Possibilities of using Cubic Spline functions in estimation of live weight of Kıvırcık lambs fed with different silage types

doi:10.21203/rs.3.rs-1251628/v1

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Possibilities of using Cubic Spline functions in estimation of live weight of Kıvırcık lambs fed with different silage types

https://doi.org/10.21203/rs.3.rs-1251628/v1

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In this study, the possibilities of using Cubic Spline functions in the estimation of live weights of Kıvırcık lambs fed with different silage types were investigated. In the experiment, 40 male Kıvırcık lambs aged 2.5-3 months were used and the animals were fattened for 56 days. In the study, Cubic Spline functions were used as live weight estimation model in lambs fed with silage containing 5 different ratios of corn and sunflower (100% sunflower silage, 75% sunflower + 25% corn silage, 50% sunflower + 50% corn silage, 25% sunflower + 75% corn silage, 100% corn silage). For this purpose, knots points were created by considering the visual investigation of the point distribution. The coefficient of determination (R²) was used to determine the fit of the model in constructing the Cubic Spline functions. For the live weight model of lambs, 5 different Cubic Spline functions were obtained. With the help of the functions obtained, live weight estimation was made for different silage groups and for different ages, and for the days when no measurements were made. According to the estimations, while the highest live weight gains were observed on the 20th and 120th days, the lowest body weight gain was observed on the 90th day. As a result, it has been determined that Cubic Spline functions can be used successfully in the estimation of live weight of fattened lamps.

Kıvırcık

silage types

live weight

Cubic Spline

Various methods are used to create a live weight estimation model in all living things and accordingly to estimate live weight at certain ages, one of these methods is Cubic Spline functions. Examining the point distributions of dependent and independent variables by separating them from certain points (knots) is called Cubic Spline function or Piecewise regression. In Cubic Spline functions, all point distributions are expressed using interval functions instead of a single function (Şahin, 2009).

Cubic Spline functions are not yet well known by researchers working in the field of animal husbandry, and therefore they are little used. The difficulty of choosing the models suitable for some point distributions obtained, or having to choose the best one among the existing models, causes researchers to ignore the relationships to be obtained with these distributions most of the time. Especially when the shape of point distributions cannot be explained by known models, the ability to obtain functions with high fit perfection with appropriate knots increases the interest and need for Cubic Spline functions. Some studies on this subject are listed below; Çağlar (1993) examined the solutions of multi-point linear boundary value problems using spline functions and compared the results with the finite difference method. Wheatley and Gerald (1994) theoretically examined the creation of Cubic Spline functions and the coefficient equations of interval functions. Comets et al. (1999) studied octreotide absorption in rabbits with non-parametric models and used Restricted Cubic Piecewise regressions to model individual absorption profiles. White et al. (1999) used Cubic Piecewise regression and Restricted Cubic Piecewise regression models for modeling lactation curves in dairy cattle. In their study, the researchers compared the results obtained from Cubic Piecewise and Constrained Cubic Piecewise regressions with the results obtained with the Wilmink model and recommended the use of Piecewise regression models because they are more flexible. Huisman et al. (2002) used Cubic Piecewise and Polynomial regressions to examine the relationship between age, live weight gain and heredity, using data records from pigs. Druet et al. (2003) investigated genetic parameter estimation and modeling of lactation curves from first lactation dairy records in Holstein cattle with different models such as polynomial, Ali and Schaffer, Wilmink and Piecewise regression. Degroot (2004) used Cubic Piecewise regression in modeling the lactation curve and estimating genetic parameters in Holstein cattle given Somatotropin hormone. Iwaisaki et al., (2005) used Linear Piecewise regression method to estimate maternal genetic parameters in the early stages of growth in beef cattle. Cubic Piecewise regressions were used for the change in measurement values of the otoliths used in age determination in fish based on time (Black et al., 2005), modeling of lactation yields in Holstein cattle and estimation of genetic parameters (Degroot et al., 2007). In another study, Cubic Spline regression was used to model lactation curves using milk yield records from cattle (Şahin and Efe, 2010).

In this study, it was aimed to model the live weights of lambs fed with corn and sunflower silage in 5 different ratios from birth to 140 days with Cubic Spline functions and to make an estimate of live weight on days without measurement.

