Abstracts
A segmented polynomial model was used on several egg production curves in an attempt to analyze the differences between the curves in a more descriptive manner. Production curves from different commercial and experimental genetic lines of layers were used. The parameters of the model include the age and level of peak production, rate of decrease in production after the peak and time between start and peak of production. These and other derived variables were compared between the curves using contrasts. The methodology used allowed an easier interpretation of the curves, and the detection of differences in sexual maturity, uniformity and peak of production. This work validates the segmented polynomial model for use in future research dealing with analysis of egg production curves.
Egg production; layers; laying hens; model
Um modelo polinomial segmentado foi usado em diversas curvas de produção de ovos, numa tentativa de analisar as diferenças entre as curvas de um modo mais descritivo. Foram usadas curvas de produção de diferentes linhagens comerciais e experimentais de poedeiras. Os parâmetros do modelo incluem a idade e nível do pico de produção, a taxa de decréscimo na produção após o pico e o tempo entre o início e o pico de produção. Essas e outras variáveis derivadas pelo modelo foram comparadas entre as curvas usando contrastes. A metodologia usada facilitou a interpretação das curvas e a detecção de diferenças na maturidade sexual, uniformidade e pico de produção. Esse trabalho valida o modelo de polinômios segmentados para uso em pesquisas futuras que tratem de análise de curvas de produção de ovos.
Aves de postura; poedeiras; modelo; produção de ovos
Modelo Matemático para Comparar Curvas de Produção de Ovos
Mathematical Model to Compare Egg Production Curves
Autor(es) / Author(s)Fialho FB^{1}
Ledur MC^{1}
Avila VS^{1}
1 Embrapa Suínos e Aves, Concórdia, SC
Correspondência / Mail Address
Flávio Bello Fialho
Embrapa Suínos e Aves
Caixa Postal 21
89700000  Concórdia  SC  Brasil
Email: bello@cnpsa.embrapa.br
Unitermos / Keywords
Aves de postura, poedeiras, modelo, produção de ovos.
Egg production, layers, laying hens, model.
RESUMO
Um modelo polinomial segmentado foi usado em diversas curvas de produção de ovos, numa tentativa de analisar as diferenças entre as curvas de um modo mais descritivo. Foram usadas curvas de produção de diferentes linhagens comerciais e experimentais de poedeiras. Os parâmetros do modelo incluem a idade e nível do pico de produção, a taxa de decréscimo na produção após o pico e o tempo entre o início e o pico de produção. Essas e outras variáveis derivadas pelo modelo foram comparadas entre as curvas usando contrastes. A metodologia usada facilitou a interpretação das curvas e a detecção de diferenças na maturidade sexual, uniformidade e pico de produção. Esse trabalho valida o modelo de polinômios segmentados para uso em pesquisas futuras que tratem de análise de curvas de produção de ovos.
ABSTRACT
A segmented polynomial model was used on several egg production curves in an attempt to analyze the differences between the curves in a more descriptive manner. Production curves from different commercial and experimental genetic lines of layers were used. The parameters of the model include the age and level of peak production, rate of decrease in production after the peak and time between start and peak of production. These and other derived variables were compared between the curves using contrasts. The methodology used allowed an easier interpretation of the curves, and the detection of differences in sexual maturity, uniformity and peak of production. This work validates the segmented polynomial model for use in future research dealing with analysis of egg production curves.
INTRODUCTION
The use of mathematical models to accurately estimate egg production curves is of great importance for poultry research and production. These models allow the comparison of different curves, the prediction of total production using partial records, providing a more detailed analysis of the egg production cycle.
A segmented polynomial model for estimation of egg production curves in laying hens was developed and described by Fialho & Ledur (1997). This model was tested against other existing models (Wood, 1967; McMillan et al., 1970; McNally, 1971; Yang et al., 1989), predicting future production based on better partial data and estimating egg production curves as well as the other models. Besides, the parameters estimated by the segmented polynomial model are easily interpreted, which gives it an advantage in characterizing egg production curves in practice.
