A Comparative Study of the Nonlinear Methods for Estimate Body Weight by Body Measurements on Different Sample Sizes in Morkaraman Sheeps

The objective of this study was to estimate body weight of Morkaraman sheeps from body measurements with nonlinear models. Selected 110 sheeps 3-5 years were scored for body weight, body length, height at wither, chest width and pump width. For determine relationships with body weight between body measurements, correlation analysis was performed. The results of the correlation analysis indicated that the highest relationship according to the all sample sizes were body weight between body length (0.95, 0.90, 0.83, 0.81). Considering all parameters included in the model, the parameter showing the highest correlation with body weight was determined as body length according to all sample sizes. the highest correlation was found in 50 sample sizes (r:0.95). According to the small sample sizes (10-20), Logistic and Saturation growth models can be used to determine the body weight by using body length, on the other hand, Incomplete gamma model is more succesful to estimate body weight when sample size is nearly 30 and 50.


Introduction
The increase in the number and size of cells in certain time intervals in accordance with the type of animal, shaped by the interaction of the genetic structure of living things and the environmental conditions in which they are found, is expressed as growth (Şahin et al., 2014). Body measurements taken from animals give information about their morphological structures and it is known that there is a close relationship between these measurements and the live weights of animals (Çankaya et al., 2009). It is stated that if the degree of relationship between some body measurements and live weight is known, simple measurements on animals with a measuring tape or measuring stick can give the breeder an idea about the live weight of the animal (Akman, 1998). The most common estimation model used to interpret the relationship between body weight and body measurements is the multiple regression model. With multiple regression analysis, characteristics such as live weight and live weight gain can be estimated by using body measurements (Çankaya et al., 2009).
It is concluded that as an alternative to linear models, incomplete gamma and exponential models can be used to predict body weight of sheep using some body measurements (Barragan et al., 2021). Values of live weight can be estimated by taking body measurements at any stage of the fattening period (beginning, middle, end). However, the correlation coe cients to be obtained in this case depend on the fattening period can be different. For example, a correlation that is signi cant at the beginning of fattening may be insigni cant at the end of fattening. For this reason, it can be said that it would be more appropriate to calculate the live weight values in the fattening period from the average values of body measurements. At the same time, since there is a high and statistically signi cant relationship between body weight at the end of fattening and chest circumference, body weight can be estimated using only chest circumference from body measurements (Şahin et al., 2018). South Karaman breed has generally black or blackish ash and dark brown colours, males were horned and females were hornless, has a fatty tail with an "S" shaped extension at the end, there were statistically signi cant correlations between different body measurements and body weight scales and the highest correlation coe cient was determined between body weight and chest circumference, live weight can be estimated safely by measuring the chest circumference in the establishment where it is not possible to weigh the sheep (Bebek and Keskin, 2020).
Selection of the appropriate model requires a statistical decision process, since the live weight varies according to the species, environmental conditions and the trait studied. It has been reported in the literature that although a constant rate of weight gain occurs in certain periods for some characteristics of some living things, the weight increase in living things is not constant throughout their lifetime (Kshirsagar and Smith, 1995;Efe, 1990;Akbaş,1995;Kocabaş et al., 1997).
For this reason, linear models are often insu cient to model the growth of living things over the lifespan (Perotto et al., 1992;Efe, 1990). In the case of periods of different growth rates, it is useful or even necessary to use non-linear models, which are slightly more complex than linear models.
The determination and estimation of non-linear models are more di cult than linear models, and the results are determined iteratively using different methods (Draper and Smith, 1981). If a model cannot be linearized as a result of reparameterization, parameter estimates will not have desirable properties such as unbiasedness, normality, and minimum variance; For this reason, complex estimation methods such as the Marquardt (1963) method may be needed (Ratkowsky, 1983 and). These numerical approaches, which are used instead of analytical solutions, generally produce approximate results. Therefore, the aim of this study is to determine the best model for body weight -body measurements according to the different sample sizes using Allometric, Logistic, Saturation growth, Exponential, and Incomplete gamma non-linear models.

Animals
Tha data from 110 Morkaraman sheep (3-5 years) were taken from Atatürk University application and research farm in Erzurum. Morkaraman sheeps were included in the study as 10, 20, 30 and 50 separately according to sample size.

Statistical Analysis
Correlation coe cients were used to determine the relationship between parameters. In addition, it is aimed to determine the best model according to the sample size in determining the live weight by using the nonlinear models which are described below.
The models were tested for goodness of t by the (MSE) Mean Square Error and (R 2 ), adjusted coe cient of determination (R 2 adj), Akaike information criterion (AIC), Bayes information criterion (BIC) and mean squared prediction error (MEP). The statements of these evaluators are also presented in detail in Silveira et al., 2011.
Considering all parameters included in the model, the parameter showing the highest correlation with body weight was determined as body length according to all sample sizes. As indicated in Table 2, the highest correlation was found in 50 sample sizes (r:0.95). This was followed by sample sizes of 30, 20 and 10, respectively. The highest correlations for the BW parameters between BL were found 0.95, 0.90, 0.83 and 0.81 respectively. In addition, the lowest correlation values are BW between HW (r:0.46), BW between HG (r:0.51) and BW between PW (r:0.48). Considering all sample sizes, body length was included as an independent variable in nonlinear models. Table 3, the results of nonlinear models, in which ve different models for estimate besttted model for relationship between body length and body weight of Morkaraman sheeps at different sample sizes. Considering the different sample sizes, the results of the linear regression model are given in Table 4.

As shown in
According to these results, the R 2 value was 0.66 and the MSE value was 17.16 in the model with a sample size of 10, and the R 2 value was 0.69 and the MSE value was 15.44 in the model with a sample size of 20. In addition, the R 2 value was 0.81 and the MSE value was 13.11 in the model with a sample size of 30, and the R 2 value was 0.90 and the MSE value was 11.08 in the model with a sample size of 50. According to these obtained values, it was found that as the sample size increased, the R 2 value increased and the MSE value decreased. According to different sample sizes, these coe cients showed that, there are more similarity between the (linear and nonlinear) methods. Barragan et.al 2020 reported that according to all nonlinear models R 2 value is calculated higher than 0.75 for estimate body weight from body measurements.
In Table 5 It is concluded that according to the small sample sizes (10-20), Logistic and Saturation growth models can be used to determine the body weight by using body length, on the other hand, Incomplete gamma model is more succesful to estimate body weight when sample size is bigger than 20.

Declarations
Author contribution The design of the study, data collection, statistical analysis, writing of the manuscript, conversion to the journal format, and submission to the journal were done by the author. The author has approved the submitted version.

Funding
This study was not funded by any organization.

Data availability
Not applicable.
Code availability Not applicable.

Ethics approval
In the study, there is no need for an ethical approval due to the lack of blood sampling from the animals and the absence of any surgical procedures. All data were collected with the approval of the breeder.

Con ict of interest
The author declare no competing interests.  Figure 1 Relationships between BW and BL of Morkaraman sheeps according to nonlinear models