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PRODUCTION, MODELING, AND EDUCATION |
Department of Animal Science, Faculty of Agricultural Science, University of Guilan, PO Box 41635-1314, Rasht, Iran
1 Corresponding author: m_mottaghi{at}gstp.ir
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Key Words: broiler • growth model • Gompertz • Richards • hyperbolastic
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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A number of nonlinear functions have been used to describe growth in fish, poultry, and mammals (France et al., 1996; Lopez et al., 2000). Despite numerous reported growth models, derivatives of the commonly used functions failed to describe the corresponding mean growth rate curve (Taylor, 1980), emphasizing the need to examine obtained growth models with similar situations that may have appeared in field observations.
The use of growth models is usually empirical, and the form of the model is chosen by its ability to fit the data. A growth model can be characterized by some underlying physiological or biochemical mechanism or constraint (France and Thornley, 1984). Such models can be defined as rate as a function of state, in which the instantaneous growth rate is a function of the organism’s size. Unlike equations in which growth rate is purely an empirical function, an equation in this form can usually be interpreted biologically and can be ascribed by its parameters (Lopez et al., 2000).
The Gompertz and Richards models have been the most well known models for describing growth in broilers (Aggrey, 2002; Darmani Kuhi et al., 2003). In these models, the growth curves are asymmetric around the point of maximum growth rate. However, differences between them are points of inflection, which in the Gompertz model is a fixed and in the Richards model is a variable (flexible) proportion of their asymptotic growth values (Tabatabai et al., 2005).
Tabatabai et al. (2005) introduced a class of 3 and 4 parameter models, namely hyperbolastic models, to predict self-limited growth behavior including growth in tumors as well as stem cells. These models are called hyperbolastic because the outcome is a function of inverse hyperbolic sine function (arcsinh). These models are a family of flexible growth models that can predict variety of growth behaviors for continuous outputs in many fields of biological research. They introduced these hyperbolastic growth models in 3 types, including hyperbolastic growth model type 1 or H1 (generalizes logistic growth model); hyperbolastic growth model type 2 or H2 (stand alone); and hyperbolastic growth model type 3 or H3 (generalizes Weibull growth model).
The aim of present study is evaluation of 3 new flexible hyperbolastic growth models, the H1, H2, and H3, and their ability to describe relationship between live weight and age in broiler chickens. In the meantime, model outputs will be compared with the Gompertz and Richards growth models.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Growth Models
Five growth models were fitted to data using NLIN procedure (Marquart algorithm) of the SAS (SAS Institute Inc., 1999). The models are as below:
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Gompertz model (Gompertz, 1825),
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and Richards model (Richards, 1959),
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In all models, W(t) is referred to live weight (g) at age t, β is the intrinsic growth rate, 
and 
are parameters, and M represents the asymptotic or maximum growth response (final weight), which is assumed to be constant, though final weight may usually change over time.
In each model 
is defined as a function of the other parameters (M, β, and initial observed value W0 at time t0), which allows reduction of the number of parameters to be estimated and also anchors the first predicted value to the original value observed at the initial time point.
A quantitative verifying of the fit of the predictive models was made using error measurement indices commonly used to evaluate forecasting models. The accuracy of models (goodness of fit) was determined by mean squared error (MSE) and R2 value as well as residual standard deviation (RSD). Forecasting error measurements were based on the value differences between model predicted and observed BW (Oberstone, 1990).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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. As Figure 1 indicates, all growth models could be fitted to the data without difficulty by nonlinear regression, although they were sensitive to choice of initial values; however, they provided an excellent fit.
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. As the table shows, the lowest estimate of M or final weight is obtained with H1 (5,220 g for male broiler chicken). It is followed by the H3 (5,367 g), H2 (5,491 g), Richards (5,934 g), and Gompertz (6,038 g). Based on M, remarkable differences were observed among prediction of introduced models.
