Introduction
The demand for meatproducing animals increases with the augment of population. These animals are frequently subject to researches that seek to optimize and to understand the productive process. Among the most studied meatproducing animals are the mammals because of their great economic and nutritional importance. According to ^{IBGE (2018)}, in the second quarter of 2018, the production of cattle and pigs increased, in comparison to the same quarter of the previous year, which confirms the growth in the sector.
The study of growth curves of these animals by nonlinear models is very attractive, as they are flexible and summarize the characteristics of the species development in a few parameters with biological interpretation. In practice, this knowledge allows of the adoption of strategies to enhance or mitigate certain characteristics of the animals under study (^{Freitas, 2005}; ^{Cassiano & Sáfadi, 2015}; ^{Souza et al., 2017}).
Curves of animal mass growth over time are shaped in S, which is also known as sigmoid or sigmoidal curves (^{von Bertalanffy, 1957}). Several authors have reaffirmed that the body growth of most animal species can be described by a sigmoid curve and, therefore, fit by nonlinear models which are widely used for a large number of regression applications (^{Freitas, 2005}; ^{Silva et al., 2011}; ^{Carneiro et al., 2014}; ^{Teleken et al., 2017}; ^{Rodrigues et al., 2018}).
Animal development is related to metabolic rate, which directly influences mass, shape, size, and other body traits (^{von Bertalanffy, 1957}). These body variations of an individual throughout their ontogeny can be measured by allometry, using an allometric coefficient “m” (^{Freitas & Carregaro, 2013}).
Allometry provides a quantitative description of the relationship between a part and the whole, and it is important for aggregating all information into a single value. For animal growth, there is a set of theories in the literature that suggest the value m = 2/3, and others that suggest m = 3/4; however, there is no consensus on this measurement (^{Shi et al., 2014}). In his study, ^{von Bertalanffy (1957)} states that the allometric value m = 2/3 is suitable for the growth of fish and crustaceans, and suggests m = 3/4 for mammal growth.
Among the nonlinear models most used in the literature, that proposed by ^{von Bertalanffy (1957)} has a great applicability, and is one of the most suitable for describing the growth of several animal breeds, especially in situations in which growth is not symmetrical in relation to the inflection point. However, the most commonly used parameterization does not have a direct interpretation for all parameters, which may confuse the authors, making the use of nonlinear regression more complicated.
The application of a reparameterization can facilitate the interpretations of the model, mainly for people who work in the practice. The use of reparameterization is common in nonlinear regression models, so that the parameters are rearranged to show interpretations more convenient with the study area in question (^{Zeviani et al., 2012}; ^{Fernandes et al., 2015}).
The objective of this work was to evaluate how the parameterization of von Bertalanffy’s model and the application of different allometric values affect the obtention of the most adequate fit of von Bertalanffy’s model, for the description of the growth curve of meatproducing mammals (bovine, pigs, rabbits, and sheep).
Materials and Methods
Data on mass according to age of meatproducing mammals were obtained from research institutions and the literature, and are described as follows for bovine, rabbits, sheep and pigs.
For bovine animals, mass data (kg) were obtained for Angus breed cattle of the period 20132017 from the Santa Éster farm of Casa Branca Agropastoril Ltda., located in the municipality of Silvianópolis, in the state of Minas Gerais, Brazil. The method of crossdata collection was used, recording the body mass of 203 Angus males from the birth up to 46.5 months of age.
For rabbits, data on the mass (g) of 10 males and females of the breed New Zealand were collected in the second half of 2016, obtained by weighing from the 38^{th} day of birth to the 100^{th} day, in Instituto Federal Goiano (IFG), in the municipality of Rio Verde, in the state of Goiás, Brazil.
Sheep data on mass (kg) were collected in the report by ^{Falcão et al. (2015)} on 31 males of Ile de France breed, confined from the birth to the slaughter, at 120 days of age.
Pig data on mass (kg) were collected from the study by ^{Oliveira et al. (2007)} on males of the progeny Camborough 22 and AGPIC 412 TG, in good management and nutrition.
The rate of change of the animal mass as a function of time can be expressed by the following differential equation:
in which: P represents the mass of the animal as a function of time t; α and β are constants of anabolism and catabolism, respectively; and m and n are exponents that indicate the last proportional values to some potency of the body mass P (^{von Bertalanffy, 1957}). Also, according to this author, by means of physiological facts, the exponent n equal to 1 can be used without any considerable loss.
