Egg production curve fitting using nonlinear models for selected and nonselected lines of White Leghorn hens

Transcription

1 Egg producion curve fiing using nonlinear models for seleced and nonseleced lines of Whie Leghorn hens R. P. Savegnago * V. A. R. Cruz * S. B. Ramos * S. L. Caeano * G. S. Schmid M. C. Ledur L. El Faro and D. P. Munari * 1 * Deparameno de Ciências Exaas Faculdade de Ciencias Agrarias e Veerinarias (FCAV)/Universidade Esadual Paulisa (UNESP) Jaboicabal São Paulo Brazil ; Embrapa Suínos e Aves Concórdia Sana Caarina Brazil ; and Agência Paulisa de Tecnologia dos Agronegócios (APTA) Cenro Lese/ Secrearia de Agriculura e Abasecimeno (SAA) Ribeirão Preo São Paulo Brazil ABSTRACT Egg producion curves describe he laying paerns of hen populaions over ime. The objecives of his sudy were o fi he weekly egg producion rae of seleced and nonseleced lines of a Whie Leghorn hen populaion using nonlinear and segmened polynomial models and o sudy how he selecion process changed he egg-laying paerns beween hese 2 lines. Weekly egg producion raes over 54 wk of egg producion (from 17 o 70 wk of age) were measured from 1693 and 282 laying hens from one seleced and one nonseleced (conrol) geneic line respecively. Six nonlinear and one segmened polynomial models were gahered from he lieraure o invesigae wheher hey could be used o fi curves for he weekly egg producion rae. The goodness of fi of he models was measured using Akaike s informaion crierion mean square error coefficien of deerminaion graphical analysis of he fied curves and he deviaions of he fied curves. The Logisic Yang Segmened Polynomial and Grossman models presened he bes goodness of fi. In his populaion here were significan differences beween he parameer esimaes of he curves fied for he seleced and nonseleced lines hus indicaing ha he effec of selecion changed he shape of he egg producion curves. The selecion for egg producion was efficien in modifying he birds egg producion curve in his populaion hus resuling in geneic gain from he 5h o he 54h week of egg laying and improved he peak egg producion and he persisence of egg laying. Key words: asympoic egg producion geneic gain model error nesed model -es 2012 Poulry Science 91 : hp://dx.doi.org/ /ps INTRODUCTION Egg producion is an imporan par of he commercial poulry indusry. Use of mahemaical models o accuraely fi egg producion curves is necessary o make economic projecions for laying hens (Adams and Bell 1980). I is also of grea imporance in pracical poulry breeding for making predicions abou egg producion on an annual or any oher chosen period basis o faciliae early selecion of he breeder birds (Bindya e al. 2010). Nonlinear models are widely used o fi egg producion daa (Cason and Ware 1990; Miyoshi e al. 1996; Narushin and Takma 2003; Savegnago e al. 2011). Mahemaically egg producion curves can be divided ino 3 phases. The firs phase is he increase in he slope from firs laying o he peak he second is he 2012 Poulry Science Associaion Inc. Received March Acceped July Corresponding auhor: danisio@fcav.unesp.br peak and he hird is he decrease in he slope from he peak unil he end of egg producion (Fialho and Ledur 1997; Grossman e al. 2000). Mos of he nonlinear models presened in he lieraure o fi egg producion can have curve parameers wih a biological inerpreaion which makes i possible o summarize in 3 or 4 parameers wha he egg producion paern is like. However some nonlinear models such as he Wood model (Wood 1967) were found o be inappropriae for fiing egg producion curves on he basis of hens chronological ages because hese models were unable o abruply change from a posiive o a negaive slope afer an iniial seep slope and here was no inflecion poin in he iniial egg producion period hereby leading o greaer predicion errors (Congleon e al. 1981). In addiion Yang e al. (1989) repored ha he parameers of he Wood model lacked reasonable biological inerpreaion. Use of nonlinear models o fi he egg producion of differen populaions makes i possible o compare egg producion beween hem. In his manner egg produc- 2977