Material

In the study, 40 Kıvırcık male lambs aged 2.5-3 months and an average live weight of 23-25kg were used as animal material. The fattening lambs were housed in individual compartments during the experiment and individual feeding was applied to the animals during the 56-day fattening period. This study was carried out in a semi-open barn in a sheep farm belonging to Bursa Uludağ University Agricultural Application and Research Center. During the trial period, the live weights and feed consumptions of the lambs were determined individually and in 2-week periods.

During the experiment, lambs were fed 5 different silages (100% sunflower silage, 75% sunflower + 25% corn silage, 50% sunflower + 50% corn silage, 25% sunflower + 75% corn silage, 100% corn silage) as pure and mixed. Lambs housed in individual chambers consumed the silage mixtures of their groups ad libitum. In addition to the silage mixtures consumed by the lambs, 700 g of concentrate feed per animal was given in the first 4 weeks of the experiment. Later, this amount was increased to 900 g for 4 weeks, and to 1400 g in the last 2 weeks of the experiment, taking into account the daily nutrient needs of the lambs. Fresh and clean drinking water was always available in front of the lambs. During the fattening period, the live weights of the lambs were determined by control weighing made every 14 days.

Method

The Spline expression was first used by Schoenberg (1946) and she worked on the use of spline functions in differential equations and spline theory (Schoenberg, 1946; Walkley, 1999).

Splines are k-order Piecewise polynomials. The joining points of the parts are knots. In deciding the knots, the function values and the first k-1 derivative should be known. Cubic spline (k=3) is sufficient for solving practical problems. A Cubic Spline function with h number of knots $({t}_{1}<{t}_{2}<\dots <{t}_{h})$ with continuous first and second order derivatives can be written as follows (Montgomery et al., 2012).

$$S\left(x\right)=\sum _{j=0}^{3}{\beta }_{0j}{x}^{j}+\sum _{i=1}^{h}{\beta }_{i}{\left(x-{t}_{i}\right)}_{+}^{3} \left(1\right)$$

Here,${\left(x-{t}_{i}\right)}_{+}$

$${\left(x-{t}_{i}\right)}_{+}=\left\{\begin{array}{c}\left(x-{t}_{i}\right) if x-{t}_{i}>0\\ 0 if x-{t}_{i}\le 0\end{array}\right. \left(2\right)$$

is defined as (Smith, 1979; Eubank, 1984). If the positions of the knots are the parameter to be estimated, the resulting problem is a nonlinear one. Equation (1) is an application of linear least squares when the positions of the knots are known. In a Cubic Spline (1) model, if all (h+1) polynomial parts are of 3rd order then it is a Cubic Spline model without continuity constraints.

$$S\left(x\right)=\sum _{j=0}^{3}{\beta }_{0j}{x}^{j}+\sum _{i=1}^{h}\sum _{j=0}^{3}{\beta }_{i}{\left(x-{t}_{i}\right)}_{+ }^{j} \left(3\right)$$

Here (x-t)_+^0 is 1 if x>t and 0 if x≤t (Montgomery et al., 2012).

A Cubic Spline model with one knot at t and no continuity constraint can be expressed as follows (Montgomery et al., 2012). Interval functions for equation (2) is expressed as;

$$S\left(x\right)=\left\{\begin{array}{c}{\beta }_{00}+{\beta }_{01}x+{\beta }_{02}{x}^{2}+{\beta }_{03}{x}^{3}, x\le t \\ {\beta }_{00}+{\beta }_{01}x+{\beta }_{02}{x}^{2}+{\beta }_{03}{x}^{3}+{{\beta }_{10}+\beta }_{11}\left(x-t\right)+{\beta }_{12}{\left(x-t\right)}^{2}+{\beta }_{13}{\left(x-t\right)}^{3}, x>t\end{array}\right.$$

(Freund et al., 2006; Harrell, 2017).

The values of the live weight averages of the lambs fed with different silage types on different days are given in Table 1.

Table 1

Live weights of lambs fed with different silage types and at different rates according to the age groups.
Days	1st group	2nd group	3rd group	^4th group	5th group
0	4.10	3.97	4.56	4.38	4.15
84	24.29	24.23	25.10	24.41	23.65
98	25.16	25.18	26.76	24.85	25.23
112	29.06	29.96	30.33	28.95	29.53
126	33.75	34.05	34.09	32.46	33.11
140	37.44	37.36	37.38	35.94	36.25
1st group: 100% corn silage, 2nd group: 75% corn + 25% sunflower silage, 3rd group: 50% corn + 50% sunflower silage, 3rd group: 25% corn + 75% sunflower silage, 5th group: 100% sunflower silage.

