Quail growth curve model identity

In this study, we sought to apply the model identity technique to compare the influence of eight treatments on growth parameters for three broiler quail lines, estimated using a logistic nonlinear regression model. For the analysis, we used the weight and age data obtained for three lines of European broiler quails (Coturnix coturnix coturnix) in a completely randomized 2×4 factorial scheme, with two levels of metabolizable energy (2900 and 3100 kcal of ME kg-1 of diet), four levels of raw protein (22%, 24%, 26% and 28% crude protein), and six repetitions. Results obtained for model identity tests indicated that although there were no significant differences among the parameters of the model between the treatments evaluated in each strain, there were, with the exception of Treatment 5 (3100 kcal of ME kg-1 and 22% crude protein), significant differences with respect to the adult weight parameter between lines within each treatment.


Introduction
Quail production at the national level has grown in recent years, mainly due to the lower costs of poultry production, and the inherent characteristics of bird physiology, as quails are precocious birds that have a high growth rate (Karadavut et al., 2017). However, despite the intense interest, there is little information available regarding the characteristic's growth curves of European quails (Couturnix couturnix sp.) from the perspective of meat production. Such information can provide researchers with strategic knowledge that can be used to establish more efficient nutritional management and facilitate the design of dedicated selection programs for different strain (Santos et al., 2012).
Among the diverse applications of growth curves in animal production, the following are of particular importance: summarization, in three or four parameters, of the characteristics of population growth, given that some parameters of the nonlinear models have biological interpretation; evaluation of the treatment response profile over time; and the study of interactions of the responses of subpopulations or treatments with time and to identify the heaviest and the youngest animals within a specific population (Brusamarelo et al., 2020).
The growth characteristics of animals have a direct influence the quantity and quality of the meat produced. Thus, studies related to growth curves have strategic application in genetic improvement programs, contribute to the definition of selection criteria regarding the finishing precocity and speed of weight gain, and can facilitate the development of more efficient production systems for different breeds and regions with respect to the management of animals, feeding programs, and determination of breeding or lines (Souza et al., 2010;. In this regard, Arango and Van Vleck (2002) stressed that it is necessary to consider growth Development, v. 9, n. 10, e9439109328, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i10.9328 4 and maturity characteristics derived from the study of growth curves as additional information in genetic improvement programs.
In addition to facilitating verification of differences in the growth of different lines used in the production of animal protein, growth curve determinations make it possible to identify the ideal time to change diets, based on a knowledge of the parameters of nonlinear models . According to Santos et al. (2018), lines that reach adult weight at an earlier age have a higher nutritional demand than slower growing lines.
Thus, to enable a better description of the growth curve of birds, it is necessary to identify mathematical models that provide the best fits, and also to determine the parameters of the equations that facilitate characterization of the growth patterns of evaluated animals. In this regard, Silveira et al. (2011) have suggested that in addition to selection of the best model, when considering multiple populations, researchers are also interested in comparing curve parameters to identify those populations in which the growth process is most efficient, and for this purpose, Regazzi (2003) has proposed that use of the model identity technique is the most suitable approach.
In this study, we accordingly used the model identity technique, with the objective of evaluating the influence of different treatments on the estimated parameters used to describe the growth curve of different lines of European broiler quails.
In this study, we evaluated a total of 576 seven-day-old male and female chicks of three different lines of European broiler quails (Coturnix coturnix sp.). The birds were distributed in an entirely randomized experimental design, with six repetitions composed of 12 quails per experimental unit in a 2×4 factorial scheme, two metabolizable energy levels of 2900 and 3100 kcal of ME kg -1 diet and four levels of raw protein (22%, 24%, 26%, and 28% CP), with treatments representing different combinations of the different factor levels ( Table 1). The quails were weighed at 7-day intervals until reaching 42 days of age. Research, Society and Development, v. 9, n. 10, e9439109328, 2020 (CC BY 4. On the basis of the eight treatments and the six repetitions, the average weights of the quails in each of the three lines (lines 1 to 3) were calculated for each 7-day time interval.