2 2978 Savegnago e al. ion can be compared beween seleced and nonseleced lines of hens o sudy changes o egg producion curves caused by selecion processes. Despie he numerous papers in he lieraure ha have used nonlinear models for egg producion curve fiing no papers have used nonlinear models o describe changes o egg producion curves caused by selecion processes. The objecives of his sudy were o fi he weekly egg producion rae of a seleced and a nonseleced line of a Whie Leghorn hen populaion using nonlinear and segmened polynomial models and o sudy how he selecion process changed he egg-laying paern beween hese 2 lines. Birds MATERIALS AND METHODS The daa se was from a Whie Leghorn populaion conaining hens of a seleced line and is respecive nonseleced line (conrol line) defined as CC and CCc respecively. Boh of hese had been developed and were being mainained wihin a selecion breeding program by Embrapa Swine and Pouly Concórdia Sana Caarina Brazil. The CC is a female line of Whie Leghorn ha was iniially seleced in 1989 mainly for egg producion and qualiy of eggs. Boh he CC and he CCc line originaed from a common founder populaion. The nonseleced line (CCc) was esablished from he CC line in accordance wih he mehodology proposed by Gowe e al. (1959) by randomly maing one male from each sire family wih an unrelaed female from each dam family. The nonseleced line was used as a reference populaion o compare he effecs of selecion for egg producion in he CC line (Schmid and Figueiredo 2005). In his sudy he sevenh generaion of he hens was used. Furher deails abou he CC line can be found in Schmid and Figueiredo (2005). All he birds were kep in single cages and hey received food and waer ad libium. The meabolizable energy (ME) and CP concenraions varied depending on he birds phase of life. A commercial die conaining 2850 kcal of ME/kg and 20% CP (1 o 6 wk of age) 2700 kcal of ME/kg and 14% CP (7 o 16 wk of age) and 2800 kcal of ME/kg and 15% CP (17 o 70 wk of age) was provided. The weekly egg producion records over a 54-wk egglaying period (from 17 o 70 wk of age) relaing o 1693 hens from he seleced line and 282 hens from he nonseleced line were used for curve fiing. The birds included were he ones ha survived unil a leas he 70h week of age. The egg producion daa were gahered on 5 d per week. According o Whea and Lush (1961) his measuremen has a correlaion of 0.99 wih weekly egg producion. The individual egg producion was expressed as he weekly egg producion rae where weekly egg producion raes of zero and 1 corresponded o he producion of zero and 5 eggs per week for each hen respecively. Nonlinear Models The weekly egg producion rae for each hen was used o fi he mean populaion curve for CC and for CCc by means of he ieraive Gauss-Newon leassquares mehod as described by Harley (1961) wih a nonlinear regression procedure (NLIN) wihin he SAS 9.2 sofware (SAS Insiue Inc. 2008). The nonlinear models applied o fi he egg producion daa were 1. Logisic (Nelder 1961): d c y = a + e e ( ) 1 x where y = egg producion rae a weeks of laying; a = asympoic value of egg producion a he peak of egg-laying; c = consan; d = mean egg producion week in which egg producion reaches is peak; x = rae of producion decrease afer he peak (eggs/hen-day decrease per week). 2. Comparmenal I (McMillan e al. 1970ab): c d x y = a e ( ) [ 1 ] e where y = egg producion rae a weeks of laying; a = asympoic value of egg producion a he peak of egg-laying; c = weekly rae of increase in egg producion; d = mean iniial week of egg-laying; x = rae of producion decrease afer he peak (eggs/hen-day decrease per week). 3. McNally model (McNally 1971): y (.) 05 a b c + d = e where y = egg producion rae a weeks of laying; a = asympoic value of egg producion a he peak of egg-laying; b c and d = consans. 4. Comparmenal II (McMillan 1981): y = a e e x b ( ) where y = egg producion rae a weeks of laying; a = asympoic value of egg producion a he peak of egg-laying; b = insananeous rae of weekly increase in egg-laying; x = rae of producion decrease afer he peak (eggs/hen-day decrease per week). 5. Yang model (Yang e al. 1989): y ae = + 1 e x c ( d ) where y = egg producion rae a weeks of laying; a = asympoic value of egg producion a