The model divides the production curve into three segments. In segment one, before sexual maturity, egg production is zero. Between the beginning of the laying period and the peak of production, the curve is represented by an ascending cubic function. In segment three, after the peak, production decreases linearly. The model is represented by the following equations:
where Y is the level of production in eggs/hen/day, x is the age of the hen in weeks, x_{o} is the age at start of production and x_{p} is the age at the peak of production.
After applying continuity restrictions and redefining the parameters, the proposed model can be expressed as:
where x_{p} is the age of the hens in weeks at the peak of production, P is the level of production at the peak, in eggs/hen/day, s is the rate of production decrease after the peak, in eggs/hen/day per week, and t is the time period between start and peak of production, in weeks. These four parameters characterize egg production curves better than a simple expression of the total number of eggs produced.
In poultry research, it is common to test the effect of different treatments (diets, management, genotypes) on egg production. Total egg production has little information about the production curve. In order to increase the quality of this information, it is common to divide total production into several periods (for instance, production every 4 weeks). However, this has some inconveniences. Although they provide more information, the interpretation of partial productions is not always clear, making it difficult to compare curves. The use of a model describing the production curve can reduce such problems.
The model described above can be used to determine parameters with practical meaning. Peak egg production, for example, describes useful information to evaluate any flock of laying hens. The characteristics of each curve can be described by the parameters of the curve. These parameters may be compared between different curves, to determine the effect of treatments on egg production.
The objective of this paper is to demonstrate, with a practical example, the use of the described model to test and compare egg production curves of different treatments.
MATERIAL AND METHODS
Four White Leghorn strains, two experimental E1 and E2 and two commercial C1 and C2 were used in an egg production trial carried out between 1990 and 1991 (Trial 1). The birds were reared and brooded in boxes. At 14 weeks of age, the hens were transferred into two experimental buildings and housed 3 per cage, in a randomized block design. Each building was divided into 4 blocks and the lines were randomly distributed within each block. Each of the 8 blocks were composed of 4 experimental units (one from each strain) of 42 hens, in a total of 8 x 4 x 42 = 1344 hens.
Between 1991 and 1992 (Trial 2), new generations of lines C1 and E1 were used, along with a third group of E1 hens submitted to a different management, indicated by E1*. Lines C1 and E1 were fed ad libitum, while the feed for line E1* was controlled during rearing and brooding. At 15 weeks of age, the hens were transferred into an experimental building and housed 3 per cage, in a randomized block design, as in Trial 1, but with 8 replicates of 24 hens per strain, in a total of 8 x 3 x 24 = 576 hens. Line E1* was reared in a different location and had to be transported for approximately 10 hours before the trial.
Water and feed were supplied ad libitum until the end of the trials (67 weeks of age in Trial 1; 70 weeks in Trial 2). Management of light followed the recommendation for commercial line C2 on Trial 1 and that for line C1 on Trial 2.
Weekly egg production and mortality were measured from 14 to 67 weeks of age in Trial 1 and from 18 to 70 weeks of age in Trial 2. Mean daily production per hen survived (PHS) and per hen housed (PHH) were evaluated and expressed in number of eggs. PHS was calculated dividing the number of eggs produced in the week by 7 times the number of hens at the beginning of the week. PHH was calculated in the same way, using the number of hens in the first week of the laying period, in order to consider the effects of mortality.
The production curve of each flock was estimated by a segmented polynomial model described by Fialho & Ledur (1997), using the NLIN procedure of SAS (1996). The parameters estimated by the model were the age of the hens at the peak of production (x_{p}, weeks), the peak level of production (P, eggs/hen/day), the rate of production decrease after the peak (s, eggs/hen/day per week) and the time period between start and peak of production (t, weeks). Using these parameters, other variables could be calculated and were also used in the analysis. For each flock, the age at the beginning of production (x_{o}, weeks) and total production up to 40, 50, 60, 70 and 80 weeks (A_{40}, A_{50}, A_{60}, A_{70} and A_{80}, eggs) were estimated by the formulae:
The R^{2} of each curve was calculated, in order to assess the adequacy of the model. The values of x_{p}, P, s, t, x_{o} and A_{40} through A_{80} were then used as response variables in a simple analysis of variance to test differences between line/trial groups, using the GLM procedure of SAS (1996). The effects of line and trial on each of the parameters were tested using contrasts considered to be meaningful.