Birth weight, 84, 98, 112, 126 and 140. day live weights were used to create the Cubic Spline function of the experimental groups. Cubic Spline function models were investigated for different types of silage (pure or mixed). The coefficient of determination (R²) was calculated as 0.9999 in all of these models evaluated for silage types. It has been possible to make the best estimation by creating node values with the Cubic Spline function. Cubic Spline functions of Kıvırcık lambs fed with silage containing corn and sunflower in different ratios from birth to 140 days are summarized as follows;

Live weight of lambs fed 100% corn silage

Cubic Spline functions were applied for the model showing the growth of Kıvırcık lambs fed with 100% corn silage and these functions are given below;

$$f\left({x}_{1}\right)=-0.0001 {\left(x-0\right)}^{3}+0.0079 {\left(x-0\right)}^{2}+4.1 0\le x<84$$

$$f\left({x}_{2}\right)=0.0006 {\left(x-84\right)}^{3}-0.0073 {\left(x-84\right)}^{2}+0.0554 \left(x-84\right)+24.29 , 84\le x<98$$

$$f\left({x}_{3}\right)=-0.0006 {\left(x-98\right)}^{3}+0.0160 {\left(x-98\right)}^{2}+0.1774 \left(x-98\right)+25.16 , 98\le x<112$$

$$f\left({x}_{4}\right)=0.0011 {\left(x-112\right)}^{3}-0.0103 {\left(x-112\right)}^{2}+0.2572 \left(x-112\right)+29.06, 112\le x<126$$

$$f\left({x}_{5}\right)=-0.0046 {\left(x-126\right)}^{3}+0.0372 {\left(x-126\right)}^{2}+0.6346 \left(x-126\right)+33.75, 126\le x<140$$

At any time (days) desired to be estimated in these functions, the live weight estimation of lambs can be calculated as follows. For example, the Cubic Spline function can be used to estimate the live weight of a lamb at day 90.

$$f\left({x}_{2}\right)=0.0006 {\left(x-84\right)}^{3}-0.0073 {\left(x-84\right)}^{2}+0.0554 \left(x-84\right)+24.29 , 84\le x<98$$

Because the calculation of the live weight estimation for the 90. day is suitable for the definition range of this function. By replacing the 90 value in this function, the live weight estimation value can be calculated approximately.

$$f\left({x}_{2}\right)=0.0006 {\left(90-84\right)}^{3}-0.0073 {\left(90-84\right)}^{2}+0.0554 \left(90-84\right)+24.29 , 84\le x<98$$

$$f\left({x}_{2}\right)=0.0006 {\left(6\right)}^{3}-0.0073 {\left(6\right)}^{2}+0.0554 \left(6\right)+24.29 , 84\le x<98$$

$$f\left({x}_{2}\right)=0.0006 {\left(6\right)}^{3}-0.0073 {\left(6\right)}^{2}+0.0554 \left(6\right)+24.29$$

$$f\left({x}_{2}\right)=0.0006*216-0.0073*36+0.0554*6+24.29=24.30$$

Similarly, the Cubic spline function is used to estimate body weight at day 110.

$$f\left({x}_{3}\right)=-0.0006 {\left(x-98\right)}^{3}+0.0160 {\left(x-98\right)}^{2}+0.1774 \left(x-98\right)+25.16 , 98\le x<112$$

X=110 is put in the function and calculated as follows.

$$f\left({x}_{3}\right)=-0.0006 {\left(110-98\right)}^{3}+0.0160 {\left(110-98\right)}^{2}+0.1774 \left(110-98\right)+25.16, 98\le x<112$$

$$f\left({x}_{3}\right)=-0.0006 {\left(12\right)}^{3}+0.0160 {\left(12\right)}^{2}+0.1774 \left(12\right)+25.16 , 98\le x<112$$

$$f\left({x}_{3}\right)=-0.0006*1728+0.0160*144+0.1774*12+25.16=28.39$$

For X=120,

$$f\left({x}_{4}\right)=0.0011 {\left(x-112\right)}^{3}-0.0103 {\left(x-112\right)}^{2}+0.2572 \left(x-112\right)+29.06 , 112\le x<126$$

The Cubic Spline function is used.

$$f\left({x}_{4}\right)=0.0011 {\left(120-112\right)}^{3}-0.0103 {\left(120-112\right)}^{2}+0.2572 \left(120-112\right)+29.06 , 112\le x<126$$

$$f\left({x}_{4}\right)=0.0011 {\left(8\right)}^{3}-0.0103 {\left(8\right)}^{2}+0.2572 \left(8\right)+29.06 , 112\le x<126$$

$$f\left({x}_{4}\right)=31.78$$

was calculated as.