We used the following nonlinear logistic model for the estimation of growth curve parameters: Where yi is the body weight at age xi; β1 is the asymptotic weight when t tends to infinity (this parameter is interpreted as weight at adulthood or weight at maturity); β2 is a constant of integration, related to the initial weights of the animal and without defined biological interpretation, the value of which is established by the initial values of y and x; and β3 is interpreted as the maturation rate or growth speed, which represents the change in weight in relation to the weight at maturity, and is used as an indicator of the speed with which the animal approaches its adult size.
The identity method of nonlinear regression models proposed by Regazzi (2003) and Regazzi and Silva (2010) was applied in order to verify differences in parameter estimates between treatments analyzed for each strain and between lines analyzed for each treatment.
To employ this method, we used a logistic model plus a dummy variable (Puiatti et al., 2020;Safari and Erfani, 2020) to represent each of the eight treatments for each of the three lines. This model, which we refer to as the complete model, is as follows: Research, Society and Development, v. 9, n. 10, e9439109328, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i10.9328 Where K: varies from 1 to 8 for the treatments for each strain, and from 1 to 3 for the lines in each treatment; For all parameters, comparisons were made using the H0 hypothesis test. This required the use of the complete model for the treatment (k = 8) and strain (k = 3) groups: Comparisons also required a reduced model for each parameter βj, where j = 1, 2, or 3, for the treatment (k = 8) and lines groups (k = 3) groups: Similar complete and reduced models were constructed for the other two parameters β2 and β3.
Research, Society and Development, v. 9, n. 10, e9439109328, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i10.9328 7 The sum of squares of the waste from the complete and reduced model adjustments, represented respectively by RSSΩ and RSSω, were used to obtain the following chi-square statistic: χ 2 calculated = N ln (RSSΩ /RSSω). The decision rule is to reject H0 at a level of significance α if χ 2 calculated ≥ χ 2α(ν) , where ν = pΩ -pω is the number of degrees of freedom, and pΩ and pωo are the number of parameters estimated in the complete and reduced models, respectively.
Details of the application of the likelihood ratio test, with approximation using chisquare statistics, have been described by Regazzi and Silva (2010). Model identity has been applied in plant and animal development studies, in order to verify the possibility of adjusting common equations for different groups of individuals (Santos et al., 2012).
On the basis of a comparison of the two approximations, i.e., χ 2 and F statistics obtained by data simulation, Regazzi and Silva (2010) concluded that for a sufficiently large total number of observations (N ≥ 120), the two approximations are almost equivalent, and that for smaller samples, it is preferential to use the approximation given by the F statistic, as the type I error rate is invariably lower, regardless of the value of N.
Rejection of the H0 hypothesis indicates that there is at least a difference between the estimates of the parameter βj, and in this event a new set of hypotheses is proposed in order to identify which strain in each treatment group (k = 1, 2, 3) is equal to or different from the parameter β1: For this purpose, we used formulae 5 and 6 for the complete and reduced models respectively, which are illustrated for a comparison between lines 1 and 2 as follows: The logistic model was adjusted to the quail weight-age data for treatments for each strain and the lines in each treatment based on the Gauss-Newton method using PROC MODEL in SAS (Statistical Analysis System, version 9.0, 2002) with the aid of dummy variables.

