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Modeling the growth curve of Iranian Shall sheep using non-linear growth models
Accepted Manuscript Title: Modeling the growth curve of Iranian Shall sheep using non-linear growth models Author: Navid Ghavi Hossein-Zadeh PII: DOI: Reference:
S0921-4488(15)30019-5 http://dx.doi.org/doi:10.1016/j.smallrumres.2015.07.014 RUMIN 4987
To appear in:
Small Ruminant Research
Received date: Accepted date:
26-12-2014 19-7-2015
Please cite this article as: Hossein-Zadeha , N.G.,Modeling the growth curve of Iranian Shall sheep using non-linear growth models, Small Ruminant Research (2015), http://dx.doi.org/10.1016/j.smallrumres.2015.07.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights (for review)
Highlights 1. Six non-linear equations were compared to describe the growth curve in Shall sheep.
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2. Richards model was the best growth model for males, females and all lambs.
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3. Negative exponential model provided the worst fit of growth curve among models.
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4. Current models were adequate in describing the growth pattern in Shall sheep.
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*Manuscript (latest version) Click here to view linked References
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Modeling the growth curve of Iranian Shall sheep using non-linear growth models
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Navid Ghavi Hossein-Zadeh
Department of Animal Science, Faculty of Agricultural Sciences, University of Guilan, Rasht, Iran
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Corresponding author: Navid Ghavi Hossein-Zadeh. Department of Animal Science, Faculty of Agricultural Sciences, University of Guilan, Rasht, Iran, P. O. Box: 41635-1314
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E-mail addresses: [email protected]
[email protected] Tel: +98 13 33690274
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Fax: +98 13 33690281
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Abstract The objective of this study was to describe the growth pattern in Iranian Shall sheep
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using non-linear models. For this purpose, six non-linear mathematical functions (Brody,
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Negative exponential, Logistic, Gompertz, Von Bertalanffy and Richards) were used. The data
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set used in this study were obtained from the Animal Breeding Center of Iran and comprised
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57000 body weight records of lambs which were collected from birth to 400 days of age during
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1982 to 2012. Each model was fitted separately to body weight records of all lambs, male and
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female lambs using the NLIN and MODEL procedures in SAS. The models were tested for
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2 goodness of fit using adjusted multiple coefficient of determination ( R adj ), root means square
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error (RMSE), Durbin-Watson statistic (DW), Akaike’s information criterion (AIC) and
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Bayesian information criterion (BIC). Richards model provided the best fit of growth curve in
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males, females and all lambs due to the lower values of RMSE, AIC and BIC and generally
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2 greater values of R adj than other models. The negative exponential model provided the worst fit
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of growth curve for males, females and all lambs. According to the moderate values of DW
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obtained from fitting different models of growth curve it was concluded that there was positive
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autocorrelation between the residuals for all models, but this autocorrelation was more obvious
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for negative exponential model than the other equations. The negative correlation of −0.99 to
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-0.49 between a and k parameters obtained from fitting different growth models implied that the
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animal with smaller mature weight will be maturing faster. Evaluation of different growth
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models used in this study indicated the potential of the non-linear functions for fitting body
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weight records of Shall sheep.
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Key words: Body weight; Fat-tailed sheep; Growth function; Model fitting
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Introduction The Shall sheep is a local sheep breed of Iran with a population of more than 600000
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heads. The breed is fat-tailed, large-size, predominantly black or brown with white spots in front
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of head, well adapted to the harsh climate and raised mainly for its meat that is most important
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source of protein in Iran and sale of its surplus lambs is the main source of cash income for
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farmers. Ewes were randomly exposed to the rams at the age of 18 months and lambing was
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occurred in one season, from mid-January to mid-March. Ewes were kept in the flock up to 6
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years old. Rams were kept until a male offspring was available for replacement. During the
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breeding season, single sire pens were used allocating 20-25 ewes per ram. Ewes usually lamb
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thrice every two years. Lambs remained with their dam until weaning. The flock was mainly
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kept on range and was fed by cereal pasture, but supplemental feed, including alfalfa and wheat
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straw, are provided especially around mating season (Amou et al., 2013).
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Growth is one of the most important features of livestock and is defined as increase in
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live weight or body dimension against age. Changes in live weight or dimension for a period of
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time are explained by the growth curves (Keskin et al., 2010). Analysis of the animal growth
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performance through their life span is helpful to establish appropriate feeding strategies and the
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best slaughter age. Studies focusing on growth curves have increased at recent years due to the
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development of new computational methods for faster and more accurate analyses as well as the
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availability of new models to be tested (Souza et al., 2013). In order to increase farmers’ income,
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there needs to be improvement in the production of these animals. Slow growth rate resulting in
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low market weight has been identified to be one of the factors limiting profitability in any
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production system (Noor et al., 2001; Abegaz et al., 2010). Growth rate is related to rate of
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maturing and mature weight and these latter traits have been suggested to have relationship with
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other lifetime productivity parameters in sheep (Bedier et al., 1992; Abegaz et al., 2010). Non-linear mathematical functions, empirically developed by depicting body weight
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against age, have been appropriate to characterize the growth curve in different animal classes
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(Malhado et al., 2009). Different growth models have been applied extensively in different
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species to describe the development of body weight, allowing information from multiple
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measurements to be combined into a few parameters with biological meaning, in order to
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facilitate both the interpretation and the understanding of the phenomenon (Lambe, 2006;
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Malhado et al., 2009). Growth curve parameters provide potentially helpful criteria for changing
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the association between body weight and age through selection and breeding (Kachman and
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Gianola, 1984) and an optimum growth curve can be obtained by selection for desired values of
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growth curve parameters (Bathaei and Leroy, 1998). The potential of changing the growth curve
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shape by breeding may be an attractive aspect for livestock producers through increasing early
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growth but restricting mature size, and hence maintenance requirements (Lambe et al., 2006).
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Growth curves supply several applications to animal husbandry, such as analysis of the
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interaction between subpopulations (or treatments) and time; evaluation of the response to
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different treatments over time; and identification of heavier animals at younger ages within a
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population (Bathaei and Leroy, 1996; Freitas, 2005; Malhado et al., 2009).
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No previous studies have been conducted on growth curve characteristics of the Shall
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sheep. Therefore, the objective of this study was to characterize the growth pattern of Shall sheep
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using non-linear models. For this purpose, six mathematical models (Brody, Negative
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exponential, Logistic, Gompertz, Von Bertalanffy and Richards) were compared to evaluate their
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efficiency in describing the growth curve of Shall sheep.
Materials and methods
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The data set used in this study were obtained from the Animal Breeding Center of Iran
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and comprised 57000 body weight records which were collected on 11400 lambs from birth to
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400 days of age during 1982 to 2012. The data were screened several times and defective and out
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of range records were deleted.
