Comparison of nine growth curve models to describe growth of partridges(Alectoris chukar)

ABSTRACT In this study, nine non linear growth curve models were used to determine the goodness of fit by the body weight measurements of the total number of 178 partridges(Alectoris chukar), 93 females, and 85 males, respectively. The R2(coefficients of determination) values for the total partridges, females and males in Brody, Gompertz, Logistic, von Bertalanffy, asymptote regression,exponential, Monomolecular, Richards and Weibull-type were 0.985, 0.980 and 0.984, 0.997, 0.998 and 0.998, 0.996, 0.999 and 0.999, 0.995, 0.995 and 0.996, 0.985, 0.980 and 0.984, 0.891, 0.871 and 0.892, 0.985, 0.980 and 0.984, 0.997, 0.999 and 0.999, 0.997, 0.999 and 0.999, respectively. The R2 values for Gompertz, Logistic, von Bertalanffy, Richards and Weibull-type were >0.99, while the exponential (<0.90) had the lowest. What’s more, the Gompertz, Logistic, Richards and Weibull-type models best described the data because of lower MSE (mean square error), AIC(Akaike’s information criteria) and BIC(Schwarz Bayesian information criterion), higher adj. R2(Adjusted coefficient of determination) and r(the correlation coefficient between measured body weight and estimated body weight) and there was not an autocorrelation between the residual values. As a result, based on goodness of fit criteria; R2, adj.R2, MSE, r, AIC, BIC values, the Weibull-type model best described live weight data of the Partridges(Alectoris chukar).


Introduction
Partridges (Alectoris chukar), belong to the phasiandae family and phasiandae subfamily group in taxonomy (Cetin and Kirikci 2000). The partridges include seven closely related inter-fertile species, distribute in Eurasia, China and southern Arabia (Johnsgard 1988). In China, the common partridges in the market are Alectoris chukar selected and bred by American scientists. Partridges are characterized with fast growth performance, high prolificacy and superior meat quality, accordingly, they are best used for commercial production. The domestication and the post-selection on growth traits provide an excellent source of protein for humans (Queiroz et al. 2004).
Growth is one of the well-known features in biological creatures. The evolution of body weight during growth is of particular importance in both breeding and management. In livestock and poultry, the fitting and analysis of growth curves of parameters is the basic work for breeding and production, and is also one of the main methods for studying the growth and development. The growth curve parameters determination, effectively, describe issues such as growth, livestock performance, and optimum slaughter age, as well as preparing an appropriate feeding process and selection (Sariyel et al. 2017). There are many mathematical models that describe the growth curve of livestock and poultry, and quite a few different models applied to fit the growth curve in other poultry including duck and Japanese quail and forth on (Maruyama et al. 1999;Raji et al. 2014). Growth curves illustrating these changes allow the data to be summarized by a few number of parameters known as growth curve parameters (Firat et al. 2016). It could also be expected that the conversion of the collected data into actual breeding practices increase the productivity of partridges breeding (Sariyel et al. 2017). Although there are a number of reports on the growth curve model of partridge, there is conflicting reports on the appropriate model for describing growth in the partridge. Therefore, sound and comprehensive models were used in this study to fit the growth curve of partridge on the basis of previous studies.