The estimated live weight of lambs every 10 days can also be calculated, fed on silages containing all different proportions of corn and sunflower in the 10-150 day range. The Cubic Spline function graph showing the growth in Kıvırcık lambs is given in Figure 1.

Live weight of lambs fed 75% corn + 25% sunflower silage

A model describing the growth of Kıvırcık lambs fed with 75% corn + 25% sunflower silage was created using the Cubic Spline function. Obtained Cubic Spline functions are as follows;

$$f\left({x}_{1}\right)=-0.0001 {\left(x-0\right)}^{3}+0.0081 {\left(x-0\right)}^{2}+3.97 0\le x<84$$

$$f\left({x}_{2}\right)=0.0007 {\left(x-84\right)}^{3}-0.0076 {\left(x-84\right)}^{2}+0.0411 \left(x-84\right)+24.23 , 84\le x<98$$

$$f\left({x}_{3}\right)=-0.0009 {\left(x-98\right)}^{3}+0.021 {\left(x-98\right)}^{2}+0.2283 \left(x-98\right)+25.18 , 98\le x<112$$

$$f\left({x}_{4}\right)=0.0014 {\left(x-112\right)}^{3}-0.0178 {\left(x-112\right)}^{2}+0.2736 \left(x-112\right)+29.96, 112\le x<126$$

$$f\left({x}_{5}\right)=-0.0046 {\left(x-126\right)}^{3}+0.0395 {\left(x-126\right)}^{2}+0.578 \left(x-126\right)+34.05, 126\le x<140$$

The graph of live weights is presented in Figure 2.

Live weight of lambs fed 50% corn + 50% sunflower silage

Live weight function of Kıvırcık lambs fed 50% corn + 50% sunflower silage was determined by Cubic Spline function as follows;

$$f\left({x}_{1}\right)=-0.0001 {\left(x-0\right)}^{3}+0.0073 {\left(x-0\right)}^{2}+4.56 0\le x<84$$

$$f\left({x}_{2}\right)=0.0004 {\left(x-84\right)}^{3}-0.0059 {\left(x-84\right)}^{2}+0.1187 \left(x-84\right)+25.10 , 84\le x<98$$

$$f\left({x}_{3}\right)=-0.0006 {\left(x-98\right)}^{3}+0.0118 {\left(x-98\right)}^{2}+0.201 \left(x-98\right)+26.76 , 98\le x<112$$

$$f\left({x}_{4}\right)=0.0012 {\left(x-112\right)}^{3}-0.012 {\left(x-112\right)}^{2}+0.1979 \left(x-120\right)+30.33, 112\le x<126$$

$$f\left({x}_{5}\right)=-0.0045 {\left(x-126\right)}^{3}+0.0392 {\left(x-126\right)}^{2}+0.5782 \left(x-126\right)+34.09, 126\le x<140$$

The graph of the live weights in the 3rd group is presented in Figure 3.

Live weight of lambs fed 25% corn + 75% sunflower silage

The live weight function of Kıvırcık lambs fed 25% corn + 75% sunflower silage was explained by the Cubic Spline function as follows;

$$f\left({x}_{1}\right)=-0.0001 {\left(x-0\right)}^{3}+0.0084 {\left(x-0\right)}^{2}+4.38 0\le x<84$$

$$f\left({x}_{2}\right)=0.0007 {\left(x-84\right)}^{3}-0.0082 {\left(x-84\right)}^{2}+0.012 \left(x-84\right)+24.41 , 84\le x<98$$

$$f\left({x}_{3}\right)=-0.0009 {\left(x-98\right)}^{3}+0.0206 {\left(x-98\right)}^{2}+0.1855 \left(x-98\right)+24.85 , 98\le x<112$$

$$f\left({x}_{4}\right)=0.0015 {\left(x-112\right)}^{3}-0.0182 {\left(x-112\right)}^{2}+0.2189 \left(x-120\right)+28.95 , 112\le x<126$$

$$f\left({x}_{5}\right)=-0.0047 {\left(x-126\right)}^{3}+0.0433 {\left(x-126\right)}^{2}+0.5697 \left(x-126\right)+32.46, 126\le x<140$$

The graph of the live weights in this group is presented in Figure 4.