Then, by substituting n = 1 and developing the differential equation, the general model is obtained as follows:
With this generalization, we have that P(t) is the mass of the animal as a function of time t; β is the catabolism constant; m is the allometric value; P_{0} is the initial mass; and P_{∞} is the the asymptotic mass of the animal, obtained by
The allometric coefficient m is directly related to the development pattern of the animal, and the correct use of this value allows of better estimates for the study on the growth curves. However, there is much discussion in the literature on the value of the more adequate metabolic rate, or allometric coefficient, to describe the body growth. There are theories suggesting the value m = 2/3, and others suggesting m = 3/4; however, the authors have not yet reached a consensus (^{West et al., 2002}; ^{Ohnishi et al., 2014}).
Developing the general model gives the following expression:
in which:
The model of von Bertalanffy shows three parametrizations, for i = 1, 2, ..., t, in which: y_{i} is the i^{th} observation of the dependent variable; x_{i} is the i^{th} observation of the independent variable; a is the asymptotic value, that is, the expected value for the maximum growth of the object under study, where x_{i}→ ∞; b is associated with the abscissa of the inflection point; is an index of maturity, or precocity, and is associated with growth, and the greater its value, the lesser the time necessary for the object under study to reach the asymptotic value a; ε_{j} is the random error associated with the i^{th} observation, which is assumed to be independent and identically distributed, following a normal distribution of zero mean and constant variance, that is, εi ~ N(0,I σ^{2}) (Table 1).
Model 1 

Model 2 

Model 3 

Equation terms: x_{i} is the i^{th} observation of the independent variable; a is the asymptotic value, that is, the expected value for the maximum growth of the object under study; b is associated with the abscissa of the inflection point; k is an index of maturity or precocity; ε_{i} is the random error associated with the i^{th} observation.
Model 1 shows the parameterization of the model of von Bertalanffy, usually found in the literature, for the fitting of growth curves with the allometric value of m = 2/3. Although it is widely used, this parameterization does not provide a direct interpretation for parameter b, and the abscissa of the inflection point is given by transformation 1/k × ln(3b). That is, in this parameterization the estimate of b is not directly the abscissa of the inflection point.
Model 2 still considers the allometric value m = 2/3; however, it is a reparameterization of model 1, so that parameter b has a direct interpretation as the abscissa of the inflection point.
Model 3 uses the allometric value of ^{von Bertalanffy (1957)}, and ^{West et al. (2002)} state that it is the allometry suitable for the description of mammalian growth m = 3/4; in this parameterization, all parameters have a direct and practical interpretation similar to Model 2.
The parameters of all models were estimated for the description of growth curves of the four meatproducing mammals, in each of the three parameterizations studied (Table 1). The estimates of these parameters were obtained by the iterative method of GaussNewton implemented in the nls( ) function of the software R (^{R Core Team, 2017}). The significance of the parameters (θi ≠ 0) was checked by the ttest, at 5% probability. Initially, all assumptions about the errors (ε) were considered as met. From the error vector of this fit, the analysis of residuals was performed based on statistical tests. The statistical tests of ShapiroWilk, DurbinWatson and BreuschPagan were applied to check normality, independence, and residual homoscedasticity, respectively. If any of the assumptions were not met, the deviation were corrected or incorporated into the parameter estimation process.
In order to evaluate the fit quality of the models, the nonlinearity of Bates and Watts curvatures, described by ^{Souza et al. (2010)}, were obtained using the rms×curv( ) function of the R software. Concomitantly to the nonlinearity measures, the quality evaluators  coefficient of determination (R^{2}) and corrected Akaike’s information criterion (AICc)  were used as follows:
in which: SSR is the square sum of residuals; SST is the total square sum; n is the sample size; p is the number of parameters; and AIC is the Akaike’s information criterion (below)
in which: ln is the natural logarithm operator. These evaluators were obtained using the functions Rsq( ) and AICc( ) of the AICcmodavg package, in the software R.
Results and Discussion
At first, the parameters of the three models were estimated for each mammalian species (Table 1) and the residuals were analyzed with the ShapiroWilk (SW), BreuschPagan (BP) and DurbinWatson (DW) tests, considering 5% significance level.
It was observed that the tests were nonsignificant for all mammals, except for pig data. For the other species this fact indicates that there are no problems of violation of residual assumptions, and that the residuals showed normality, constant variances, evidencing that they were independent.