3 he peak of egg-laying; c = reciprocal indicaor of he variaion in week of producion of firs egg; d = mean week of egg producion a sexual mauriy; x = rae of producion decrease afer he peak (eggs/hen-day decrease per week). 6. Segmened polynomial model (Fialho and Ledur 1997): EGG PRODUCTION CURVE FITTING 2979 (AIC) mean square error (MSE) R 2 quaniaive error measuremens (model error and mean model error) and graphical analysis. AIC. Akaike s informaion crierion (Akaike 1974) ha can be approximaed o he leas mean square mehod (Moulsky and Chrisopoulos 2003) is calculaed as follows: y = 0 for < p ip p p y = Peak Peak Peak ip for p ip ip < p y = P s( p ) for p where y = egg producion rae a weeks of laying; Peak = peak producion level (%egg/henday); p = age of hens in weeks a he peak; s = rae of producion decrease afer he peak (eggs/ hen-day decrease per week); ip = ime inerval beween sar and peak of producion. 7. Persisency model (Grossman e al. 2000): y Peak Peak = Ln e p 03 p 1 + e / 03. ( s Ln e e p P)/ 03. ( P e p + )/ p /03. p / e p / 03. where y = egg producion rae a weeks of laying; Peak = peak producion level (%egg/henday); p = age of hens in weeks a he peak; s = rae of producion decrease afer he peak (eggs/hen-day decrease per week); P = number of weeks during which a consan egg producion level is mainained afer he peak. Higher values for parameer x and parameer s indicae shorer persisence of egg laying and vice versa. The Logisic model (Nelder 1961) used in his sudy was a reparamerizaion of he original model described by Brown e al. (1976). The erm e x where e is he neperian number x is he parameer associaed wih he rae of weekly decrease in producion and is he week of egg laying was suggesed by Brody e al. ( ). I was added o he Logisic model by Cason and Ware (1990) o describe he rae of weekly decrease of egg producion afer he peak egg producion because his model was firs developed o fi growh daa which does no have rae of decrease afer he asympoic value as occurs wih egg laying. Saisical Crieria o Evaluae he Fied Curves The goodness of fi of each nonlinear mode was evaluaed by means of Akaike s informaion crierion 3 SSError AIC = n ln k n + 2 where n is he number of daa poins k is he number of parameers in he model and SS Error is he sum of he squared error. MSE. The MSE is calculaed as follows: MSE = n m i= 1 = 1 2 ( yi yˆ i ) nm p where y i and ŷ i are he observed and prediced weekly egg producion raes respecively of hen i a week of laying n is he oal number of hens m is he oal number of weeks of egg laying evaluaed nm is he oal number of observed values in he daa se and p is he number of model parameers. Coefficien of Deerminaion (R 2 ). The R 2 is calculaed as follows: SS R 2 = SS mod el oal where SS model is he sum of he squares of he model and SS oal is he oal sum of he squares. Model Error and Mean Model Error. The fied curve may presen a good fi as indicaed by saisical measuremens; ha is small values of MSE and high values of R 2. However hese indicaors by hemselves do no indicae he rend of he fied curve. The model error (MER) a weeks of egg laying is defined as MER y y = ˆ y where ŷ is he average prediced egg producion rae a week of laying and y is he average real egg producion rae a week of laying. The MER is expressed as he deviaion beween he average prediced egg producion rae minus he average observed egg producion rae on he basis of he average observed egg producion rae. When ŷ (prediced) and y (observed) are equal he deviaion of he adjused egg producion rae a week is zero. When ŷ (prediced) is higher han y (observed) he predicion of egg producion a is overesimaed (posiive model error). When ŷ (prediced) is smaller han y (observed) he predicion of egg producion a is under-