Material and methods
The study was carried out at Wenzhou Yongzheng Agricultural cooperation company (Wenzhou,). A total of 93 female and 85 male partridge chicks were randomly selected in this study. Sex identification of newly hatched chicks was used pre-laboratory studies. The chicks were wing-banded and individually weighed with a digital scale with a sensitivity of 0.01 g every two weeks. Ten chicks were then housed in cages (55 × 65 cm) fitted with improvised feeders and drinkers, providing with water and feed for ad libitum consumption (Raji et al. 2014). The feed for partridges contained 18% CP and 2800 kcal of ME/kg (NRC.1994). On the first day, The ambient temperature was controlled at around 34°C and was lowered by 3°C per week until reaching 20°C. (Ozek et al. 2003;Arslan 2004;Yildiz et al. 2005). Chicks were exposed to 24-hour light during the first three weeks and then exposed to natural light (16L: 8D) in the rest of the experiment. All procedures were based on the guiding principles for the care and use of research animals (Cetin et al. 2007).
The following growth curve models were used to determine the parameters belonging to the growth curves (Table 1). Parameter predictions belonging to the models were used Levenberg-Marquardt nonlinear least squares algorithm in SPSS software by taking advantage of weekly average weight of partridges (Akbas et al. 1999;. During the iteration procedure, when any parameter value at a current iteration did not change in the successive iteration, the procedure stopped since the relative reduction between the sum of squares of consecutive residuals is at most SSCON = 1.00E−008 and curve parameters obtained (Raji et al. 2014). The goodness of fit of each model is determined by the following factors, coefficients of determination (R 2 ), mean square error (MSE), DW, and the correlation coefficient between measured body weight and estimated body weight (r). These calculated values are also used to estimate the goodness of fit, Coefficient of determination (R 2 ) = 1-(SSE/SST) where SSE is the sum of square of errors, and SST is the total sum of squares. Adjusted coefficient of determination (adj. R 2 ) = R 2 −((k−1)/(k−n))(1−R 2 ) where n is the number of observations, and k is the number of parameters ). Akaike's information criteria (AIC) = n*ln(SSE/n) + 2k, where n; the number of observations, SSE; sum of square of errors, k; the number of parameters. Schwarz Bayesian information criterion (BIC) = n*ln(SSE/n) + k*ln(n), where n; the number of observations, SSE; sum of square of errors, k; the number of parameters (Narinc, Aksoy, Karaman, Curek et al. 2010;Narinc et al. 2013) The Durbin-Watson (DW) value is compared to the lower and upper critical values, dL and dU. If the calculated DW is lower than dL, there is a positive autocorrelation (DW closes to 0) in the error terms. If the calculated DW is higher than dU, there is not an autocorrelation (DW closes to 2) or there is a negative autocorrelation (DW closes to 4) in the error Table 1. Growth curve models and the age at point of inflection and weight at point of inflection in partridge (Alectoris chukar).

Model
Function Ti Yi The asymptotic weight or maximum growth response (g); B = The biological constant; K = The growth rate;n = The shape parameter; t = The age in days; Ti = The age at point of inflection; Yt = The body weight (g) at age (t); Yi = Weight at point of inflection. terms and if the calculated DW is between dL and dU, the test is inconclusive (Cetin et al. 2007).