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shows observed and predicted values of broiler BW using 5 models, as well as values of model error measurements and determined residuals. The results show that the H3 (MSE = 364, R2 = 0.99995, RSD = 19.2) and Richards (MSE = 603.3, R2 = 0.999919, RSD = 24.7) models provide the best fit to broiler growth data. It is followed by Gompertz (MSE = 627.7, R2 = 0.999916, RSD = 25.2), H2 (MSE = 1,294, R2 = 0.99983, RSD = 34.9), and H1 (MSE = 2,533, R2 = 0.99966, RSD = 48.8). All growth models are powerful to describe the broiler growth pattern (see Figure 1
).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Although models are closely related, the parameter values may have appeared quite different when these models were fitted to a single set of data. The Gompertz model has 2 parameters (M and β) and can fit asymmetric growth, though it is not very flexible when compared with the H3 and Richards model. The Richards model used in this study was formed with 3 parameters (M, β, and ). It is an asymmetric model and also more flexible than the Gompertz model. The H1 has one more parameter () than the Gompertz functions, which allows more flexibility and can fit asymmetric growth patterns with enough ability to detect increasing and decreasing growth kinetics. The H2 has the same number of parameters as H1 and can fit asymmetric curves, but it cannot fit decreasing growth patterns, so it seems to be less flexible. The H3 model has the same flexibility as the H1 function at the expense of one more parameter () similar to the Richards equations. More flexibility of H3 and Richards may lead to more accurate prediction and better fit to the broiler growth data set than other models (Darmani Kuhi et al., 2003). This means that the fixed point of inflection (less flexibility) in the growth models may act as a limitation for data fitting. Consequently, consideration of flexible growth models (such as the H3 and Richards) as an alternative to the Gompertz is worthwhile due to the following reasons: 1) they are easy to fit and fitting them to growth data leads to biologically meaningful parameters; 2) they have more flexibility, i.e., very often give a closer fit to data points and therefore a smaller MSE and RSD and higher R2 values than the simpler models; and 3) they encompass simpler models for additional parameters. This is especially important when the behavior of a particular data set is not defined previously (Darmani Kuhi et al., 2003).
Despite unstable value, the asymptote can be regarded as the final weight. In this study, to predict final weight (M) some bias was observed among models (see Table 1). It has been established in different species that estimation of final weight is a function of algorithm fitting, and its accuracy judging is possible when a precise final weight is available (Ricklefs, 1985; Lopez et al., 2000).
The new growth models, and H3 in particular, clearly demonstrate with valid data fitting for broiler and better goodness of fit than others introduced in the present study. Based on calculated R2 for the same data set (Roush et al., 2006), the results revealed that the H3 model (R2 = 0.99995) has more accurate prediction than artificial neural network of broiler growth model (R2 = 0.999938), though in H3 model higher MSE value was observed (364 vs. 163.1).
In the present study, the H3 model showed lower residuals distribution (in terms of RSD) than that of Gompertz and Richards. This is in agreement with Tabatabai et al. (2005) when classical and hyperbolastic models were compared, to evaluate cancer, craniofacial, and stem cell growth data fitting. Their suggestion was that choosing a flexible and highly accurate predictive model such as hyperbolastic can significantly improve the outcome of a study, and the accuracy of a model determines its utility.
In conclusion, hyperbolastic growth models can be used to fit a broiler growth data set, and the results indicate higher prediction accuracy for the H3. However, it is reasonable to compare growth models before making a decision to select the more accurate one.
Received for publication February 20, 2007. Accepted for publication May 9, 2007.
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REFERENCES |
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Brown, J. E., H. A. Fitzhugh, Jr., and T. C. Cartwright. 1976. A comparison of nonlinear models for describing weight-age relationships in cattle. J. Anim. Sci. 42:810–818.
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Gompertz, B. 1825. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Phil. Trans. Royal Soc. 182:513–585.
Lopez, S., J. France, M. S. Dhanoa, F. Mould, and J. Dijkstra. 2000. A generalized Michaelis-Menten equation for the analysis of growth. J. Anim. Sci. 78:1816–1828.
Oberstone, J. 1990. Management Science—Concepts, Insights, and Applications. West Publ. Co., New York, NY.
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Roush, W. B., W. A. Dozier III, and S. L. Branton. 2006. Comparision of Gompertz and neural networks models of broiler growth. Poult. Sci. 85:794–797.
SAS Institute Inc. 1999. SAS/STAT User’s Guide. Version 8. SAS Institute Inc., Cary, NC.
Tabatabai, M., D. K. Williams, and Z. Bursac. 2005. Hyperbolastic growth models: Theory and application. Theor. Biol. Med. Model. 2:14.[CrossRef][Medline]
Taylor, St C. S. 1980. Genetically standardized growth equations. Anim. Prod. 30:167–175.[ISI]
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