4 2980 Savegnago e al. Table 1. ANOVA and F-es on he differences in curve parameer esimaes beween he seleced and nonseleced lines of hens Model Sum of squares (SS) Degrees of freedom (DF) Null hypohesis (Reduced Model) SS reduced DF reduced Alernaive hypohesis (Full Model) SS full DF full Difference SS reduced SS full DF reduced DF full Relaive difference (SS reduced SS )/SS full (DF reduced DF full )/DF full ( SSreduced SS full) /SSfull Raio (F) DF DF /DF ( ) reduced full full esimaed (negaive model error). The mean model error (MME) is obained as he mean of all he model errors (MER). Graphical Evaluaion of Curve Fiing The fied curve may presen a good fi as indicaed by saisical measuremens bu hese indicaors by hemselves do no indicae he rend of he fied curve. Plos of real and fied daa made i possible o evaluae he flexibiliy of he model ha is is capaciy o fi a a place where he curve changed slope and he deviaions of he daa. Comparison of Nonlinear Trends Beween he Seleced and Nonseleced Lines of Hens The es described in Moulsky and Chrisopoulos (2003) SAS Insiue Inc. (2008) and Schabenberger (2009) was used o compare nesed nonlinear models. This es ascerained wheher here was any difference in he curves fied for egg producion beween he CC and CCc hen lines. The null hypohesis was ha here would be no significan difference (P > 0.05) beween he effecs of he seleced and nonseleced hens on he egg producion rae and consequenly here would be no differences beween he curve parameers fied for he seleced and nonseleced lines. Thus if he null hypohesis were rue here would only be one curve wih he same parameers called he reduced model o fi he egg producion rae of boh he seleced and he nonseleced hens and he differences beween he sample means would be due o chance. The alernaive hypohesis was ha here would be a significan difference (P < 0.05) beween he effecs of he seleced and nonseleced hens on he egg producion rae ha would imply he exisence of 2 differen ses of parameer esimaes called full model: one for each line for curve fiing he weekly egg producion rae. This es was carried ou using he SAS 9.2 sofware (SAS Insiue Inc. 2008). The ANOVA scheme for his es is shown in Table 1. The idea of he es was ha if he reduced model (null hypohesis) was correc he relaive increase in he sum of he squares would be expeced o be approximaely equal o he relaive increase in degrees of freedom. If he full model (alernaive hypohesis) was correc he relaive increase in he sum of he squares would be expeced o be greaer han he relaive increase in degrees of freedom. An alernaive and more common presenaion of he raio (F) is (SSreduced SS full)/(dfreduced DF full) F = SS /DF and he probabiliy of he F-es is given by full Prob > F = 1 Prob ( DF DF DF ). full F reduced full full To calculae he probabiliy of he F-es value he SAS PROBF funcion (SAS Insiue Inc. 2008) was used which reurns he probabiliy of an F disribuion wih (DF reduced DF full ) degrees of freedom o he numeraor and DF full degrees of freedom o he denominaor. Geneic Gain The expeced differences in egg producion rae for each week beween he seleced and nonseleced lines would be due o he hen selecion process for improving egg producion. This was defined as he geneic gain calculaed as ΔG = MS MC where ΔG is he geneic gain in egg producion rae a week of egg laying and MS and MC are he observed average egg producion raes a for he seleced and nonseleced lines respecively. A -es was performed o ascerain wheher here were any significan differences (P < 0.05) in he egg producion rae beween he seleced and nonseleced hens in each week of egg producion. This es was calculaed by means of he TTEST procedure of he SAS 9.2 sofware (SAS Insiue Inc. 2008). RESULTS AND DISCUSSION Descripion of Egg Producion Curve Fiing and Saisical Crieria for Model Selecion The descripive analysis on egg producion for he seleced and nonseleced lines is shown in Table 2. The

