Animals and flock management
Sangsari sheep are primarily raised in Iran for mutton production, making them the primary source of protein in the country. This breed comes in various colors, including black, pea, light brown, and dark brown. Usually, rams and ewes are hornless, and males have horns that appear as small appendages. First mating occurs at 18 months for ewes, while at 18 months or two years of age for rams. The mating season commenced in August and concluded in September. A fertile ram mated with about 10–12 ewes. The lambing season starts in January and lasts until February. Pedigree information, sex, date, and type of birth of the lambs were registered immediately after birth. In approximately 3 months, lambs were weaned. When the pasture conditions are suitable, the sheep are taken daily to pastures around the station, eat the fodder at the pastures during the day, and return to their station in the evening. Usually, sheep are kept indoors and hand-fed in late autumn and winter because of the harsh weather conditions.
Data analysis
Test significance of the environmental factors to be involved in the model was conducted using the general linear model procedure of the SAS program (SAS, 2009). Preliminary analysis shows that the year of lambing (22 classes: 1995 to 2016) and the age of ewe at lambing (6 classes: 2–7 years old) have a significant influence on all reproductive traits (p < 0.01). No significant interaction was found among the fixed effects (p > 0.05), therefore it was not incorporated into the final model. The influence of various fixed effects on reproductive traits was assessed using the following general linear model:
Yijk = µ + Bi + Dj + eijk
In this equation, Yijk, µ, and eijk are the reproductive traits of the ewe, the overall mean of each trait, and residual error, respectively, Bi represents the year of lambing and Dj denotes the age of the ewe at lambing. The least-square mean (LSM) was used to compare the means among subgroups. Utilizing the average information restricted maximum likelihood (AIREML) algorithm, the genetic parameters for the traits under investigation were calculated in the Wombat program (Meyer, 2012). The animal models in use were as follows:
Model 1: y = Xb + Zaa + e
Model 2: y = Xb + Zaa + Wpepe + e
Model 3: y = Xb + Zaa + Zss + e
Model 4: y = Xb + Zaa + Zss + Wpepe + e
Where:
y = Vector of observations for the studied traits; b, a, s, pe, and e = Vectors of fixed effects, direct additive genetic effects, service sire effects, maternal permanent environmental effects, and residual effects, respectively; X, Za, Zs, and Wpe = Design matrices relating fixed effects, direct additive genetic effects, service sire effects, and maternal permanent environmental effects to observations, respectively. The following matrix shows the (co)variance structure:
$$\text{V}\text{a}\text{r}\left(\begin{array}{c}\text{a}\\ \text{s}\\ \begin{array}{c}\text{p}\text{e}\\ \text{e}\end{array}\end{array}\right)=\left(\begin{array}{cc}\begin{array}{c}{\text{A}{\sigma }}_{\text{a}}^{2}\\ 0\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}0\\ {{\text{I}}_{\text{s}}{\sigma }}_{\text{s}}^{2}\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ 0\end{array}\\ {{\text{I}}_{\text{d}}{\sigma }}_{\text{p}\text{e}}^{2}\\ 0\end{array}& \begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ {{\text{I}}_{\text{n}}{\sigma }}_{\text{e}}^{2}\end{array}\end{array}\end{array}\end{array}\end{array}\right)$$
In the structure of this matrix, \({{\sigma }}_{\text{a}}^{2}\), \({{\sigma }}_{\text{s}}^{2}\), \({{\sigma }}_{\text{p}\text{e}}^{2}\), and \({{\sigma }}_{\text{e}}^{2}\) = Variances of direct additive genetic, sire service, maternal permanent environmental, and residual, respectively, and A = Additive relationship matrix. Is, Id, and In = Identity matrices for the numbers of sires, ewes, and records, respectively. It is supposed that the direct additive genetic effects, service sire effects, permanent environmental effects, and residual effects follow a normal distribution with a mean of 0 and variances of \({\text{A}{\sigma }}_{\text{a}}^{2}\), \({{\text{I}}_{\text{s}}{\sigma }}_{\text{s}}^{2}\), \({{\text{I}}_{\text{d}}{\sigma }}_{\text{p}\text{e}}^{2}\), and \({{\text{I}}_{\text{n}}{\sigma }}_{\text{e}}^{2}\), respectively. Based on the model following parameters were calculated:
Direct heritability as: \({\text{h}}^{2}= \frac{{{\sigma }}_{\text{a}}^{2}}{{{\sigma }}_{\text{p}}^{2}}\), service sire varaiance ratio as: \({\text{s}}^{2}= \frac{{{\sigma }}_{\text{s}}^{2}}{{{\sigma }}_{\text{p}}^{2}}\), maternal environmental variance ratio as: \({\text{p}\text{e}}^{2}= \frac{{{\sigma }}_{\text{p}\text{e}}^{2}}{{{\sigma }}_{\text{p}}^{2}}\), and repeatability as: \(\text{r}=\frac{{{\sigma }}_{\text{a}}^{2}+{{\sigma }}_{\text{p}\text{e}}^{2}}{{{\sigma }}_{\text{P}}^{2}}\).
Where, \({{\sigma }}_{\text{a}}^{2}\) = direct genetic variance, \({{\sigma }}_{\text{s}}^{2}\) = service sire variance, \({{\sigma }}_{\text{p}\text{e}}^{2}\) = ewe permanent environmental variance, and \({{\sigma }}_{\text{P}}^{2}\) = phenotypic variance, and, \({{\sigma }}_{\text{P}}^{2}={{\sigma }}_{\text{a}}^{2}+ {{\sigma }}_{\text{s}}^{2}+ {{\sigma }}_{\text{p}\text{e}}^{2}+ {{\sigma }}_{\text{e}}^{2}\). A selection of the best-fitted model was carried out by using Akaike's information criterion (AIC), as illustrated below:
AICi = -2 log Li + 2pi
In this equation, log Li and pi are maximized of the model at convergence and the number of parameters derived from each model, respectively. The best model was considered to be the one with the lowest AIC. Genetic (rg) and phenotypic (rp) associations between reproductive traits were determined using bivariate analyses, incorporating the same fixed effects as the univariate models.