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The non-linear growth models used to describe the growth curves of Shall sheep are
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presented in Table 1. The Brody, Negative exponential, Logistic, Gompertz, Von Bertalanffy and
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Richards functions were fit to the data to model the relationship between body weight and age.
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Each model was fitted separately to body weight records of all lambs, male and female lambs
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using the NLIN and MODEL procedures in SAS (SAS Institute, 2002) and the parameters were
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estimated. The NLIN procedure provides least squares or weighted least squares estimates of the
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parameters of a non-linear model. For each non-linear model to be analyzed, the model (using a
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single dependent variable) and the names and starting values of the parameters to be estimated
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must be specified (SAS Institute, 2002). When non-linear functions were fitted, the Gauss-
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Newton method was used as the iteration method. To begin this process the NLIN procedure first
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evaluates the starting value specifications of the parameters. If a grid of values is specified,
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NLIN procedure evaluates the residual sum of squares at each combination of parameter values
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to determine the set of parameter values producing the lowest residual sum of squares. These
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parameter values are used for the initial step of the iteration (SAS Institute, 2002). The MODEL
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procedure analyzes models in which the relationships among the variables comprise a system of
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one or more nonlinear equations. Primary uses of the MODEL procedure are estimation,
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simulation, and forecasting of nonlinear simultaneous equation models (Ghavi Hossein-Zadeh,
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2014). The models were tested for goodness of fit (quality of prediction) using adjusted
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2 coefficient of determination ( R adj ), residual standard deviation or root means square error
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(RMSE), Durbin-Watson statistic (DW), Akaike’s information criterion (AIC) and Bayesian
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information criterion (BIC).
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2 R adj was calculated using the following formula:
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n 1 2 2 Radj 1 1 R n p
Where, R 2 is the multiple coefficient of determination ( R 2 1
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RSS ), TSS is total sum TSS
of squares, RSS is residual sum of squares, n is the number of observations (data points) and p is
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the number of parameters in the equation. The R 2 value is an indicator measuring the proportion
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of total variation about the mean of the trait explained by the growth curve model. The
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coefficient of determination lies always between 0 and 1, and the fit of a model is satisfactory if
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R 2 is close to unity.
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RMSE is a kind of generalized standard deviation and was calculated as follows:
RSS n p 1
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Where, RSS is residual sum of squares, n is the number of observations (data points) and
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p is the number of parameters in the equation. RMSE value is one of the most important criteria
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to compare the suitability of used growth curve models. Therefore, the best model is the one with
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the lowest RMSE.
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DW was used to detect the presence of autocorrelation in the residuals from the
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regression analysis. In fact, the presence of autocorrelated residuals suggests that the function
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may be inappropriate for the data. The Durbin-Watson statistic ranges in value from 0 to 4. A
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value near two indicates non autocorrelation; a value toward 0 indicates positive autocorrelation;
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a value toward 4 indicates negative autocorrelation (Ghavi Hossein-Zadeh, 2014). DW was
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calculated using the following formula:
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e e DW e n
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t n
Where, e t is the residual at time e, and et 1 is residual at time t-1.
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AIC was calculated as using the equation (Burnham and Anderson, 2002):
AIC n ln RSS 2 p
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AIC is a good statistic for comparison of models of different complexity because it
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adjusts the RSS for number of parameters in the model. A smaller numerical value of AIC
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indicates a better fit when comparing models.
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BIC combines maximum likelihood (data fitting) and choice of model by penalizing the (log) maximum likelihood with a term related to model complexity as follows: 7
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RSS BIC n ln p ln n n
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A smaller numerical value of BIC indicates a better fit when comparing models.
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After selecting the best fitted function, the absolute growth rate (AGR) was calculated
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based on the first derivative of the function in relation to time (
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the weight gained per time unit which, in this case, equals the daily weight gain estimated
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throughout a growth period; thereby it corresponds to the mean animal growth rate within a
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population (Malhado et al., 2009).
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Estimated parameters of non-linear growth models for Shall sheep are presented in Table
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2. Also, goodness of fit statistics for the six growth models fitted to body weight records are
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2 presented in Table 3. Radj values had little differences among the models for all lambs, males and
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2 females, but Richards equation provided the greatest Radj value for all lambs, males and females
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2 and negative exponential model provided the lowest values of Radj for males, females and all
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lambs (Table 3). Also, negative exponential function had the lowest values of DW for males,
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females and all lambs, but other functions had little differences in this regard and Richards
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function provided the greatest value. Richards equation provided the lowest values of RMSE,
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AIC and BIC for males, females and all lambs. On the other hand, negative exponential model
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provided the greatest values of RMSE, AIC and BIC for all lambs, males and female lambs.
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Therefore, Richards equation provided the best fit of growth curve in males, females and all
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lambs. The negative exponential model provided the worst fit of growth curve for males, females
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and all lambs. The parameter a is an estimate of asymptotic weight, which can be interpreted as
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mature, or adult, weight. For the data set studied here Richards and logistic functions provided
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the greatest and the lowest estimates of a parameter among different models, respectively (Table
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2). The parameter k, which represents the maturation rate, is another important feature to be
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considered, since it indicates the growth speed to reach the asymptotic weight. In this study,
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females generally showed higher values for this parameter than males. The negative correlation
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of −0.99 to -0.49 between a and k parameters obtained between different models for all lambs.
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Observed body weights of animals from 1 to 400 days of age are depicted in Figure 1. As shown,
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there was a sigmoidal trend for body weights along with increase in age. Also, predicted body
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weights (kg) as a function of age (days) obtained with different growth models for all lambs,
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males and females are presented in Figure 2. The growth curves estimated were typically
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sigmoid. The absolute growth rates (AGR) based on the first derivative of Richards function in
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relation to time are shown in Figure 3. The AGR values declined gradually with increasing time
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for males, females and all lambs.
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Discussion
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Over the last few years, growth curves were applied to analyze the growth in farm
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animals for improving their production system. The growth curves are being used in different
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aspects of management in producing meat type animals such as designing optimum feeding
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programs and determining optimum slaughtering age (Bahreini Behzadi et al., 2014). Once an 9
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appropriate model of the growth curve is selected, selection emphasis can then be directed
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exclusively to the level of the growth curve. It is important to develop an optimal strategy to
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achieve a desired growth shape through changing the growth model parameters. In previous
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studies, a broad range of growth models have been selected, depending on how accurately they
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fit the data. Similar to the results of this study, Lambe et al. (2006) selected the Richards and
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Gompertz models for their accuracy of fit among four competing models (Gompertz, logistic,
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Richards and the exponential model). Also similar to the current results, Goliomytis et al. (2006)
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reported the excellent fit of the Richards function to the weight–age data of Karagouniko sheep.