5 EGG PRODUCTION CURVE FITTING 2981 Table 2. Descripive analysis on egg producion for he seleced and nonseleced lines 1 Line Number of birds Number of daa lines Mean of oal egg producion rae SD of oal egg producion rae CV (%) of oal egg producion rae Weeks of egg producion a peak 2 Age of hens when egg producion Egg producion reaches he peak 3 rae a peak Age a firs egg (days of age) Week of egg producion a firs egg 2 Seleced hens (0.003) o o (0.023) (0.216) (0.002) Nonseleced hens (0.009) o o (0.034) (0.550) (0.004) 1 Values in brackes are he SEM. 2 Week of egg producion basis (from 1s o 54h week of egg producion). 3 Week of age basis (from 17h o 70h week of age). Table 3. Saisical crieria for evaluaing he curve fiing of he models for he weekly egg producion rae for he seleced and nonseleced lines 1 Seleced line of hens Nonseleced line of hens Model AIC MSE R 2 MME Min MME Max MME (week) 2 (week) 2 AIC MSE R 2 MME Min MME Max MME (week) 2 (week) 2 1. Nelder (1961) (1) (54) (1) (5) 2. McMillan e al. (1970ab) (1) (54) (1) (5) 3. McNally (1971) (1) (54) (1) (53) 4. McMillan (1981) (1) (54) (1) (5) 5. Yang e al. (1989) (1) (54) (1) (5) 6. Fialho and Ledur (1997) (1) (54) (1) (5) 7. Grossman e al. (2000) (1) (54) (1) (5) 1 AIC = Akaike s informaion crierion; MSE = mean square error; R 2 = coefficien of deerminaion; MME = mean model error; Min = minimum mean model error; Max = maximum mean model error. 2 Number inside parenheses is he week in which he minimum or maximum MME occurred.

6 2982 Savegnago e al. Figure 1. Average weekly egg producion rae for he seleced and nonseleced lines over 54 wk of egg producion. = average observed egg producion for he seleced line. = average observed egg producion for he nonseleced line. average weekly egg producion rae for he seleced and nonseleced lines over he 54-wk period of egg producion is shown in Figure 1. The seleced line presened higher oal egg producion rae sooner peak of egg producion higher egg producion a he peak and laer age a firs egg compared wih he nonseleced line. Alhough he nonseleced line had fewer birds han he seleced line he sandard errors of he mean were low hus indicaing accurae averages. According o AIC R 2 MSE and MME he Logisic model was he one ha bes fied he egg producion curves for boh he seleced and he nonseleced lines (Table 3) and he Comparmenal II model was he wors in comparison wih he ohers. However he differences beween he values for he goodness of fi crieria were generally so small ha in pracice he models had he same curve fiing. The MME values indicaed ha all he models had residual errors close o zero and ha he minimum and maximum deviaions along he egg producion cycle were almos he same for all he nonlinear models. On he oher hand he graphical analysis showed ha Comparmenal I McNally and Comparmenal II models (Figure 2) did no presen enough flexibiliy a he poin of inflecion o properly fi he egg producion rae a he peak. This is he reason why he esimaes for parameer a of models Comparmenal I and Comparmenal II surpassed he maximum egg producion which was 1 (100%) for he seleced line (Table 4). The McNally model had he wors esimae for parameer a for boh lines in comparison wih he esimaes for he oher models. Moreover his model which is a modificaion of he Wood model did no have a good fi afer he egg-laying peak: he fied curve of his model showed furher increase afer he peak as he hens go older which did no occur in he observed daa (Figure 2). This resul was also repored by Adams and Bell (1980). Despie he values of saisical crieria being similar for all models (Table 3) when he flexibiliy of inflecion poin (Figure 2) and asympoic egg producion parameer a (Table 4) were aken ino accoun in model evaluaion he Comparmenal I McNally and Comparmenal II models were no appropriae o fi he egg producion curve for boh lines in comparison wih he oher models. The curve fiing of Logisic Yang Segmened Polynomial and Persisency models are shown in Figure 3. Persisency model (Grossman e al. 2000) presened he esimaes for he peak parameer (Peak) ha were closes o he observed egg producion a he peak (Table 2) for boh lines in comparison wih he esimaes for he asympoic value provided by he oher models. I was found ha he fied curves for weekly egg producion rae for he seleced