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Comparison of Non-Linear Functions to Describe the Growth in Mengali Sheep Breed of Balochistan.

Byline: Mohammad Masood Tariq, Farhat Iqbal, Ecevit Eyduran, Masroor Ahmad Bajwa, Zil E Huma and Abdul Waheed

Abstract.- The objectives of this study were to estimate and compare the goodness of fit six non-linear growth models, i.e. Gompertz, Logistic, Von Bertalanffy, Richards, Weibull and Morgan-Mercer-Flodin (MMF), in Mengali sheep of Balochistan. For this purpose, monthly body weight data from 2377 sheep from birth up to 360 days of age were used. The average body weights in each period were used to define the weight - age relationship in these sheep.

The models parameters, adjusted coefficient of determination (R2 ), root mean square error (RMSE), Akaike's and Bayesian information criteria (AIC and BIC), predicted body weights and residuals were calculated for each model. Among six growth functions, the MMF model was found appropriate for its accuracy of fit according to the highest Adj (0.9991) and the lowest RMSE (0.3348) followed by the Von Bertalanffy and Weibull models. The MMF model was also selected the best based on the lowest values of AIC (9.6516) and BIC (10.0489).

The results of this study suggest that the MMF model can be used to accurately predict the growth in body weight of Mengali sheep.

Key words: Mengali sheep, weight-age relationship, growth models, non-linear regression models


Among many different breeds of sheep in Balochistan, the Mengali sheep is one of the important breed found in black or brown color with white patches on the body, prominent head with black face but white forehead (Khan et al., 2007). According to Kakar and Ahmad (2004) the mature body weight is 34 kg in the rams and 27 kg in the ewes. Though the source of origin of Mengali sheep is not known, it is mostly raised by native "Mengal" tribe (most populated tribe of Kalat Division area) therefore the breed is famous as Mengali (local farmers also name it Budi) Khadkucha (Mastung) is identified to be a hub of this breed (Tariq et al.,2011).

Growth is one of the most important traits in farm animals and defined as an increase in body size per unit time (Topal et al., 2004). Body growth is related to an increase in cell number and volume.

Age related changes can be observed in the weight or size of any organ, in the composition of tissues, in cell size and number as well as in live weight (Eisen, 1976). Growth curves are also used for investigating optimum feeding programmes, determining optimum slaughtering age and the effects of selection on curve parameters and on live weight at a certain age (Blasco and Gomes, 1993).

Among statistical procedures available for analyzing growth data, fitting a non-linear function offers an opportunity to summarize the information contained in the entire sequence of size-age points into a small set of parameters that can be interpreted biologically and used to derive other growth traits (Perotto et al., 1992). Numerous growth equations have been developed to describe and fit the nonlinear sigmoid relationship between growth and time. Changes in live weight or dimension for a period of time are explained by the growth curves such as, Brody, Gompertz, Logistic and Von Bertalanffy models in Turkish Kivircik and Daglic male lambs (Akbas et al., 1999), in Awassi lambs (Tekel et al., 2005), in Akkaraman and German black headed mutton x Akkaraman (Kucuk and Eyduran, 2009) and in Mengali sheep (Tariq et al.,2011).

The objectives of this study were to compare the goodness of fit of non-linear function (Gompertz, Logistic, Von Bertalanffy, Richards, Weibull and Morgan-Mercer-Flodin) and to provide a specific shape of the growth curve according to the environmental effects on monthly weight from birth to 360 days of age in Mengali sheep.

Where; Yijklm = Trait of interest observed; u = Population mean; Fi = Effect of the flock (1-4); YOBj = Year of birth; SOBk = Season of birth (Spring, Autumn); TOBl = Type of birth(Single, twin); SEXm = Male and female; b= regression coefficient; Pijklmn = maternal effect of dam, eijklmn = random error associated with each observation. It is assumed to be normally distributed with mean zero and variance +e.2.


The pedigree and performance data on Mengali sheep of Balochistan recorded on 2377 lambs descended from 581 ewes and 56 rams in four flocks at three different locations were available for the present study. Data on four sheep flocks maintained at three places Mastung (Khadkocha), Nushki (Peer Wala) and Quetta (Killi Hassni) and Experimental station at Centre for Advanced Studies in Vaccinology and Biotechnology (CASVAB) (ESC), University of Balochistan (UoB) Brewery Road, Quetta were collected monthly from birth to 360 days of age during Jan 2005 to Dec 2009. Table I shows the average body weight of these sheep for each period. The same data set was used by Tariq et al. (2011) and the Gompertz curve was fitted to the Mengali sheep data.

Table I.- Average body weight of Mengali sheep by age.












Table II represents the estimated parameters along with standard errors of different growth models. All parameters are found significant (P<0.05) except for two parameters of Richards model (l and ). Hence, we exclude the Richards model for further evaluation.