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On the other hand, da Silva et al. (2012) observed the Richards model was problematic during
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the process of convergence in Santa Inês sheep. Tariq et al. (2013) selected the Morgan-Mercer-
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Flodin model as the best fitted equation for growth curve in Mengali sheep breed of Balochistan.
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Gbangboche et al. (2008) selected the Brody model as the best function to fit the growth curve of
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African Dwarf sheep. On the other hand, Freitas (2005) reported that Logistic, Von Bertalanffy
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and Brody functions were more versatile to fit growth curves in sheep. In the Bergamasca sheep
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in Brazil (McManus et al., 2003), among the fitted growth models (Brody, Richards, and
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Logistic), the Logistic model showed the goodness of fit. The Gompertz and Von Bertalanffy
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models showed the best fit in Morkaraman and Awassi lambs (Topal et al. 2004); Gompertz
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model in Suffolk sheep (Lewis et al., 2002). Malhado et al. (2009) reported both Gompertz and
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Logistic functions presented the best fit of growth curve in Dorper sheep crossed with the local
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Brazilian breeds, Morada Nova, Rabo Largo, and Santa Inês. Bathaei and Leroy (1996),
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evaluating the growth in Mehraban Iranian fat-tailed sheep, selected the Brody function because
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of simplicity of interpretation and ease of estimation. Sarmento et al. (2006) observed that
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Gompertz function presented the best adjustment when compared to the other models in studies
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of growth curves of Santa Inês sheep herds in the state of Paraíba, Brazil. Akbas et al. (1999)
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studied the fitting performance of Brody, Negative exponential, Gompertz, Logistic and
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Bertalanffy models to data on weight-age of Kývýrcýk and Daglýc male lambs and found that
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the Brody model was the best equation for describing the growth of lambs. A same function
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might present variable results according to the breed, population or feature tested (Malhado et
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al., 2009).
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According to the values of DW obtained from fitting the non-linear models of growth
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curve in the current study it was concluded that there was slight positive autocorrelation between
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the residuals for all models. But negative exponential model provided the most positive
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autocorrelation and Richards model provided the lowest one among different models. It seems
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that residual autocorrelation presented no problem, or only a very slight one in the current study.
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Positive autocorrelation is a serial correlation in which a positive error for one observation
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increases the chances of a positive error for another observation (Ghavi Hossein-Zadeh, 2014).
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The relatively large estimates of a parameter, obtained from the best fitted models in this study,
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may indicate that animals are heavier as adults and may be considered slow-growth, as these
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sheep require more time to reach maturity compared to other breeds (da Silva et al., 2012). The
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definition of an optimum adult weight is controversial, once it depends on the species, breed,
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selection method, management system and environmental conditions (Malhado et al., 2009).
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Similar to the current results, da Silva et al. (2012) reported high asymptotic weight estimates in
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Santa Inês sheep. Also, Lôbo et al. (2006) and McManus et al. (2003) obtained greater a values
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in their study on Santa Inês sheep and Bergamácia sheep, respectively. Malhado et al. (2008)
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obtained lower a values ranging from 29.14 to 32.16 kg in Texel×Santa Inês crossed sheep and
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Malhado et al. (2009) reported a values of 29.35 to 32.41 in the fit of different growth functions
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for local Brazilian breeds, Morada Nova, Rabo Largo, and Santa Inês sheep. The parameter b
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represents an integration constant, related to the initial animal weight but lacking a clear
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biological interpretation (Malhado et al., 2009). Animals with high k values show a precocious
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maturity in relation to those with lower k values and similar initial weight. The greater values of
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k parameter for female lambs in this study indicating higher maturity rates (i.e., they reached
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mature weight earlier). Because of the narrow deviation range in the weight at birth, the variation
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among the k values becomes a reliable predictor of the growth rate (Malhado et al., 2009).
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Therefore, animals with higher k values reach asymptotic weight earliest. Bathaei and Leroy
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(1996) estimated much higher k values for body weight than the current study, for males and
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females of the Mehraban Iranian fat tailed sheep, when applying the Brody growth model, over a
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48-month period. This difference could be attributed to the time unit they used, month instead of
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day as in the present study, and the different growth function used (Goliomytis et al., 2006).
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Goonewardene et al. (1981), Rogers et al. (1987) and Perotto et al. (1992) obtained different k
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values when applying different growth functions to the same weight–age data indicating that the
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model of choice affects parameter estimates. The most important biologically correlation for a
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growth curve is between a and k parameters. The negative correlation between these parameters
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in the current study implies that the earliest animals are less likely to exhibit high adult weight;
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i.e., animals that have higher adult weights generally present lower growth rates than animals
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with a lower adult weight (da Silva et al., 2012). Also, similar to the sigmoidal patterns of
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growth curves in this study were obtained by other studies (Goliomytis et al. 2006; Malhado et
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al. 2009; da Silva et al. 2012). The AGR was decreased over the time in the current study and
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this declining trend can be explained by the low nutritional level and poor quality of the pasture
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at the breeding station of Shall sheep (Bahreini Behzadi et al., 2014). Therefore, good nutritional
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plans are needed to minimize the negative effects of diet changes on the AGR losses especially
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after weaning. Sarmento et al. (2006) pointed out that the decrease in the AGR might result from
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improper management practices that fail to match the increasing nutritional demands as long as
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the animals grow. Consistent with the current study, Bahreini Behzadi et al. (2014) reported a
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declining trend of AGR over time in Baluchi sheep. Malhado et al. (2009) reported the AGR
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values were increasing up to reaching a maximum value of 200, 156 and 143 grammas per day
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for Dorper×Santa Inês, Dorper×Morada Nova and Dorper×Rabo Largo sheep in Brazil,
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respectively, and then decreased afterwards.
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Conclusions
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The six non-linear models evaluated in the present study were adequate in describing the
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growth pattern in Shall sheep. However, Richards model provided the best fit of growth curve
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for males, females and all lambs due to the lower values of RMSE, AIC and BIC and greater
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2 values of Radj than other models. The results of this study can help planning farm management
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strategies and decision making regarding the culling of poor producers and selecting the highly
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productive animals just by viewing their growth curve. After selecting a desired model of the
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growth curve for this breed of sheep, it is worth noting to propose an optimal strategy to obtain
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an appropriate growth shape through modifying the growth model parameters.
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Acknowledgement
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The corresponding author would like to acknowledge the University of Guilan for financial
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support of this research (Project No. 989). Also, the Animal Breeding Center of Iran was
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acknowledged for providing the data for this study.
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Lewis, R.M., Emmans, G.C., Dingwall, W.S., 2002. A description of the growth of sheep and its genetic analysis. Anim. Sci. 74, 51–62.
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Lôbo, R.N.B., Villela, L.C.V., Lôbo, A.M.B.O., Passos, J.R.S., Oliveira, A.A., 2006. Genetic
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parameters of estimated characteristics of the growth curve of Santa Inês sheep. R. Bras.