Live weight of lambs fed 100% sunflower silage

The live weight function of Kıvırcık lambs fed with 100% sunflower silage is explained by the Cubic Spline function as follows;

$$f\left({x}_{1}\right)=-0.0001 {\left(x-0\right)}^{3}+0.0072 {\left(x-0\right)}^{2}+4.15 0\le x<84$$

$$f\left({x}_{2}\right)=0.0005 {\left(x-84\right)}^{3}-0.0061 {\left(x-84\right)}^{2}+0.094 \left(x-84\right)+23.65 , 84\le x<98$$

$$f\left({x}_{3}\right)=-0.0008 {\left(x-98\right)}^{3}+0.0161 {\left(x-98\right)}^{2}+0.2353 \left(x-98\right)+25.23 , 98\le x<112$$

$$f\left({x}_{4}\right)=0.0014 {\left(x-112\right)}^{3}-0.0169 {\left(x-112\right)}^{2}+0.2249 \left(x-120\right)+29.53 , 112\le x<126$$

$$f\left({x}_{5}\right)=-0.0046 {\left(x-126\right)}^{3}+0.0404 {\left(x-126\right)}^{2}+0.5538 \left(x-126\right)+33.11, 126\le x<140$$

The graph of the live weights in this group is presented in Figure 5.

Estimated live weights at 10-day intervals for all experimental groups in the 10-150-day age range are presented in Table 2 and Figure 6.

Table 2

Estimated live weights for different age periods.
Days	1st group	2nd group	3rd group	^4th group	5th group	Max. weight	Min. weight
10	7.85	7.69	7.94	8.34	7.36	4. grup	5. grup
20	11.24	11.05	11.06	11.90	10.31	4. grup	5. grup
30	14.26	14.07	13.90	15.05	13.01	4. grup	5. grup
40	16.92	16.72	16.49	17.78	15.47	4. grup	5. grup
50	19.22	19.03	18.84	20.06	17.72	4. grup	5. grup
60	21.15	20.98	20.95	21.90	19.70	4. grup	5. grup
70	22.72	22.58	22.83	23.28	21.49	4. grup	5. grup
80	23.91	23.83	24.49	24.18	23.07	3. grup	5. grup
90	24.30	24.11	25.57	24.60	23.95	3. grup	5. grup
100	25.56	25.71	27.18	25.29	25.74	3. grup	4. grup
110	28.39	29.23	29.77	28.32	28.90	3. grup	4. grup
120	31.78	32.48	32.52	31.09	31.72	3. grup	4. grup
130	34.97	35.01	35.09	33.36	33.99	3. grup	4. grup
150	38.65	40.04	39.31	39.61	39.07	2. grup	1. grup

Lambs fed with 100% corn silage had higher live weight gain at both the measured and estimated ages (Table 2). In lambs fed with 75% corn + 25% sunflower silage, it was observed that the live weight gain was higher after the 100. day. Lambs fed 50% corn + 50% sunflower, 25% corn + 75% sunflower and 100% sunflower silages showed a faster increase in live weight from the 110. day compared to other periods.

Except for the measurement days (between 10-150 days), live weight gains were calculated using the estimated live weight values at 10-day intervals between days and are given in Table 3.

Table 3

Values of the live weight gains of the lambs belonging to the experimental groups.
Days	1st group	2nd group	3rd group	4th group	5th group
20	3.39	3.36	3.12	3.56	2.95
30	3.02	3.02	2.84	3.15	2.7
40	2.66	2.65	2.59	2.73	2.46
50	2.3	2.31	2.35	2.28	2.23
60	1.93	1.95	2.11	1.84	2.0
70	1.57	1.6	1.88	1.38	1.79
80	1.19	1.25	1.66	0.9	1.58
90	0.39	0.28	1.08	0.42	0.88
100	1.26	1.6	1.61	0.69	1.79
110	2.83	3.52	2.59	3.03	3.16
120	3.39	3.25	2.75	2.77	2.82
130	3.19	2.53	2.57	2.27	2.27
140	2.47	2.35	2.29	2.58	2.26
150	1.21	2.68	1.93	3.67	2.82