RESULTS AND DISCUSSION
The means of the parameters estimated by the model and the estimates of accumulated partial productions up to 40, 50, 60, 70 and 80 weeks of age are shown in Table 1 for production per hen survived (PHS), and on Table 2 for production per hen housed (PHH). In the same tables are the significance levels of the contrasts used to compare different combinations of line and trial. Mortality data and the respective contrasts are presented on Table 3.
The model's R^{2} was, on average, .964 for the PHS curves and .956 for the PHH curves. In all flocks, the R^{2} was above .925, except for some flocks of the E1* line in Trial 2, which had R^{2} as low as .771 (this happened because E1* had a very atypical curve).
Considering the average of the two trials, line E1 produced fewer eggs than line C1, in both PHS and PHH, in all periods considered (A_{40} through A_{80}). Line C1 had smaller x_{p}, meaning that the birds reached the peak of production before the E1 birds. This has more effect on partial production up to shorter periods (A_{40} and A_{50}) and tends to be less important when longer periods are considered. However, line C1 also had smaller s, which means better persistence of laying during the production cycle than line E1. This was the main reason for the difference on the level of production between the two lines. The effect of persistence is greater in longer laying periods, as opposed to what happens with the moment of the peak. Thus, production in line C1 was greater, initially because the birds reached the peak earlier (smaller x_{p}) and later because they maintained high production levels for a longer period of time (smaller s). There was no significant difference in the production level at the peak (P). This indicates that the maximum production potential of line E1 is adequate, and that a breeding program for this line should concentrate on improving laying persistence.
As seen on Table 3, line E1 had significantly higher mortality than line C1. This was reflected on the variation of the s parameter between PHS and PHH. When mortality is high, PHH decreases sharply with time as the birds die, while PHS is not affected. Comparing the values of s from Table 1 with those from Table 2, one can see that the increase in s from PHS to PHH is greater in line E1 (.00043 on Trial 1 and .00159 on Trial 2) than in line C1 (.00028 on Trial 1 and .00020 on Trial 2). This reduces PHH relative to PHS, especially towards the end of the production cycle. On line C1, with low mortality, the average difference between PHS and PHH is .3 eggs up to 40 weeks and 1.4 eggs up to 80 weeks. In contrast, on line E1, with higher mortality, the difference between PHS and PHH rises from 1.8 eggs up to 40 weeks to 7.6 eggs up to 80 weeks. This shows the (obvious) effect of mortality on the reduction of egg production.
Lines C1 and E1, as opposed to the others, were present in both trials. For this reason, the production from one year to the other was compared using only these two lines. There was no significant interaction between trial and line in any of the variables studied (parameters and production estimates), indicating that both lines varied in a similar manner from one year to the next. Therefore, the comparison between trials was made using both lines simultaneously.
There was a significant increase in egg production from Trial 1 to Trial 2 in lines C1 and E1, as a consequence of genetic selection. The improvement was due to a significant increase in the peak level of production, estimated by parameter P. As demonstrated by Fialho & Ledur (1997), an increase in P with the other parameters kept constant causes a uniform increase in production during the entire laying cycle.
There was no significant difference between lines E1 and E2 on parameters or production estimates during Trial 1. Between lines C1 and C2 there was significant difference on egg production per hen survived, but not per hen housed. This was due to the greater mortality in C2, as can be observed on Table 3. The estimated production per hen survived up to 80 weeks was 18.3 eggs greater for C2 than for C1. However, production per hen housed, which includes the effects of mortality, was only 8.8 eggs more, which was not a significant difference.
With respect to the parameters used to explain production curve behavior, there was significant difference in x_{p}, P and t for PHS, and only x_{p} and t for PHH. Line C2 started to lay about 1.2 weeks before C1, reaching peak of production about 1.7 weeks after it. This shows less uniformity of age at sexual maturity for C2 birds. Parameter t (time between start and peak of production), which measures the lack of uniformity in sexual maturity, was about 2.9 weeks greater on line C2 than on line C1. Peak PHS level P was significantly greater in line C2, increasing total production. However, the relatively high mortality in line C2 reduced sharply P in PHH, while this did not happen in C1. Consequently, the difference between both lines on peak and total egg production per hen housed was smaller, due to mortality.