Results and discussion
The growth curves of both sexes of chukar partridges appeared very similar to general sigmoid shape of typical growth curve. The growth patterns of male and female were different (Soner Balcioğlu et al. 2009), therefore, data set was analyzed for each gender and population. Estimated growth curve parameters of total partridges, females and males for the nine growth models utilized were presented on Table 2. Asymptotic weight is directly related with genotypic and environmental effects (Sariyel et al. 2017). The asymptotic weight parameter represents the maximum growth response for animals . Different growth models used in this study had different estimated asymptotic weight parameters. As shown in Table 2, the parameter A, which is represented the asymptotic weight or maximum growth response in total partridges, females and males, had higher value among the three models of brody ( Sariyel et al. (2017). This could be due to the fact that asymptote weight parameters is directly related to genotype and environmental effects, hence different partridges genotypes fed in different environment would have different asymptote weight (Raji et al. 2014).
The parameter K is a maturing index and indicates the rate of maturity, which had the highest value in the Asymptote regression (0.980, 0.970 and 0.985) and the lowest value in the Weibull-type (0.012, 0.009 and 0.009) ( Table 2). The values of both female and male in Gompertz and Logistic model of this study are consistent with those reported by previous researches (Cetin et al. 2007;Soner Balcioğlu et al. 2009;Sariyel et al. 2017). The parameter n is the shape parameter determining the position of the inflection point of the curve. In this research, only Richards (0.293, 0.723 and0.520) and Weibull-type (1.873, 2.076 and 2.016) have the parameter n   (Table 2), which are lower than the values in Richards reported by Cetin et al. (2007).
The growth curve parameter Ti (the age at the point of inflection) and Yi (the weight of point of inflection) were found to be 6.9 weeks and 172.9 g for total partridges, 6.5 weeks and 162.1 g for females and 7.1 weeks and 188.6 g for males in the Gompertz model, respectively, and 8.2 weeks and 210.3 g for total partridges, 7.9 weeks and 201.3 g for females and 8.4 weeks and 228.7 g for males in the Logistic model, respectively. As for von Bertalanffy model, the growth curve parameter Ti and Yi were found to be 6.1 weeks and 152.0 g for total partridges, 5.7 weeks and 139.7 g for females and 6.4 weeks and 166.3 g for males, respectively, while in the Richards model, the growth curve parameter Ti and Yi were found to be 2.2 weeks and 415.7 g for total partridges, 3.3 weeks and 276.0 g for females and 3.9 weeks and 382.2 for males, respectively ( Table 2).
The estimated growth curves for total partridges, females and males determined by nine different growth curve models are shown in Figures 1-3. It can be seen that four models including Gompertz, Logistic, Richards and Weibull-type present a better fit in this study. Used goodness of fit criteria; coefficients of determination (R 2 ), Adjusted coefficient of determination (adj. R 2 ), mean square error (MSE), the correlation coefficient between measured body weight and estimated body weight(r), Akaike's information criteria (AIC), Schwarz Bayesian information criterion (BIC) and DW to evaluated the goodness of fitting. It is known that the model with the highest R 2 value and the lowest MSE value explains the change in live weight depending on age in studies in which R 2 and MSE values are evaluated together (Keskin and Dag 2006;Sahin et al. 2014). Some researchers used the Durbin-Watson (DW) statistic, along with R 2 and MPE (Norris et al. 2007;Keskin et al. 2009) to evaluate whether growth curve models accord with actual values, As it can be seen in Table 3, for the model selection criteria, the R 2 values for the total partridges, females and males in Brody, Gompertz,Logistic,von Bertalanffy,asymptote regression,exponential,Monomolecular,0.980 and 0.984,0.997,0.998 and 0.998,0.996,0.999 and 0.999,0.995,0.995 and 0.996,0.985,0.980 and 0.984,0.891,0.871 and 0.892,0.985,0.980 and 0.984,0.997,0.999 and 0.999,0.997,0.999 and 0.999,respectively. The R 2 values for Gompertz, Logistic, von Bertalanffy, Richards and Weibull-type were >0.99, while the exponential (<0.90) had the lowset. The R 2 values obtained in this study are higher than those values reported by Cetin et al. (2007) for the Gompertz, Logistic, and Richards models, those values reported by Soner Balcioğlu et al. (2009) for the Gompertz, Logistic and von Bertalanffy models, those values reported by Aourir et al. (2014) for the Gompertz model and those values reported by Sariyel et al. (2017) for Brody, Gompertz, Logistic and von Bertalanffy models. The values that Tholon et al. (2006) found for partridges (Rhynchotus rufescens) in the Gompertz model, Tholon and Queiroz (2007) obtained for tinamous (R. rufescens) in the Brody, Gompertz, Logistic, and von Bertalanffy models and Raji et al. (2014) gained for Japanese quail (Coturix japonica) in the asymptote regression, Gompertz, Logistic, Monomolecular, Richards and Weibull-type models are lower than the coefficient of determination obtained in this study. The Weibull-type function shows the minimum MSE(54.98 for total partridges, 12.76 for females and 21.68 for males), AIC and BIC values (209 and 221 for total partridges, 37 and 48 for females, 88 and 97 for males, respectively).

Conculsion
In this study, nine models were used to analyze growth curve of total partridges and different sexes. Via comparing and analyzing the R 2 , Adj.R 2 , MSE, r, AIC, BIC and DW values of each model, the four models containing Gompertz, Logistic, Richards and Weibull-type were all assumed to be able to describe partridges live weight change with age. However, based on goodness of fit criteria; R 2 , Adj.R 2 , MSE, r, AIC, BIC and DW values, the Weibulltype model is in best accordance with the actual live weight data of partridges (Alectoris chukar).