line presened slighly lower esimaes for he rae of producion decrease afer he peak (parameer x of Logisic Comparmenal I McNally Comparmenal II and Yang models and parameer s of Segmened Polynomial and Persisency models) in comparison wih he nonseleced line hus indicaing ha here was lile improvemen in persisence of egg producion afer he laying peak especially beween he 11h and 27h weeks of egg producion in he seleced line (Figure 1). The Persisency model had an ineresing parameer P which esimaed he number of weeks during which he level of consan egg producion was mainained afer he peak. This made i possible o compare he persisence of egg laying. For he seleced line i was esimaed ha egg producion persised for 19 wk afer he peak versus 6 wk for he nonseleced line. This provided an indicaion ha he selecion process was efficien in improving he persisency of egg laying. The parameer d of Logisic model and parameer p of Segmened Polynomial and Persisency models pro-

7 EGG PRODUCTION CURVE FITTING 2983 Figure 2. Fied curves for weekly egg producion rae using Comparmenal I (a) McNally (b) and Comparmenal II (c) models for he seleced and nonseleced lines of hens. = observed egg producion rae for he seleced line; fied curve for egg producion rae for he seleced line; = observed egg producion rae for he nonseleced line; fied curve for egg producion rae for he nonseleced line.

8 2984 Savegnago e al. Table 4. Curve parameer esimaes and SE of models used o fi weekly egg producion raes for he seleced and nonseleced lines Seleced line of hens Nonseleced line of hens Model Parameer Esimae SE Esimae SE 1. Nelder (1961) a c d x McMillan e al. (1970ab) a c d x McNally (1971) a b c d McMillan (1981) a b x Yang e al. (1989) a c d x Fialho and Ledur (1997) Peak p ip s Grossman e al. (2000) Peak p s P vided esimaes for he week of egg producion in which he hens reach heir peak. Logisic model esimaed he peak producion o be one week ahead of he esimaes obained from Segmened Polynomial and Persisency (Table 4). These esimaes were close o he observed week of egg producion in which he hens reached heir peak (Table 2). The esimaes for parameer c of Yang model indicaed ha he average week of producion in which he hens laid heir firs egg were close o wha was observed which was around he firs week of producion (he hens 17h week of age). The parameer d of Yang model indicaed ha on average he age of sexual mauriy was reached around 3 wk afer he birds laid heir firs egg. According o Morris (1966) cied by Koops and Grossman (1992) sexual mauriy occurs when he hens are able o reproduce (age a sexual mauriy) bu his is no he same as he age a which he hens are able o produce (age a firs egg). I can herefore be expeced ha he age a firs egg will be earlier han he age a sexual mauriy and his was observed as differences beween he esimaes for parameers c and d in Yang model hus indicaing ha he age a sexual mauriy (parameer d) occurred on average 3 wk afer he age a firs egg (parameer c). However age a firs egg usually is an indicaion of age a sexual mauriy. Therefore his daa se did no have any records of he hens real age of sexual mauriy. Yang model provided an indicaion of when he hens reached he age of sexual mauriy based on egg producion and age a firs egg. Segmened Polynomial model provided he lengh of ime from he beginning of egg producion o he peak ( ip ) and he parameer p which was he week of peak egg producion. The difference beween p and ip can provide an idea of he beginning of egg producion (Table 4). Table 5. Comparison of nonlinear esimaes for curve parameers beween he seleced and nonseleced lines of hens using he F-es for nesed models 1 Model SS reduced DF reduced SS full DF full F-es P-value 1. Nelder (1961) < McMillan e al. (1970ab) < McNally (1971) < McMillan (1981) < Yang e al. (1989) < Fialho and Ledur (1997) < Grossman e al. (2000) < SS reduced = sum of squares of he Reduced Model; SS full = sum of squares of he Full Model; DF reduced = degrees of freedom of he Reduced Model; DF full = degrees of freedom of he Full Model.