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Zootec. 35 (3), 1012–1019.
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Malhado, C.H.M., Carneiro, P.L.S., Santos, P.F., Azevedo, D.M.M.R., Souza, J.C., Affonso,
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P.R.A.M., 2008. Curva de crescimento emovinos mestic¸ os Santa Inês×Texel criados no
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Sudoeste do Estado da Bahia. Rev. Bras. Saúde Prod. An. 9, 210–218.
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Malhadoa, C.H.M., Carneiroa, P.L.S., Affonso, P.R.A.M., Souza Jr., A.A.O., Sarmento, J.L.R.,
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2009. Growth curves in Dorper sheep crossed with the local Brazilian breeds, Morada
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Nova, Rabo Largo, and Santa Inês. Small Rumin. Res. 84, 16-21.
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McManus, C., Evangelista, C., Fernandes, L.A.C., Miranda, R.M., Moreno-Bernal, F.E., Santos,
335
N.R., 2003. Growth curves of Bergamácia Sheep raised in the Federal District. R. Bras.
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Zootec. 32 (5), 1207–1212.
340 341 342 343
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growth curve of three genotypes of dairy cattle. Can. J. Anim. Sci. 72, 773–782. Rogers, S.R., Pesti, G.M., Marks, H.L., 1987. Comparison of three nonlinear regression models for describing broiler growth curves. Growth 51, 229–239.
Sarmento, J.L.R., Regazzi, A.J., Souza, W.H., Torres, R.A., Breda, F.C., Menezes, G.R.O., 2006. Study of growth curve of Santa Inês sheep. R. Bras. Zootec. 35 (2), 435–442.
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Perotto, D., Cue, R.I., Lee, A.J., 1992. Comparison of nonlinear functions for describing the
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weight of Javanese Fat Tailed sheep. Arch. Tierz. 44, 649-659.
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Noor, R.R., Djajanegara, A., Schüler, L., 2001. Selection to improvement birth and weaning
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SAS, 2002. SAS User’s guide v. 9.1: Statistics. SAS Institute, Inc, Cary, NC.
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Souza, L.A., Carneiro, P.L.S., Malhado, C.H.M., Silva, F.F., Silveira, F.G., 2013. Traditional
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and alternative nonlinear models for estimating the growth of Morada Nova sheep. R. Bras.
348
Zootec. 42(9), 651-655.
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345
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Tariq, M., Iqbal, F., Eyduran, E., Bajwa, M.A., Huma, Z.E., Waheed, A., 2013. Comparison of
350
non-linear functions to describe the growth in Mengali sheep breed of Balochistan. Pak. J.
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Zool. 45(3), 661-665.
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Topal, M., Ozdemir, M., Aksakal, V., Yildiz, N., Dogru, U., 2004. Determination of the best
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nonlinear function in order to estimate growth in Morkaraman and Awassi lambs. Small
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Rumin. Res. 55, 229–232.
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363 364 365 366
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367 368
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Table 1. Functional forms of equations used to describe the growth curve of Shall sheep Functional form
Brody
y a 1 be kt
Negative exponential
y a ae kt
y
Logistic
Von Bertalanffy
y a 1 be kt
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y ae be
y a 1 be kt
378 379 380 381 382
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y= represents body weight at age t (day); a= represents asymptotic weight, which is interpreted as mature weight; and b= is an integration constant related to initial animal weight. The value of b is defined by the initial values for y and t; k= is the maturation rate, which is interpreted as weight change in relation to mature weight to indicate how fast the animal approaches adult weight; m= is the parameter that gives shape to the curve by indicating where the inflection point occurs.
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377
3
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376
kt
Gompertz
Richards 370 371 372 373 374 375
a 1 be ct
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Equation
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383 384 385 386 19
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Table 2. Parameter estimates (standard errors are in parentheses) for the different growth models in Shall sheep
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387 388
Parameter
Brody
Negative exponential
a
40.31 (0.08)
39.13 (0.07)
b
0.90 (0.001)
k
0.0072 (0.00004)
m
-
a
42.85 (0.13)
b
0.90 (0.002)
Gompertz
Von Bertalanffy
Richards
36.17 (0.04)
37.40 (0.05)
38.08 (0.05)
44.99 (0.44)
0.0086 (0.00004)
5.61 (0.06)
2.06 (0.01)
0.51 (0.002)
0.98 (0.002)
-
0.0217 (0.0001)
0.0138 (0.00007)
0.0114 (0.00006)
0.0039 (0.0002)
-
-
-
-
-0.60 (0.01)
41.51 (0.10)
38.18 (0.06)
39.54 (0.07)
40.30 (0.08)
48.80 (0.80)
0.0084 (0.00006)
5.64 (0.08)
2.07 (0.01)
0.51 (0.002)
0.98 (0.003)
ce pt
ed
All lambs
Male lambs
Logistic
M an
Item
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Model
0.0070 (0.00006)
-
0.0213 (0.0002)
0.0135 (0.00009)
0.0112 (0.00008)
0.0035 (0.0003)
m
-
-
-
-
-
-0.59 (0.02)
37.84 (0.09)
36.82 (0.08)
34.20 (0.05)
35.31 (0.05)
35.91 (0.06)
41.51 (0.45)
0.89 (0.002)
0.0089 (0.00006)
5.60 (0.07)
2.04 (0.01)
0.51 (0.002)
0.98 (0.003)
k
0.0075 (0.00006)
-
0.0221 (0.0001)
0.0141 (0.00009)
0.0117 (0.00007)
0.0042 (0.0002)
m
-
-
-
-
-
-0.61 (0.02)
a b Female lambs
Ac
k
20
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Table 3. Comparing goodness of fit for different growth curves in Shall sheep Model
Female lambs
Gompertz
2 Radj
0.9675
0.9634
0.9643
0.9663
DW
1.16
1.03
1.06
1.12
RMSE
6650
7496
7304
6904
AIC
813840
820658
819187
815974
BIC
189671
196480
195017
2 Radj
0.9670
0.9630
DW
1.10
RMSE
391
Richards
0.9668
0.9677
1.14
1.17 6611
815056
813505
191805
190887
189345
0.9636
0.9656
0.9662
0.9672
0.98
0.99
1.05
1.07
1.10
5269
5904
5809
5480
5389
5234
AIC
387780
391005
390545
388895
388418
387597
BIC
97544
100760
100308
98658
98182
97369
2 Radj
0.9738
0.9695
0.9708
0.9727
0.9732
0.9740
DW
1.13
0.97
1.02
1.09
1.11
1.14
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6794
RMSE
3411
3972
3802
3561
3496
3389
AIC
380564
384928
383674
381800
381266
380380
86162
90518
89272
87398
86864
85987
BIC 390
Von Bertalanffy
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Logistic
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Negative exponential
d
Male lambs
Brody
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All lambs
Statistics
Ac ce p
Item
2 Radj : Adjusted coefficient of determination; RMSE: Root means square error; DW: Durbin–
Watson; AIC: Akaike information criteria; BIC: Bayesian Information Criteria
392 393 21
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(a)
(b) (b)
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(a)
394
397 398 399 400 401
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(c)
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402 403 404
Figure 1. Observed body weights for all lambs (a), males (b) and female lambs (c)
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Page 23 of 26
(a)
(b) (b)
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(a)
(a)
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409 410 411 412 413 414
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(c)
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Figure 2. Predicted body weights (kg) as a function of age (days) obtained with different growth models for all lambs (a), males (b) and female (c) lambs 23
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Figure 3. Absolute growth rate (AGR) for all lambs, males and female lambs based on Richards model
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*Conflict of Interest Statement
Dear Editor-in-Chief of Small Ruminant Research
Conflict of interest statement
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The author confirms that there are no known conflicts of interest associated with this manuscript
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which have influenced its outcome.
