FITTING OF A PARTIALLY REPARAMETERIZED GOMPERTZ MODEL TO BROILER DATA N. Okendro Singh Associae Professor (Ag. Sa.), College of Agriculure, Cenral Agriculural Universiy, Iroisemba 795 004, Imphal, Manipur (India) ABSTRACT Differen non-linear growh models were fied o he growh daa of broiler and Gomperz model was found appropriae o he daa under inensive sysem. One of he esimaed parameers of he fied Gomperz model showed a non-linear behavior. Thus, he concep of parial reparameerizaion by expeced-value parameers was used o miigae nonlinear behavior of he esimaed parameer. The growh daa of broiler refied o an explici form of he parially reparameerized version of Gomperz model showed superior resuls. Keywords: Curvaure Effecs, Growh, Nonlinear Model, Parameer and Reparameerizaion. I. INTRODUCTION In he Norh-Easern Region (NEH) of India, here is a high demand of mea (animal proeins). Broiler farming will play an imporan role o mee he demand of food and nuriion in his paricular region. Growh parameers are imporan no only as selecion crieria bu also in erms of feed managemen echniques. Therefore, i is preferable o model he growh rend ha defines periodic changes in he underlying characerisic. There have been quie a few sudies underaken oward he deerminaion of growh rend in broilers in he NEH region of India. Furher, he nonlinear models fied o maximum of he daa usually resuled in highly nonlinear behavior of he esimaed parameers. Many auhors highlighed he imporance of reparameerizaion in nonlinear model fiing especially o ackle he issue of nonlinear behavior of he esimaed parameers ([-5]). In fac, a lile aenion is given o he various reparameerizaions and consequenly, he parameer esimaes hardly saisfy any of he opimum properies. The presen sudy aims o esimae growh rae curves and heir parameers using differen nonlinear growh models o deermine he age-live weigh relaionship of broiler under inensive sysem. The suiabiliy of reparameerized model o miigae he nonlinear behavior of esimaed parameer is also demonsraed. II. MATERIALS AND METHODS The following nonlinear models will provide a reasonable represenaion of average weigh a ime whose f, : model funcion is of he form Logisic: 84 P a g e
Gomperz: Von-Beralanffy: exp () exp exp () () exp here, and are he parameers o be esimaed. The parameer represens he limiing growh value or asympoic size, he scaling parameer and, he rae of mauriy. If is likely o be an offensive parameer say, in equaion (), hen i can be parially reparameerized by expeced-value parameer. To obain an expeced-value parameer from above equaion (), we need o choose value of he regressor variable, wihin he observed range of. Then, we ge he expeced value from equaion () as follows: exp exp Solving his equaion for he parameer only, we ge exp exp Subsiuing back ino he original equaion (), we ge exp exp (4) exp exp The above model is proposed o eliminae he nonlinear behaviour of he esimaed parameer. Here, he likely offensive parameer is reparameerized by expeced-value parameer while he oher parameers are no changed.. Crieria for Model Selecion To examine model performance, summary saisics like roo mean square error (RMSE) and mean absolue error (MAE) are generally used: where n RMSE Ŵ MAE n Ŵ / n n, and ; Ŵ Prediced weigh of h observaion; 85 P a g e
Average weigh; n Number of observaions,,,..., n. The beer model will have he leas values of hese saisics. I is, furher, recommended for residual analysis o check he model assumpions such as independence or he randomness assumpion of he residuals and he normaliy assumpion. To es he independence assumpion of residuals, run es procedure is available in he lieraure ([6]). Furher, Shapiro-ilk s es was applied o check he normaliy assumpion bu, i is no so sringen for selecing nonlinear models because heir residuals may no follow normal disribuion. Moreover, he curvaure in a nonlinear model consiss of wo componens: he inrinsic (IN) curvaure and parameer effecs (PE) curvaure. Deails of he roo mean square (RMS) IN and PE measures of curvaure and curvaure criical value are given in Baes and as ([7-8]). According o Rakowsky ([9]), he IN curvaure is ypically smaller han he PE curvaure, which can be affeced by alering he parameerizaion of he model. Severe curvaure effecs are indicaed by values of IN and PE exceeding he criical value i.e., F, p n 0.05 p, p is he number of parameers involved in he model. In usual, PE is compued when IN is wihin permissible limis and a lower value of PE suggess ha he model exhibis close-o-linear behavior ([6]). Hougaard s measure of skewness, g, can also be employed o assess wheher a parameer is close o linear or wheher i conains considerable nonlineariy. Hougaard s measure is compued as follows: E p ˆ E ˆ MSE L jk L kll jl jkl where he sum is a riple sum over he number of parameers, L J J, n J H, jkl mj m mkl J is he Jacobian marix, J m is he Jacobian vecor, H is he Hessian marix, H m is is componen evaluaed a observaion m and Hougaard s measure of skewness as: g jkl is he h parameer. This hird momen is normalized using he sandard error o give E ˆ E ˆ. MSE L According o Rakowsky ([6]), if g 0., he esimaor ˆ of parameer behavior and, if 0. g 0. 5 kjl lkj is very close-o-linear in, he esimaor is reasonably close-o-linear. If 0.5 g is very apparen. For g, he nonlinear behavior is considerable., he skewness. Descripion of Daa The weekly average growh daase of body weigh/ broiler in gm under inensive sysem observed by Fanai ([0]) was considered. The experimen was conduced o compare he growh performance of broiler under 86 P a g e