For the pig data, the DW test was significant, indicating that the residuals showed dependence, so a new fit was made by the generalized least squares method, incorporating into the model a first order autoregressive parameter (ɸ) [AR(1)]. This was made because, according to ^{Nascimento et al. (2011)} and ^{Muniz et al. (2017)}, it is necessary to model this autocorrelation, and to incorporate it into the model, to obtain adjustments that produce more accurate predictions, and that allow of the analysis of measured series in few temporal observations, thus guaranteeing a greater precision in the estimates and a better quality of fit.
Based on the fit of the three parameterizations of von Bertalanffy’s model, all parameters were significant, by the ttest, at 5% probability (Table 2).
Mammal  Parameter^{(2)}  Model 1  SE  Model 2  SE  Model 3  SE 
Bovine  a  658.20  (25.52)  658.20  (25.52)  652.60  (24.02) 
b  0.5978  (0.0365)  6.605  (0.5433)  7.127  (0.5504)  
k  0.0884  (0.0092)  0.0884  (0.0092)  0.0935  (0.0094)  
Rabbit  a  2588.00  (68.07)  2588.00  (68.07)  2561.00  (64.26) 
b  1.252  (0.0934)  40.20  (0.4919)  41.34  (0.5102)  
k  0.03291  (0.0019)  0.03291  (0.0019)  0.0346  (0.002)  
Sheep  a  47.9336  (2.6283)  47.9336  (2.6283)  46.6372  (2.294) 
b  0.5485  (0.0117)  32.2988  (2.7776)  34.157  (2.6287)  
k  0.0154  (0.0013)  0.0154  (0.0013)  0.0167  (0.0013)  
Pig  a  259.0049  (13.9763)  259.0049  (13.9763)  242.9691  (10.4726) 
b  0.9264  (0.0311)  121.0277  (5.0752)  121.0709  (4.0281)  
k  0.0084  (0.0005)  0.0084  (0.0005)  0.0096  (0.0005)  
ɸ  0.9425    0.9425    0.9396   
^{(1)}Model 1, von Bertalanffy’s model with m = 2/3, without direct interpretation of b; Model 2, von Bertalanffy’s model with m = 2/3, with direct interpretation of b; Model 3, von Bertalanffy’s model with m = 3/4, with direct interpretation of b. ^{(2)}Parameters: a is the asymptotic value, that is, the expected value for the maximum growth of the object under study; b is associated with the abscissa of the inflection point; k is the index of maturity or precocity; ɸ is a firstorder autoregressive parameter.
The practical interpretation of the parameters is very important for the understanding of growth studies (^{Freitas, 2005}). Based on the estimates of parameter a by the models (Table 2), it is possible to observe that the mass estimations by Model 3 were more consistent with the real values, whose maximum observed were 599.5 kg (bovines), 154.8 kg (pigs), 2,258.3 g (rabbits), and 36.9 kg (sheep).
Models 1 and 2 overestimated the final mass of the meatproducing mammals, as verified by ^{Silva et al. (2011)} in the study on cows, ^{Falcão et al. (2015)} in the study on sheep, and ^{Jacob et al. (2015)} in the study on rabbits, using the model of von Bertalanffy.
Therefore, it can be concluded that Model 3 provided a minimization of overestimation of parameter a, and also showed a smaller amplitude of the standard error.
^{Lopes et al. (2016)} state that, for bovine growth, Brody’s model was the one that most approached the adult mass, observed with the estimate of parameter a. However, ^{Silva et al. (2011)} and ^{Santana et al. (2016)} showed that, in addition to the Brody’s model, that of von Bertalanffy also showed adequate results to describe cattle growth curves. Thus, the relationship between the asymptotic estimation of the model and the real values is explicit, and models that overestimate the adult mass are usually discarded in practice.
In the parameterization used in Model 1, which is one of the most found in the literature, b is a location parameter that maintains the sigmoidal shape of the model, and is associated with the abscissa of the inflection point of the curve, but it has not a direct, practical interpretation (^{Silveira et al., 2011}; ^{Teixeira Neto et al., 2016}). However, some authors, such as ^{Mischan & Pinho (2014)} and ^{Louzada et al. (2014)}, present mathematical transformations which can find the abscissa value of the inflection point, through the estimation of Model 1.
Thus, a great advantage of the parameterization of Models 2 and 3 is related to parameter b, since its estimates are interpreted directly as the abscissa of the inflection point, that is, it indicates the age at which the growth of the animal slows down. For instance, for pigs, it is expected that up to approximately 121 days their growth will be more accentuated and, after this period, it will decrease until stabilization (Table 2).