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Navid Ghavi Hossein-Zadeh
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Corresponding author
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26 December 2014
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S0921-4488(15)30019-5 http://dx.doi.org/doi:10.1016/j.smallrumres.2015.07.014 RUMIN 4987
To appear in:
Small Ruminant Research
Received date: Accepted date:
26-12-2014 19-7-2015
Please cite this article as: Hossein-Zadeha , N.G.,Modeling the growth curve of Iranian Shall sheep using non-linear growth models, Small Ruminant Research (2015), http://dx.doi.org/10.1016/j.smallrumres.2015.07.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights (for review)
Highlights 1. Six non-linear equations were compared to describe the growth curve in Shall sheep.
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2. Richards model was the best growth model for males, females and all lambs.
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3. Negative exponential model provided the worst fit of growth curve among models.
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4. Current models were adequate in describing the growth pattern in Shall sheep.
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*Manuscript (latest version) Click here to view linked References
1
Modeling the growth curve of Iranian Shall sheep using non-linear growth models
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Navid Ghavi Hossein-Zadeh
Department of Animal Science, Faculty of Agricultural Sciences, University of Guilan, Rasht, Iran
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Corresponding author: Navid Ghavi Hossein-Zadeh. Department of Animal Science, Faculty of Agricultural Sciences, University of Guilan, Rasht, Iran, P. O. Box: 41635-1314
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E-mail addresses: [email protected]
[email protected] Tel: +98 13 33690274
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Fax: +98 13 33690281
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Abstract The objective of this study was to describe the growth pattern in Iranian Shall sheep
25
using non-linear models. For this purpose, six non-linear mathematical functions (Brody,
26
Negative exponential, Logistic, Gompertz, Von Bertalanffy and Richards) were used. The data
27
set used in this study were obtained from the Animal Breeding Center of Iran and comprised
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57000 body weight records of lambs which were collected from birth to 400 days of age during
29
1982 to 2012. Each model was fitted separately to body weight records of all lambs, male and
30
female lambs using the NLIN and MODEL procedures in SAS. The models were tested for
31
2 goodness of fit using adjusted multiple coefficient of determination ( R adj ), root means square
32
error (RMSE), Durbin-Watson statistic (DW), Akaike’s information criterion (AIC) and
33
Bayesian information criterion (BIC). Richards model provided the best fit of growth curve in
34
males, females and all lambs due to the lower values of RMSE, AIC and BIC and generally
35
2 greater values of R adj than other models. The negative exponential model provided the worst fit
36
of growth curve for males, females and all lambs. According to the moderate values of DW
37
obtained from fitting different models of growth curve it was concluded that there was positive
38
autocorrelation between the residuals for all models, but this autocorrelation was more obvious
39
for negative exponential model than the other equations. The negative correlation of −0.99 to
40
-0.49 between a and k parameters obtained from fitting different growth models implied that the
41
animal with smaller mature weight will be maturing faster. Evaluation of different growth
42
models used in this study indicated the potential of the non-linear functions for fitting body
43
weight records of Shall sheep.
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Key words: Body weight; Fat-tailed sheep; Growth function; Model fitting
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Introduction The Shall sheep is a local sheep breed of Iran with a population of more than 600000
47
heads. The breed is fat-tailed, large-size, predominantly black or brown with white spots in front
48
of head, well adapted to the harsh climate and raised mainly for its meat that is most important
49
source of protein in Iran and sale of its surplus lambs is the main source of cash income for
50
farmers. Ewes were randomly exposed to the rams at the age of 18 months and lambing was
51
occurred in one season, from mid-January to mid-March. Ewes were kept in the flock up to 6
52
years old. Rams were kept until a male offspring was available for replacement. During the
53
breeding season, single sire pens were used allocating 20-25 ewes per ram. Ewes usually lamb
54
thrice every two years. Lambs remained with their dam until weaning. The flock was mainly
55
kept on range and was fed by cereal pasture, but supplemental feed, including alfalfa and wheat
56
straw, are provided especially around mating season (Amou et al., 2013).
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Growth is one of the most important features of livestock and is defined as increase in
58
live weight or body dimension against age. Changes in live weight or dimension for a period of
59
time are explained by the growth curves (Keskin et al., 2010). Analysis of the animal growth
60
performance through their life span is helpful to establish appropriate feeding strategies and the
61
best slaughter age. Studies focusing on growth curves have increased at recent years due to the
62
development of new computational methods for faster and more accurate analyses as well as the
63
availability of new models to be tested (Souza et al., 2013). In order to increase farmers’ income,
64
there needs to be improvement in the production of these animals. Slow growth rate resulting in
65
low market weight has been identified to be one of the factors limiting profitability in any
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production system (Noor et al., 2001; Abegaz et al., 2010). Growth rate is related to rate of
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maturing and mature weight and these latter traits have been suggested to have relationship with
68
other lifetime productivity parameters in sheep (Bedier et al., 1992; Abegaz et al., 2010). Non-linear mathematical functions, empirically developed by depicting body weight
70
against age, have been appropriate to characterize the growth curve in different animal classes
71
(Malhado et al., 2009). Different growth models have been applied extensively in different
72
species to describe the development of body weight, allowing information from multiple
73
measurements to be combined into a few parameters with biological meaning, in order to
74
facilitate both the interpretation and the understanding of the phenomenon (Lambe, 2006;
75
Malhado et al., 2009). Growth curve parameters provide potentially helpful criteria for changing
76
the association between body weight and age through selection and breeding (Kachman and
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Gianola, 1984) and an optimum growth curve can be obtained by selection for desired values of
78
growth curve parameters (Bathaei and Leroy, 1998). The potential of changing the growth curve
79
shape by breeding may be an attractive aspect for livestock producers through increasing early
80
growth but restricting mature size, and hence maintenance requirements (Lambe et al., 2006).