In Trial 2, line E1 was compared to a group of the same line with different management (E1*). For PHS, there was significant difference in x_{p}, P, start of lay and production up to 40 and 50 weeks of age. Group E1* started to lay and reached the peak of production before E1. However, production at the peak (P) was smaller. As a consequence of greater precocity, E1* produced more eggs at the start of the laying cycle (A_{40} and A_{50}). This advantage was lost when greater production periods were considered, due to the low value of P. Although E1* took less time to reach the peak, its management did not allow the birds to express their maximum peak potential, jeopardizing the rest of the cycle.
All parameters and production estimates from PHH curves differed significantly between E1 and E1*. Mortality in line E1*, which had different management, was extremely high, deeply affecting production per hen housed. There was high incidence of oviduct prolapse in these birds, which was the main cause of mortality. The E1* birds were transported a long distance shortly before the beginning of production, causing stress and consequently excessive precocity and prolapse. Production peak level (P) was much smaller and persistence of lay dropped drastically (demonstrated by the increase in s), resulting in low egg production in all periods considered.
When a nonlinear model is used, as is the case of the model being studied, the analysis normally uses numerical methods, which approximate the solution asymptotically starting from initial estimates of the parameters. A good choice of these initial estimates is very important. In some circumstances, the choice of initial estimates, which are too different from the real values, may prevent the analysis program from converging to a solution. It is also possible for a parameter set to minimize errors locally, but not globally, which means a nonideal solution may be found.
For these reasons, it is advisable that the initial estimates of the parameters be carefully chosen. An indication that some curve was not well adjusted is a high value for the sum of squares of the errors or impossible values for the parameters. In these cases, the analysis should be repeated, choosing different starting values. The proposed model seemed robust, having little tendency toward this type of problem. However, the possibility of these errors occurring must not be neglected, especially if the shape of the observed production curve differs from the usual.
One example of the type of distortion that may occur can be seen by comparing the PHS (Table 1) and PHH (Table 2) curves of line C1 on Trial 2. Opposite to the expected, the estimate of P and the estimated productions are slightly higher for PHH than for PHS. Of the 8 flocks of this group, 5 had zero mortality, making PHH equal to PHS. The other three had little mortality, which occurred only on the last 3 weeks, having no effect on production most of the time. The reduction on the number of birds at the end of the production cycle caused an increase in s and a slight increase in P to compensate. As a result there was a very slight (.2%) increase in estimated production. This is normal, due to the approximation errors in the parameter adjustments, and does not cause any problems.
Another consequence of the adjustment of the curves is that the start of production estimated by PHS and PHH are not exactly the same, with small differences (.18 weeks on average) between the two estimates.
CONCLUSIONS
The interpretation of different production curves was facilitated by the use of the proposed model. The commercial lines had distinct production curves, showing differences in sexual maturity, uniformity and peak of production. The selection and management of E1 should be directed towards improving persistence of lay. The distinct management given to E1* was deleterious to hens performance, causing excessive precocity, high mortality and reduced egg production. The parameters estimated by the model have practical meaning and can be used to identify the causes of eventual changes in total egg production due to genotype, nutrition, management, health or any other factor, when comparing experimental treatments.
 Fialho FB, Ledur MC. Segmented polynomial model for estimation of egg production curves in laying hens. British Poultry Science 1997; 38:6673.
 McMillan I, Fitz Earle M, Robson DS. Quantitative genetics of fertility. I. Lifetime egg production of Drosophila melanogaster ž Theoretical. Genetics 1970; 65: 349353.
 McNally DH. Mathematical model for poultry egg production. Biometrics 1971; 27: 735738.
 SAS Institute Inc. System for Microsoft Windows, Release 6.12, Cary, North Carolina, USA, 1996 (1 CDROM).
 Wood PDP. Algebraic model of the lactation curve in cattle. Nature 1967; 216: 164165.
 Yang N, Wu C, McMillan I. New mathematical model of poultry egg production. Poultry Science 1989; 68: 476481.
Publication Dates

Publication in this collection
16 Aug 2002 
Date of issue
Dec 2001