9 EGG PRODUCTION CURVE FITTING 2985 Figure 3. Fied curves for weekly egg producion rae using Logisic (a) Yang (b) Segmened Polynomial (c) and Persisency (d) models for he seleced and nonseleced lines of hens. = observed egg producion rae for he seleced line; fied curve for egg producion rae for he seleced line; = observed egg producion rae for he nonseleced line; fied curve for egg producion rae for he nonseleced line.

10 2986 Savegnago e al. Figure 4. Geneic gain in each week of egg producion. Nonsignifican differences (P > 0.05) are marked wih * and significan differences (P < 0.01) are marked wih. Comparison of Nonlinear Trends Beween he Seleced and Nonseleced Lines of Hens There were highly significan differences according o he F-es (P < ) beween he esimaes for he curve parameers fied in he Full and Reduced models. This indicaed ha he effec of selecion over 7 generaions had changed he shape of he egg producion curves beween he 2 lines (Table 5) especially in relaion o he parameer a. Geneic Gain In his populaion he differences in egg producion in each week beween he 2 lines can be explained by he geneic gain obained hrough he selecion process (Figure 4). Geneic gain occurred when he average weekly egg producion rae was higher in he seleced line han in he nonseleced line in his populaion. Geneic gain was observed in his populaion from he 5h o he 54h week of egg producion hus indicaing ha he selecion had no been effecive in improving egg producion during he firs weeks of he producion cycle. The -es was significan for almos all weeks of egg producion excep for he 1s 5h and 10h weeks of egg producion in his populaion. Evaluaion of he models using saisical crieria (Table 3) graphical crieria (Figures 2 and 3) and biological inerpreaion of he parameers (Table 4) showed ha Logisic Yang Segmened Polynomial and Persisency models were appropriae for egg producion curve fiing for he seleced and nonseleced lines in his populaion. These models presened parameer esimaes wih biological inerpreaions of imporance for he poulry indusry and for research like peak egg producion persisence of egg laying afer he peak rae of decrease in egg producion afer he peak he week in which egg producion reached is peak he lengh of ime from he sar o he peak of egg producion age a firs egg and age a sexual mauriy (Table 4). The selecion of hens for egg producion was efficien in his populaion for modifying he egg producion curve (Table 5) hus resuling in geneic gains afer he fourh week of laying consising of improved peak egg producion and persisence of egg producion in his populaion (Figure 4). ACKNOWLEDGMENTS Financial suppor was provided by he Brazilian Agriculural Research Corporaion (Embrapa; Empresa Brasileira de Pesquisa Agropecuária). R. P. Savegnago and S. L. Caeano were graned scholarships by he São Paulo Research Foundaion (FAPESP; Fundação de Amparo à Pesquisa do Esado de São Paulo). V. A. R. Cruz and S. B. Ramos received scholarships from he Coordinaion Office for Advancemen of Universiy-level Personnel (CAPES; Coordenação de Aperfeiçoameno de Pessoal de Nível Superior) in conjuncion wih he Posgraduae Program on Geneics and Animal Breeding Faculdade de Ciências Agrárias e Veerinárias Universidade Esadual Paulisa (FCAV - UNESP). D. P. Munari held a produciviy research fellowship from he Naional Council for Scienific and Technological Developmen (CNPq; Conselho Nacional de Desenvolvimeno Cienífico e Tecnológico). REFERENCES Adams C. J. and D. D. Bell Predicing poulry egg producion. Poul. Sci. 59: Akaike H A new look a he saisical model idenificaion. IEEE Trans. Auoma. Conr. 19:

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