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Growth curves supply several applications to animal husbandry, such as analysis of the
82
interaction between subpopulations (or treatments) and time; evaluation of the response to
83
different treatments over time; and identification of heavier animals at younger ages within a
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population (Bathaei and Leroy, 1996; Freitas, 2005; Malhado et al., 2009).
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No previous studies have been conducted on growth curve characteristics of the Shall
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sheep. Therefore, the objective of this study was to characterize the growth pattern of Shall sheep
87
using non-linear models. For this purpose, six mathematical models (Brody, Negative
4
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88
exponential, Logistic, Gompertz, Von Bertalanffy and Richards) were compared to evaluate their
89
efficiency in describing the growth curve of Shall sheep.
Materials and methods
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The data set used in this study were obtained from the Animal Breeding Center of Iran
93
and comprised 57000 body weight records which were collected on 11400 lambs from birth to
94
400 days of age during 1982 to 2012. The data were screened several times and defective and out
95
of range records were deleted.
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The non-linear growth models used to describe the growth curves of Shall sheep are
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presented in Table 1. The Brody, Negative exponential, Logistic, Gompertz, Von Bertalanffy and
98
Richards functions were fit to the data to model the relationship between body weight and age.
99
Each model was fitted separately to body weight records of all lambs, male and female lambs
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using the NLIN and MODEL procedures in SAS (SAS Institute, 2002) and the parameters were
101
estimated. The NLIN procedure provides least squares or weighted least squares estimates of the
102
parameters of a non-linear model. For each non-linear model to be analyzed, the model (using a
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single dependent variable) and the names and starting values of the parameters to be estimated
104
must be specified (SAS Institute, 2002). When non-linear functions were fitted, the Gauss-
105
Newton method was used as the iteration method. To begin this process the NLIN procedure first
106
evaluates the starting value specifications of the parameters. If a grid of values is specified,
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NLIN procedure evaluates the residual sum of squares at each combination of parameter values
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to determine the set of parameter values producing the lowest residual sum of squares. These
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parameter values are used for the initial step of the iteration (SAS Institute, 2002). The MODEL
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procedure analyzes models in which the relationships among the variables comprise a system of
111
one or more nonlinear equations. Primary uses of the MODEL procedure are estimation,
112
simulation, and forecasting of nonlinear simultaneous equation models (Ghavi Hossein-Zadeh,
113
2014). The models were tested for goodness of fit (quality of prediction) using adjusted
114
2 coefficient of determination ( R adj ), residual standard deviation or root means square error
115
(RMSE), Durbin-Watson statistic (DW), Akaike’s information criterion (AIC) and Bayesian
116
information criterion (BIC).
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2 R adj was calculated using the following formula:
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n 1 2 2 Radj 1 1 R n p
Where, R 2 is the multiple coefficient of determination ( R 2 1
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RSS ), TSS is total sum TSS
of squares, RSS is residual sum of squares, n is the number of observations (data points) and p is
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the number of parameters in the equation. The R 2 value is an indicator measuring the proportion
122
of total variation about the mean of the trait explained by the growth curve model. The
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coefficient of determination lies always between 0 and 1, and the fit of a model is satisfactory if
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R 2 is close to unity.
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RMSE is a kind of generalized standard deviation and was calculated as follows:
RSS n p 1
RMSE 126 6
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Where, RSS is residual sum of squares, n is the number of observations (data points) and
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p is the number of parameters in the equation. RMSE value is one of the most important criteria
129
to compare the suitability of used growth curve models. Therefore, the best model is the one with
130
the lowest RMSE.
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DW was used to detect the presence of autocorrelation in the residuals from the
132
regression analysis. In fact, the presence of autocorrelated residuals suggests that the function
133
may be inappropriate for the data. The Durbin-Watson statistic ranges in value from 0 to 4. A
134
value near two indicates non autocorrelation; a value toward 0 indicates positive autocorrelation;
135
a value toward 4 indicates negative autocorrelation (Ghavi Hossein-Zadeh, 2014). DW was
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calculated using the following formula:
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e e DW e n
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2
t 1
2 t 1 t
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t n
Where, e t is the residual at time e, and et 1 is residual at time t-1.
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AIC was calculated as using the equation (Burnham and Anderson, 2002):
AIC n ln RSS 2 p
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AIC is a good statistic for comparison of models of different complexity because it
142
adjusts the RSS for number of parameters in the model. A smaller numerical value of AIC
143
indicates a better fit when comparing models.
144 145
BIC combines maximum likelihood (data fitting) and choice of model by penalizing the (log) maximum likelihood with a term related to model complexity as follows: 7
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RSS BIC n ln p ln n n
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A smaller numerical value of BIC indicates a better fit when comparing models.
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After selecting the best fitted function, the absolute growth rate (AGR) was calculated
y ). In fact, the AGR represents t
based on the first derivative of the function in relation to time (
150
the weight gained per time unit which, in this case, equals the daily weight gain estimated
151
throughout a growth period; thereby it corresponds to the mean animal growth rate within a
152
population (Malhado et al., 2009).
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Estimated parameters of non-linear growth models for Shall sheep are presented in Table
156
2. Also, goodness of fit statistics for the six growth models fitted to body weight records are
157
2 presented in Table 3. Radj values had little differences among the models for all lambs, males and
158
2 females, but Richards equation provided the greatest Radj value for all lambs, males and females
159
2 and negative exponential model provided the lowest values of Radj for males, females and all
160
lambs (Table 3). Also, negative exponential function had the lowest values of DW for males,
161
females and all lambs, but other functions had little differences in this regard and Richards
162
function provided the greatest value. Richards equation provided the lowest values of RMSE,
163
AIC and BIC for males, females and all lambs. On the other hand, negative exponential model
164
provided the greatest values of RMSE, AIC and BIC for all lambs, males and female lambs.
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Therefore, Richards equation provided the best fit of growth curve in males, females and all
166
lambs. The negative exponential model provided the worst fit of growth curve for males, females
167
and all lambs. The parameter a is an estimate of asymptotic weight, which can be interpreted as
168
mature, or adult, weight. For the data set studied here Richards and logistic functions provided
169
the greatest and the lowest estimates of a parameter among different models, respectively (Table
170
2). The parameter k, which represents the maturation rate, is another important feature to be
171
considered, since it indicates the growth speed to reach the asymptotic weight. In this study,
172
females generally showed higher values for this parameter than males. The negative correlation
173
of −0.99 to -0.49 between a and k parameters obtained between different models for all lambs.