Results and Discussions
Using the average weights obtained for each strain in response to the eight treatments (Table 2), we performed equality tests for the parameters of the logistic model (Table 3), and noted that for all treatments, there were no significant differences among the three lines with respect to adult weight (β1), the integration constant (β2) of the model, or the maturation rate (β3), and therefore the estimates for each parameter were considered equal (Table 4).  Research, Society and Development, v. 9, n. 10, e9439109328, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i10.9328 The growth curves obtained from the estimates in the complete model (Table 4) analyzed graphically for the treatments for lines1 to 3 are presented in Figure 1A to 1C, respectively. Similarly, the model identity test was carried out using χ 2 statistics (N = 144) for the lines within each treatment (Table 5) and, based on the tests performed, we concluded that with the exception of Treatment 5, for each treatment studied, the lines showed differences only in terms of adult weight (β1). In contrast, responses to Treatment 5 (3100 kcal of ME kg -1 feed and 22 % RP) showed no significant differences. We subsequently performed the model identity test in order to compare the adult weight parameter (β1) of the lines within each treatment (Table   6), which, with the exception of Treatment 5, did not indicate a significant difference. RSSΩ -residual sum of squares full model; RSSῳ -residual sum of squares reduced models; *H1indicates that there is at least one difference between in lines relation to each of the curve parameters within each studied diets; βij where: i = parameter; j = lines; N= 3 lines x 6 time intervals = 18; v = 9 -7 = 2; X 2 tabulated -5 % (2) = 5.99. Source: Authors. RSSΩ -residual sum of squares full model; RSSῳ -residual sum of squares reduced models; *H1-indicates that there is at least one difference between in lines relation the parameter β1 of the curve. within each studied diet; N= Growth curves obtained on the basis of estimates using the complete model (Table 7) are presented graphically for treatments 1, 2, 3, 4, 6, 7, and 8 in figures 2A, 2B, 2C, 2D, 3A, 3B and 3C, respectively. Estimates of the adult weight parameter (β 1 ) for each strain in response to the different treatments indicated that Line 1 had a higher adult weight when compared with the other two lines (Table 7). With the exception of Treatment 5, we detected a strong inverse relationship between adult weight (β 1 ) and the maturation rate (β 3 ) in response to all other treatments, and therefore it can be inferred that the different nutritional treatments had no significant effect on the growth curve of European quails.  We observed a strong inverse relationship (r = -0.94) between adult weight (β 1 ) and maturation rate (β 3 ) only in Strain 1, which was expected, given that the higher the adult weight, the lower is the rate of maturation . Biologically, this correlation can be interpreted as indicating that animals with higher growth rates are less likely to reach higher weights at maturity than those that grow more slowly in early life; that is, birds that are heavier at maturity tend to have a lower growth rate (Karadavut et al., 2017). It should be noted that this relationship is biologically the most important (Kaplan and Gürcan, 2018).
These differences in adult weight highlight the fact that the lines examined in this study have been selected for slaughter weight. In contrast, weight gain does not appear to have been a focus of selection for these lines, as we detected no significant differences with respect to the maturation rate parameter. Previously, Santos et al. (2018) have shown that European quail lines specifically bred for meat production have a higher adult weight range than that of Japanese quail lines, whereas Sezer and Tarhan (2005) have observed distinct growth behavior in the first and second growth phases of three quail lines, which indicates that different sets of genes can determine differences in the early and late growth of lines. Additionally, a difference in the growth curve with respect later weights was considered to reflect a significant effect of the adult weight parameter for the different lines evaluated. However, the same authors failed to observe any significant difference in the growth rate, which is consistent with the findings of the present study.
Some authors have studied the effect of selection or the nutritional levels of diets Santos et al., 2018) on the growth curves of quails, and have inferred that further studies should investigate the effects of diet on the quail growth curve. Furthermore, using standard and reaction models, Mota et al. (2015) examined the effects of genotype × environment interactions with respect to different diets on the body weight of quails, and accordingly observed that nutritional levels affected body weight only under less favorable environmental conditions. Similar to the present study, Bonafé et al. (2011) evaluated the identity of nonlinear regression models to assess the growth of two broiler quail lines, adjusted Richards' nonlinear regression model to the data, and performed a parameter equality test, and accordingly deduced that two curves are necessary, as the parameters were significantly different for the two lines.
Moreover, in a comparison of the growth of two generations of Japanese quails using the Gompertz model, Santos et al. (2018) performed a parameter equality test of the model and concluded that a common equation should not be used to describe the growth of the two generations.

Conclusion
On the basis of the findings of this study, we can conclude that the use of a single curve is inappropriate for characterizing the growth of different quail lines, evaluated with respect to different nutritional management. We found that adult weight was the parameter that contributed to the observed differences found. The model identity technique is considered to represents a strategic approach that can be used to determine more efficient nutritional management and facilitate the design of selection programs for specific lines.