Table III shows the predicted values and residuals obtained after fitting each model to the body growth data. The goodness of fit measures (R2, R2Adj, RMSE, AIC and BIC) are also displayed. The determination coefficient for the MMF model was found the highest (0.9995) and the lowest for Logistic model (0.9943). The values for Bertalanffy and Weibull models were similar (0.9993) followed by the Gompertz model (0.9943). The adjusted coefficient of determination, in descending order, for MMF, Bertalanffy, Weibul, Gompertz and Logistic models were 0.9991, 0.9990, 0.9987,0.9921, and 0.9792, respectively.

The MMF,Bertalanffy and Weibull models showed the highest values. When models are evaluated based on RMSE, the MMF model showed the lowest values (0.3348) followed by Bertalanffy (0.3577), Weibull (0.3988), Gompertz (0.9994) and Logistic (1.6167). The MMF model outperformed other competing models according to goodness of fit by RMSE. This model also showed the minimum AIC and BIC values (AIC=9.6516, BIC=10.0489) followed by Bertalanffy (AIC=10.4654, BIC=10.8132) and Weibull (AIC=12.4501, BIC=12.8473). The values of AIC and BIC we found very high for Gompertz (AIC=26.9336, BIC=27.2514) and Logistic model (AIC=34.6289, BIC=34.9467).

Based on various evaluation measures we found that the MMF model has the best fitting to the body weight data of Mengali sheep. The Bertalannfy and Weibull models showed higher accuracy. The performance of Gompertz model had smaller R2Adj and larger RMSE and AIC/BIC as compared to the three modes mentioned above. The Logistic model had the least performance as compared to other non- linear growth models used in this study showing least values for adjusted coefficient of determination and the highest value for root mean square error, AIC and BIC.

The rank of the model was determined based on three evaluation criteria (R2Adj RMSE, AIC, BIC) and showed as a superscript on models in Table III. We can say that the best three models for describing the growth rate of Mengali sheep are Morgan- Mercer-Flodin, Von Bertalanffy and Weibull model, in descending order.

The Brody function was found to be sufficient for the growth in Morkaraman, Awassi and Tushin sheep by Esenbuga et al. (2000) and Bilgin and Esenbuga (2003). The Gompertz function was found to be appropriate for describing the growth curve of Suffolk sheep (Lewis et al.,2002). The Gompertz and Bertalanffy were found the best fitted functions for the Morkaraman and Awassi breed, respectively, by Topal et al. (2004). On the other hand, Tekel et al. (2005) concluded that Logistic, Gompertz and Bertalanffy models described growth of Awassi lambs better than Brody and Negative exponential models.

For Kivircik breed, the reported value of R2 was 0.993 (Akbas et al., 1999), for Morkaraman, Awassi and Tushin lambs were 0.99, 0.99 and 0.98, respectively (Esenbuga et al., 2000) for Awassi lamb the reported values was 0.98 (Tekel et al.,2005) and for Norduz female lamb it was 0.997 (Kum et al., 2010). Tariq et al. (2011) fitted the Gompertz curve to the same Mengali sheep data and the reported values of R2 and RMSE were 0.992 Gompertz curve reliably explained relationship between weight and age in Mengali sheep. The R2 and RMSE values of Gompertz curve in this study are in line with Tariq et al. (2011). In addition we calculated the adjusted determination coefficient and model selection criteria (AIC and BIC). A slight difference in RMSE value may be due to different software we used for estimating non-linear model. The Gompertz curve though fit the data reasonably as mentioned by Tariq et al. (2011) but ranked 4 out of 5 models considered in the present study.

Gompertz growth curve fitted to body weight.from birth to 360 days of age. The fitted curves of Von Bertalanffy and Weibull model showed the same trend as MMF and therefore not displayed. The Gompertz curve is plotted in order to compare its difference with that of the best fitted curve (MMF) in this study. It is evident from the figure that the MMF growth curve predicted the body weight of Mengali sheep better than the Gompertz growth curve. This further supports our findings that the MMF model describes the growth of body weight better than the Gompertz model.

In this study, six different growth models (Gompertz, Logistic, Von Betalanffy, Richards, Weibull and Morgan-Mercer-Flodin) were studied to determine the growth in the body weight of Mengali sheep breed from birth to 1 year of age. The MMF model was found to be the best model based on various performance measures such as adjusted coefficient of determination, root mean square error and model selection criteria (AIC and BIC).


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Author:Tariq, Mohammad Masood; Iqbal, Farhat; Eyduran, Ecevit; Bajwa, Masroor Ahmad; Huma, Zil E.; Waheed,
Publication:Pakistan Journal of Zoology
Article Type:Report
Geographic Code:9PAKI
Date:Jun 30, 2013
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