174
Observed body weights of animals from 1 to 400 days of age are depicted in Figure 1. As shown,
175
there was a sigmoidal trend for body weights along with increase in age. Also, predicted body
176
weights (kg) as a function of age (days) obtained with different growth models for all lambs,
177
males and females are presented in Figure 2. The growth curves estimated were typically
178
sigmoid. The absolute growth rates (AGR) based on the first derivative of Richards function in
179
relation to time are shown in Figure 3. The AGR values declined gradually with increasing time
180
for males, females and all lambs.
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Discussion
183
Over the last few years, growth curves were applied to analyze the growth in farm
184
animals for improving their production system. The growth curves are being used in different
185
aspects of management in producing meat type animals such as designing optimum feeding
186
programs and determining optimum slaughtering age (Bahreini Behzadi et al., 2014). Once an 9
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appropriate model of the growth curve is selected, selection emphasis can then be directed
188
exclusively to the level of the growth curve. It is important to develop an optimal strategy to
189
achieve a desired growth shape through changing the growth model parameters. In previous
190
studies, a broad range of growth models have been selected, depending on how accurately they
191
fit the data. Similar to the results of this study, Lambe et al. (2006) selected the Richards and
192
Gompertz models for their accuracy of fit among four competing models (Gompertz, logistic,
193
Richards and the exponential model). Also similar to the current results, Goliomytis et al. (2006)
194
reported the excellent fit of the Richards function to the weight–age data of Karagouniko sheep.
195
On the other hand, da Silva et al. (2012) observed the Richards model was problematic during
196
the process of convergence in Santa Inês sheep. Tariq et al. (2013) selected the Morgan-Mercer-
197
Flodin model as the best fitted equation for growth curve in Mengali sheep breed of Balochistan.
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Gbangboche et al. (2008) selected the Brody model as the best function to fit the growth curve of
199
African Dwarf sheep. On the other hand, Freitas (2005) reported that Logistic, Von Bertalanffy
200
and Brody functions were more versatile to fit growth curves in sheep. In the Bergamasca sheep
201
in Brazil (McManus et al., 2003), among the fitted growth models (Brody, Richards, and
202
Logistic), the Logistic model showed the goodness of fit. The Gompertz and Von Bertalanffy
203
models showed the best fit in Morkaraman and Awassi lambs (Topal et al. 2004); Gompertz
204
model in Suffolk sheep (Lewis et al., 2002). Malhado et al. (2009) reported both Gompertz and
205
Logistic functions presented the best fit of growth curve in Dorper sheep crossed with the local
206
Brazilian breeds, Morada Nova, Rabo Largo, and Santa Inês. Bathaei and Leroy (1996),
207
evaluating the growth in Mehraban Iranian fat-tailed sheep, selected the Brody function because
208
of simplicity of interpretation and ease of estimation. Sarmento et al. (2006) observed that
209
Gompertz function presented the best adjustment when compared to the other models in studies
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of growth curves of Santa Inês sheep herds in the state of Paraíba, Brazil. Akbas et al. (1999)
211
studied the fitting performance of Brody, Negative exponential, Gompertz, Logistic and
212
Bertalanffy models to data on weight-age of Kývýrcýk and Daglýc male lambs and found that
213
the Brody model was the best equation for describing the growth of lambs. A same function
214
might present variable results according to the breed, population or feature tested (Malhado et
215
al., 2009).
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According to the values of DW obtained from fitting the non-linear models of growth
217
curve in the current study it was concluded that there was slight positive autocorrelation between
218
the residuals for all models. But negative exponential model provided the most positive
219
autocorrelation and Richards model provided the lowest one among different models. It seems
220
that residual autocorrelation presented no problem, or only a very slight one in the current study.
221
Positive autocorrelation is a serial correlation in which a positive error for one observation
222
increases the chances of a positive error for another observation (Ghavi Hossein-Zadeh, 2014).
223
The relatively large estimates of a parameter, obtained from the best fitted models in this study,
224
may indicate that animals are heavier as adults and may be considered slow-growth, as these
225
sheep require more time to reach maturity compared to other breeds (da Silva et al., 2012). The
226
definition of an optimum adult weight is controversial, once it depends on the species, breed,
227
selection method, management system and environmental conditions (Malhado et al., 2009).
228
Similar to the current results, da Silva et al. (2012) reported high asymptotic weight estimates in
229
Santa Inês sheep. Also, Lôbo et al. (2006) and McManus et al. (2003) obtained greater a values
230
in their study on Santa Inês sheep and Bergamácia sheep, respectively. Malhado et al. (2008)
231
obtained lower a values ranging from 29.14 to 32.16 kg in Texel×Santa Inês crossed sheep and
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Malhado et al. (2009) reported a values of 29.35 to 32.41 in the fit of different growth functions
233
for local Brazilian breeds, Morada Nova, Rabo Largo, and Santa Inês sheep. The parameter b
234
represents an integration constant, related to the initial animal weight but lacking a clear
235
biological interpretation (Malhado et al., 2009). Animals with high k values show a precocious
236
maturity in relation to those with lower k values and similar initial weight. The greater values of
237
k parameter for female lambs in this study indicating higher maturity rates (i.e., they reached
238
mature weight earlier). Because of the narrow deviation range in the weight at birth, the variation
239
among the k values becomes a reliable predictor of the growth rate (Malhado et al., 2009).
240
Therefore, animals with higher k values reach asymptotic weight earliest. Bathaei and Leroy
241
(1996) estimated much higher k values for body weight than the current study, for males and
242
females of the Mehraban Iranian fat tailed sheep, when applying the Brody growth model, over a
243
48-month period. This difference could be attributed to the time unit they used, month instead of
244
day as in the present study, and the different growth function used (Goliomytis et al., 2006).
245
Goonewardene et al. (1981), Rogers et al. (1987) and Perotto et al. (1992) obtained different k
246
values when applying different growth functions to the same weight–age data indicating that the
247
model of choice affects parameter estimates. The most important biologically correlation for a
248
growth curve is between a and k parameters. The negative correlation between these parameters
249
in the current study implies that the earliest animals are less likely to exhibit high adult weight;
250
i.e., animals that have higher adult weights generally present lower growth rates than animals
251
with a lower adult weight (da Silva et al., 2012). Also, similar to the sigmoidal patterns of
252
growth curves in this study were obtained by other studies (Goliomytis et al. 2006; Malhado et
253
al. 2009; da Silva et al. 2012). The AGR was decreased over the time in the current study and
254
this declining trend can be explained by the low nutritional level and poor quality of the pasture
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at the breeding station of Shall sheep (Bahreini Behzadi et al., 2014). Therefore, good nutritional
256
plans are needed to minimize the negative effects of diet changes on the AGR losses especially
257
after weaning. Sarmento et al. (2006) pointed out that the decrease in the AGR might result from
258
improper management practices that fail to match the increasing nutritional demands as long as
259
the animals grow. Consistent with the current study, Bahreini Behzadi et al. (2014) reported a
260
declining trend of AGR over time in Baluchi sheep. Malhado et al. (2009) reported the AGR
261
values were increasing up to reaching a maximum value of 200, 156 and 143 grammas per day
262
for Dorper×Santa Inês, Dorper×Morada Nova and Dorper×Rabo Largo sheep in Brazil,
263
respectively, and then decreased afterwards.
