_{d} 
0.07 – 0.15 
0.12 
0.13 
0.11 ± 0.04 
C^{2} 
0.18  0.28 
0.22 
0.20 
0.24 ± 0.03 
r 
Model 3 

0.01 – 0.07 
0.03 
0.04 
0.04 ± 0.03 
σ^{2}a 
0.14 – 0.67 
0.04 
0.2 
0.25 ± 0.21 
σ^{2}m 
0.14 – 0.17 
0.16 
0.16 
0.16 ± 0.01 
σ^{2}pe 
0.07 – 2.37 
1.02 
0.62 
1.17 ± 0.6 
σ^{2}e 
0.01 –0.07 
0.02 
0.02 
0.03 ± 0.02 
h^{2}_{d} 
0.16 – 0.50 
0.14 
0.15 
0.18 ± 0.16 
h^{2}_{m} 
0.00 – 0.22 
0.10 
0.11 
0.12 ± 0.06 
C^{2} 
0.06  0.18 
0.10 
0.11 
0.12 ± 0.3 
r 
σ^{2}a: additive genetic variance σ^{2}m: maternal additive genetic variance σ^{2}pe: maternal permanent environmental variance σ^{2}e: residual variance σ^{2}p: Phenotype variance h^{2}_{d}:_{ }direct heritability h^{2}_{m:} maternal heritability C^{2}: ratio of maternal permanent environmental effects to phenotypic variance. HPDR (95%): 95% highest posterior density region r: repeatability. 
Figure 1. Marginal posterior distributions of LS trait heritability with model1 (h^{2}_{1}), marginal
posterior distributions of LS trait heritability with model2 (h^{2}_{2}), and marginal posterior distributions of LS trait heritability with model 3(h^{2}_{3}). 
There is a
considerable range in heritability estimates of
LS
trait
for
different breeds and statistical models.
Bradford
(1985) summarized over 30 estimates for different breeds and methods and pointed
out that heritability of LS trait is quite low, and
reported a mean of 0.10. Also, Safari et al (2005) reported the average
heritability for LS to be 0.10.
Altarriba
et al (1998) obtained an estimate of h^{2} for
LS
trait
of .077 in Rasa Aragonesa sheep using a threshold model. In Brien et al (2002)
study, heritability of LS of Merino reported .08 and 0.12 with linear and
threshold model.
Estimation of direct heritability in this study (0.14) was higher than those reported by some others (Rao and Notter 2000; Rosati et al 2002:Vatankhah et al 2008; Hanford et al 2002; Hanford et al 2003; Bromley et al 2000; Ligda et al 2000). Direct heritability estimates for LS trait given by Yazdi et al (1999) for Baluchi sheep (0.43) and Snyman (1998) for Afrino sheep (0.27) was higher than this study and the estimated heritability for LS trait of Turkish Merino was also 0.025 and lower than this study (Ekiz et al 2005 ). But average modal value of 0.10 for the direct heritability of LS in this study was in the range of estimates (.02 to .45) reported in the literature by Fogarty (1995).
The mean of Marginal posterior distribution of variance due to permanent environmental effects (c^{2}) of ewe (0.11) indicates the evidence of maternal effect on LS trait in Moghani sheep. The average modal value of 0.13 in this study was higher than those reported by others ((Rao and Notter 2000; Ligda et al 2000; Rosati et al 2002; Altarriba et al 1998; ; Hanford et al 2003; Vatankhah et al 2008). But estimated value of c^{2} in this study was similar to reported value of this parameter in Ekiz et al. (2005) for Turkish Merino.
The mean of marginal posterior distribution of repeatability (r) obtained with the best model was 0.24. In Vatankhah and Talebi study (2008) this parameter was reported more than this study (0.28). But repeatability of LS trait of Turkish Merino in Ekiz et al. (2005) study was lower than this study (0.125).
As mentioned before, LS is a categorical trait, and difficulties with the statistical analysis of such traits have led to the implementation of approximate methods that have yielded a wide range of estimates (Altariba et al 2010). Threshold model and Bayesian approach with Gibbs sampler, applied in this study, avoids the use of analytic approximate. This procedure produces the Monte Carlo estimates of the marginal posterior distribution of heritability, which provides for a rich inference framework.
Many researchers have addressed major genes or chromosome location across population. Because of low heritability of LS, identification of major genes affecting LS in sheep population could have a considerable impact on genetic improvement of this trait. Several genes influencing LS have been reported in sheep (Davis 2005).The result of segregation analysis with linear and threshold mixed model are presented in tables 3. As table shows, in linear model both the left and right sides and in threshold Analysis left side of HPDR 95% are zero. Based on the results of this study, it can be concluded that, there is no significant major gene involved in the mode of inheritance of LS in this breed, and that the trait is controlled in a purely polygenic manner in it. Present result was in contrast with the earlier one that reported the existence of single gene affecting LS trait of sheep. The densities of marginal posterior estimates for polygenic variance (σ^{2}_{u}) and major gene variance (σ^{2}_{m}) are presented in figures 2 and 3.
Table 3. Estimated marginal densities for polygenic (σ^{2}_{u}) and major gene (σ^{2}_{m}) variance with both linear and threshold models. 

Parameters and Models 
Mean 
HPD (95%) left 
HPD (95%) right 
Linear model 



σ^{2} u 
0.00 
0.00 
0.00 
σ^{2} m 
0.01 
0.00 
0.06 
Threshold model 



σ^{2} u 
0.15 
0.01 
0.42 
σ^{2} m 
33.2 
0.00 
212 
σ2u :Polygenic additive variance; σ2m: major gene variance; 

Figure 2. Estimated marginal densities for polygenic (σ^{2}_{u}) and major gene (σ^{2}_{m}) variance with linear model. 
Figure 3. Estimated marginal densities for polygenic (σ^{2}_{u}) and major gene (σ^{2}_{m}) variance with Threshold model. 
Estimation of co (variance) components and heritability is necessary for genetic evaluation of sheep and choosing the best selection schemes. Although, results from this study indicated that LS has low heritability in Moghani sheep, since it is economically the most important reproductive trait, a selection based on LS performance over a long period of time could result in a moderately positive response in this trait. On the other hand, because LS trait in Moghani sheep is influenced by maternal permanent invironmental effects, the improvement of nongenetic factors in flocks, such as dam nutrition before mating and reproductive management must also be adequate to obtain satisfactory response.
The authors acknowledge Moghani sheep breed station staff (Jafar Abad Moghan, Iran) especially Mr. Bayeriyar for providing data for this research, also great thanks to Dr L. L. G. Janss for supplying iBay software.
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Received 1 February 2014; Accepted 17 April 2014; Published 1 June 2014