According to ^{Teleken et al. (2017)}, von Bertalanffy’s model was the most suitable, among others, for the description of the growth of New Zealand rabbits. ^{Jacob et al. (2015)} also state that this model described well the growth of rabbits of the Soviet Chinchilla and White Giant breeds, however, their estimates overestimated the asymptotic parameter, and did not give a direct interpretation for parameter b.
Values of the coefficient of determination (R^{2}) were higher than 0.99, indicating a satisfactory fit. Regarding the corrected Akaike’s information criterion (AICc), Model 3 showed smaller values, and the lower AICc value suggests that the model is the most adequate to describe the data. Therefore, the goodnessoffit shows the superiority of Model 3 to describe the growth curve of meatproducing mammals (Table 3).
Mammal  Criteria^{(1)}  Model 1  Model 2  Model 3 
Bovine  R^{2}  0.9938  0.9938  0.9938 
AICc  93.4718  93.4718  93.3386  
C^{ι}  0.1106  0.1106  0.1109  
C^{θ}  0.9623  0.8797  0.8141  
Rabbit  R^{2}  0.9974  0.9974  0.9974 
AICc  121.8681  121.8681  121.8064  
C^{ι}  0.0284  0.0284  0.0284  
C^{θ}  0.8958  0.6756  0.6030  
Sheep  R^{2}  0.9978  0.9978  0.9979 
AICc  31.1405  31.1405  30.6398  
C^{ι}  0.0445  0.0445  0.0446  
C^{θ}  2.0296  1.4957  1.2771  
Pig  R^{2}  0.9978  0.9979  0.9984 
AICc  74.5154  73.5078  67.8611  
C^{ι}  0.0402  0.0402  0.0301  
C^{θ}  1.7030  0.9408  0.6432 
^{(1)}R^{2}, coefficient of determination; AICc, corrected Akaike’s information criterion; C^{l} measurement of intrinsic nonlinearity; and Cθ measurement of parametric nonlinearity.
The measures of intrinsic (C^{l}) and parametric (C^{θ}) nonlinearity for each parameterization of the model are also presented (Table 3). As commented by ^{Souza et al. (2010)}, ^{Zeviani et al. (2012)}, and ^{Fernandes et al. (2015)}, lowernonlinearity values of measures indicate a better fit of the models, since more reliable estimates are obtained.
The present study shows that the values of the intrinsic curvature (C^{l}) remain practically constant in the three parameterizations, for all scenarios, and only for pigs it has a more pronounced decrease for Model 3. This change in the measure of intrinsic nonlinearity may be a result of the change in the allometric value, since it did not change for Models 2 and 3.
The parametric nonlinearity measure changes, according to the parameterization and the allometric value used (Table 3), provided worse results for Model 1, and better results for Model 3. This fact evidenced that for the description of growth characteristics of the group of animals under study, the parameterization and the value of m = 3/4 of great importance to obtain better fits.
In practice, it is common to neglect the models with a practical or biological interpretation for parameter b, as in the studies of ^{Silva et al. (2011)}, ^{Silveira et al. (2011)}, ^{Cassiano & Sáfadi (2015)}, ^{Jacob et al. (2015)}, and ^{Rodrigues et al. (2018)}. Nevertheless, if the parameterization of Model 3 is used, considering the correct allometry, a better fit can be achieved in these studies.
For all meatproducing species, it is possible to identify the sigmoidal shape in the development of the animals (Figure 1); however, this shape is more remarkable in the data of pigs, as they show a greater number of mass measurements in the study interval.
In the graphical adjustment, the three parameterizations (Figure 1) did not change practically. It is noticed that the adjustment lines for the three forms of von Bertalanffy’s model are overlapped, therefore, in the present study, the graphical analysis of the models was not taken into account to select the best model. The fact evidenced the superiority of Model 3 for R^{2} and AICc, and the considerable reduction of the parametric nonlinearity, therefore making the parameter estimates more reliable.
Conclusions
The most adequate data of meatproducing mammals (bovines, pigs, rabbits, and sheep) are the allometric value of m = 3/4, as well as the results in lower values of the corrected Akaike’s information criterion and higher values of the coefficient of determination.
Model 3 of von Bertalanffy minimizes the overestimation of parameter a, and its parameterization gives a direct interpretation for parameter b, besides the lowest values for curvature measurements, mainly the parametric ones, providing more reliable adjustments; therefore, this model can be used in the description of the growth curve of meatproducing mammals.