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Conclusions
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The six non-linear models evaluated in the present study were adequate in describing the
267
growth pattern in Shall sheep. However, Richards model provided the best fit of growth curve
268
for males, females and all lambs due to the lower values of RMSE, AIC and BIC and greater
269
2 values of Radj than other models. The results of this study can help planning farm management
270
strategies and decision making regarding the culling of poor producers and selecting the highly
271
productive animals just by viewing their growth curve. After selecting a desired model of the
272
growth curve for this breed of sheep, it is worth noting to propose an optimal strategy to obtain
273
an appropriate growth shape through modifying the growth model parameters.
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Acknowledgement
277
The corresponding author would like to acknowledge the University of Guilan for financial
278
support of this research (Project No. 989). Also, the Animal Breeding Center of Iran was
279
acknowledged for providing the data for this study.
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280 References
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weight of Javanese Fat Tailed sheep. Arch. Tierz. 44, 649-659.
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Topal, M., Ozdemir, M., Aksakal, V., Yildiz, N., Dogru, U., 2004. Determination of the best
353
nonlinear function in order to estimate growth in Morkaraman and Awassi lambs. Small
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Rumin. Res. 55, 229–232.
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363 364 365 366
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Table 1. Functional forms of equations used to describe the growth curve of Shall sheep Functional form
Brody
y a 1 be kt
Negative exponential
y a ae kt
y
Logistic
Von Bertalanffy
y a 1 be kt
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y ae be
y a 1 be kt
378 379 380 381 382
m
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y= represents body weight at age t (day); a= represents asymptotic weight, which is interpreted as mature weight; and b= is an integration constant related to initial animal weight. The value of b is defined by the initial values for y and t; k= is the maturation rate, which is interpreted as weight change in relation to mature weight to indicate how fast the animal approaches adult weight; m= is the parameter that gives shape to the curve by indicating where the inflection point occurs.
te
377
3
Ac ce p
376
kt
Gompertz
Richards 370 371 372 373 374 375
a 1 be ct
ip t
Equation
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383 384 385 386 19
Page 20 of 26
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Table 2. Parameter estimates (standard errors are in parentheses) for the different growth models in Shall sheep
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387 388
Parameter
Brody
Negative exponential
a
40.31 (0.08)
39.13 (0.07)
b
0.90 (0.001)
k
0.0072 (0.00004)
m
-
a
42.85 (0.13)
b
0.90 (0.002)
Gompertz
Von Bertalanffy
Richards
36.17 (0.04)
37.40 (0.05)
38.08 (0.05)
44.99 (0.44)
0.0086 (0.00004)
5.61 (0.06)
2.06 (0.01)
0.51 (0.002)
0.98 (0.002)
-
0.0217 (0.0001)
0.0138 (0.00007)
0.0114 (0.00006)
0.0039 (0.0002)
-
-
-
-
-0.60 (0.01)
41.51 (0.10)
38.18 (0.06)
39.54 (0.07)
40.30 (0.08)
48.80 (0.80)
0.0084 (0.00006)
5.64 (0.08)
2.07 (0.01)
0.51 (0.002)
0.98 (0.003)
ce pt
ed
All lambs
Male lambs
Logistic
M an
Item
us
Model
0.0070 (0.00006)
-
0.0213 (0.0002)
0.0135 (0.00009)
0.0112 (0.00008)
0.0035 (0.0003)
m
-
-
-
-
-
-0.59 (0.02)
37.84 (0.09)
36.82 (0.08)
34.20 (0.05)
35.31 (0.05)
35.91 (0.06)
41.51 (0.45)
0.89 (0.002)
0.0089 (0.00006)
5.60 (0.07)
2.04 (0.01)
0.51 (0.002)
0.98 (0.003)
k
0.0075 (0.00006)
-
0.0221 (0.0001)
0.0141 (0.00009)
0.0117 (0.00007)
0.0042 (0.0002)
m
-
-
-
-
-
-0.61 (0.02)
a b Female lambs
Ac
k
20
Page 21 of 26
389
Table 3. Comparing goodness of fit for different growth curves in Shall sheep Model
Female lambs
Gompertz
2 Radj
0.9675
0.9634
0.9643
0.9663
DW
1.16
1.03
1.06
1.12
RMSE
6650
7496
7304
6904
AIC
813840
820658
819187
815974
BIC
189671
196480
195017
2 Radj
0.9670
0.9630
DW
1.10
RMSE
391
Richards
0.9668
0.9677
1.14
1.17 6611
815056
813505
191805
190887
189345
0.9636
0.9656
0.9662
0.9672
0.98
0.99
1.05
1.07
1.10
5269
5904
5809
5480
5389
5234
AIC
387780
391005
390545
388895
388418
387597
BIC
97544
100760
100308
98658
98182
97369
2 Radj
0.9738
0.9695
0.9708
0.9727
0.9732
0.9740
DW
1.13
0.97
1.02
1.09
1.11
1.14
M
an
us
6794
RMSE
3411
3972
3802
3561
3496
3389
AIC
380564
384928
383674
381800
381266
380380
86162
90518
89272
87398
86864
85987
BIC 390
Von Bertalanffy
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Logistic
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Negative exponential
d
Male lambs
Brody
te
All lambs
Statistics
Ac ce p
Item
2 Radj : Adjusted coefficient of determination; RMSE: Root means square error; DW: Durbin–
Watson; AIC: Akaike information criteria; BIC: Bayesian Information Criteria
392 393 21
Page 22 of 26
(a)
(b) (b)
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(a)
394
397 398 399 400 401
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396
(c)
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402 403 404
Figure 1. Observed body weights for all lambs (a), males (b) and female lambs (c)
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Page 23 of 26
(a)
(b) (b)
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(a)
(a)
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409 410 411 412 413 414
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408
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(c)
415 416 417 418
Figure 2. Predicted body weights (kg) as a function of age (days) obtained with different growth models for all lambs (a), males (b) and female (c) lambs 23
Page 24 of 26
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421 422
Figure 3. Absolute growth rate (AGR) for all lambs, males and female lambs based on Richards model
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420
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*Conflict of Interest Statement
Dear Editor-in-Chief of Small Ruminant Research
Conflict of interest statement
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The author confirms that there are no known conflicts of interest associated with this manuscript
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which have influenced its outcome.
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Navid Ghavi Hossein-Zadeh
an
Corresponding author
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26 December 2014
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