inensive sysem and backyard sysem. A oal of 600 commercial broilers were divided ino wo groups, 00 chicks were reared in he College farm under inensive sysem following sandard managemen pracices and remaining 00 were equally disribued o 0 farmers o rear under backyard sysem. All he producion parameers like growh rae, feed conversion rae, moraliy rae and economics under backyard sysem as well as inensive sysem of managemen from birh ill weeks of age were recorded and analyzed. They observed ha overall performance of broiler was comparaively beer under inensive sysem han backyard sysem. Thus he growh daa of broiler under inensive sysem is furher considered for he presen sudy. III. FIGURES AND TABLES Table : Summary Saisics for Fiing of Various Growh Models on Broiler Daa Under Inensive Sysem Parameer Esimaes Growh Models Logisic Gomperz Reparameerized Gomperz β ( ) 505.60 (.50) 69.0 (70.60) 04.80 (.60) β 8.4 (.0) 4.76 (0.5) 4.76 (0.5) β 0.44 (0.0) 0. (0.0) 0. (0.0) Curvaure RMS IN Curvaure 0.0 0.0 0.0 RMS PE Curvaure 0.4 0.4 0. Criical Value 0.5 0.5 0.5 Hougaard s Skewness β ( ) 0.9 0.6 0.0 β 0.50 0.4 0.4 β 0. 0.05 0.05 Goodness of fi RMSE 85.48 47.7 47.7 MAE 6.44 6.67 6.67 Residual Analysis Run es Z.7 0.00 0.00 Shapiro-ilk es p-value 0.79 0. 0. Noe: Figures in parenheses are he corresponding asympoic sandard errors. 87 P a g e
Fig. : Graphical Display of Observed and Prediced Growh of Broiler Under Inensive Sysem (Reparameerized Gomperz Model is Adjudged o be he Bes fi) IV. RESULTS AND DISCUSSION The above daase was fied o differen nonlinear models using SAS 9. version available a College of Agriculure, CAU, Imphal. Differen ses of iniial parameer values have been ried so ha a global convergence crierion is me for fiing of nonlinear models. The global convergence crieria have been me for logisic and Gomperz models. The esimaes of parameers, RMSE, MAE, curvaure effecs, Hougaard s skewness coefficiens, run es saisic ( Z ) value and Shapiro-ilk es p-value for he above wo models under inensive sysem are presened in Table. Gomperz model shows beer performance han oher model when he crieria of RMSE and MAE are used o idenify he bes-fi model. Furher, independence assumpion abou residuals is saisfied since he run es Z values (lies beween 0.00.7, given in Table ) are well below he criical value of.96 of normal disribuion a 5% level of significance. Also, he significance values of Shapiro- ilk es for residuals clearly indicae (p>0.05) ha residuals are normally disribued. The asympoic weigh of broiler esimaed by he Gomperz model is approximaely 69.0 gm under inensive sysem. Moreover, RMS IN curvaure (0.0) and RMS PE curvaure of Baes and as (0.4) are less han he corresponding criical value 0.5 and hey are accepable. However, Hougaard s skewness value of he esimaed parameer of Gomperz model say, ˆ is greaer han 0.5 which shows ha he nonlinear behavior is very apparen. To recify he above problem, a parially reparameerized Gomperz model is proposed considering he parameer ˆ as an offensive parameer, given in equaion (4). The parameer is replaced by in he process of reparameerizaion as is considered o be he offensive parameer. A value of =7 was chosen and he corresponding value of = 6.5 is aken as an iniial value for compuaion of he final esimae of he parameer, which gives he bes resul in erms of leas nonlinear behavior. The reparameerized model was refied o he daa and he resuls are again presened in Table. Furher improvemens in Hougaard s skewness and curvaure effecs are also seen in his refied model. The graph of fied model along wih observed growh daa is also depiced in Fig. which shows he appropriaeness of he proposed model. 88 P a g e
V. CONCLUSION I is summarized ha Gomperz model is adjudged o be he bes fi for he presen daa se. e also conclude ha under he inensive sysem, we can expec he maximum size of approximaely 69.0 gm in weigh of broiler. As such one of he esimaed parameer of Gomperz model was showing non-linear behavior, i was parially reparameerized by expeced value parameer. The parially reparameerized model was refied o he same daa se and he resuls showed appropriaeness of he proposed model. Thus, reparameerizaion will help o miigae he nonlinear behavior of he esimaed parameers. REFERENCES [] Sarada, C. and Prajneshu (005). On appropriae reparameerizaion of a nonlinear saisical model. J. Ind. Soc. Ag. Sa., 59(): 7-4. [] Prajneshu (008). Fiing of Cobb-Douglas producion funcions: revisied. Agriculural Economics Research Review, : 89-9. [] El-Shehawy, S.A. (00). On he selecion of models in nonlinear regression. Asian Journal of Mahemaics and Saisics, (4): 54-66. [4] Ross, J.S.G., Prajneshu and Sarada, C. (00). Reparameerizaion of nonlinear saisical models: a case sudy. Journal of Applied Saisics, 7(): 05-6. [5] Singh, N.O., Paul, A.K., Kumar, S., Alam,., Singh, N.G., Singh, K.N. and Singh, P. (0). Fiing of parial reparameerized logisic growh model o oil palm yield daa. In. J. Agricul. Sa. Sci., 9(Suppl- ): 55-6. [6] Rakowsky, D.A. (990). Handbook of Nonlinear Regression Models. New York: Marcel Dekker. [7] Baes, D.M. and as, D.G. (980). Relaive curvaure measures of nonlineariy. Journal of he Royal Saisical Sociey, B, 4(): -5. [8] Baes, D.M and as, D.G. (998). Nonlinear regression analysis and is applicaions. John iley and Sons, New York. [9] Rakowsky, D.A. (98). Nonlinear Regression Modelling - A Unified Pracical Approach. New York: Marcel Dekker. [0] Fanai, R.L. (0). Comparison of broiler performance under inensive sysem and backyard sysem. CAU Research Newsleer, (): 8-9. 89 P a g e