As seen in Table 3, the highest live weight gain in lambs fed with 100% corn silage was on the 20. - 120. days (3.39 kg) and 130. day (3.19 kg), while the lowest body weight gain was on the 90. (0.39 kg) and 80. days (1.19 kg). In lambs fed 75% corn + 25% sunflower silage, the highest live weight gains were 3.36, 3.52 and 3.25 kg on the 20., 110. and 120. days, respectively, while the lowest weight gains were 0.28 kg on the 90. day and 1.25 kg on the 80. day. The highest live weight gains of lambs fed with 50% corn + 50% sunflower silage were 3.12, 2.84 and 2.75 kg on the 20., 30. and 120. days, respectively, while the lowest weight gain was 1.08 kg on the 9. day. The highest body weight gain values were determined as 3.56, 3.15 and 3.67 kg on the 20., 30. and 150. days, respectively, in the lambs fed with 25% corn + 75% sunflower silage, and the lowest weight gain was 0.42 kg on the 90. day and 0.69 kg on the 100. day. In lambs fed 100% sunflower silage, the highest body weight gain was 2.95, 3.16 and 2.82 kg on the 20., 110. and 120. days, respectively, and the lowest 0.88 kg on the 90. day.

In general, the lowest body weight gain was estimated at 90. day in all experimental groups. The highest live weight gains were determined on the 20. and 120. days, and it was determined that the said increases were seen in the first and last days of the fattening period. The graph showing the change in live weight gains according to the days is presented in Figure 7.

Some findings of different researchers regarding live weight and live weight gains in lambs are as follows; Yakan et al., (2012) reported the 180. day live weights of Akkaraman, Awassi and Kıvırcık breeds as 37.88, 36.65 and 33.86 kg, respectively. Average live weight of different genotypes showed differences according to breeds.

Alarslan and Aygün, (2019) determined the birth weight, 30., 60., 90., 120., 150. and 180. days live weights of Kıvırcık lambs as 4.49, 8.69, 1.94, 21.50, 28.58, 33.68 and 37.60 kg, respectively. In the study, they found significant the effect of maternal age and birth type on the birth weight of lambs, the effect of birth type on live weights on the 60. and 90. days, and the effect of gender on the live weights on the 90., 120., 150. and 180. days.

Öztürk et al. (2012) determined the birth weight and average live weight of 30., 60., 90., 120. and 150. days of Kıvırcık x pure (F1) (Hybrid) lambs to be 4.25, 8.76, 14.40, 21.72, 26.69 and 35.85 kg, respectively. In the study, the effect of age on body weight was found to be significant.

In another study, Demir et al., (2001), reported the 56th day live weight of Kıvırcık lambs as 28.81 - 28.86 kg. Akbaş et al. (1999) investigated the live weights of Kıvırcık and Dağlıç male lambs with different mathematical growth models. Among these models, Brody, Negative Exponential, Gompertz, Logistik and Bertalanffy models showed a good fit to the weight-age data of lambs.

In this study, the change of live weight of Kıvırcık lambs fed with corn and sunflower silage in 5 different ratios (pure and mixed) according to time was investigated. Cubic spline functions have been applied to better describe growth by examining growth periods separately. That is, 5 different Cubic Spline functions were obtained for 5 different time intervals, 0-84, 84-98, 98-112, 112-126 and 126-140 days. According to the Cubic Spline function formed according to these time intervals, the live weight estimation on any day could be made.

As a result, it has been seen that Cubic Spline functions, which are little used in agricultural areas and are not known much by researchers working on this subject, can be used successfully in the estimation of live weight of lambs. The obtained results revealed that the Cubic Spline functions will make an important contribution to the research on animal husbandry.

Author contribution

IA conceived and designed research. OS conducted experiments. ŞÇ analyzed data. ŞÇ and IA wrote the manuscript. All authors read and approved the manuscript.

Funding

No funding was received for conducting this study.

Conflict of Interest

The authors declare that they have no conflict of interest.

Ethics approval

The manuscript does not contain clinical studies or patient data.

Consent to participate

All the authors approved the final manuscript.

Consent to publication

All the authors consented the final manuscript.

Data availability

All data generated or analysed during this study are included in this published article.

Code availability

All codes analysed during this study are included in this published article.

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Possibilities of using Cubic Spline functions in estimation of live weight of Kıvırcık lambs fed with different silage types

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