PROBABILITY DENSITY FUNCTION OF THE NUMBER OF EMBRYOS COLLECTED FROM SUPEROVULATED NELORE BREED DONORS

Several models have been developed to evaluate reproductive status of cows through concentration of progesterone in milk, the effect of sex selection in the commercial production of herds and bioeconomic performance of the multiple ovulation and embryo transfer system in select herds. However, models describing the production of embryos in superovulated females have yet to be developed. A probability density function of the number of embryos collected by donors of the Nelore breed was determined. Records of 61,928 embryo collections from 26,767 donors from 1991 to 2005 were analyzed. Data were provided by the Brazilian Association of Creators of Zebu and Controlmax Consultoria e Sistemas Ltda. The probability density function of the number of viable embryos was modeled using exponential and gamma distributions. Parameter fitting was carried out for maximum likelihood using a non-linear gradient method. Both distributions presented similar level of precision: root mean square error (RMSE) = 0.0072 and 0.0071 for the exponential and gamma distributions, respectively; both distributions are thus deemed suitable for representing the probability density function of embryo production by Nelore females.


INTRODUCTION
Several models have been developed to evaluate reproductive status of cows through concentration of progesterone in milk (Friggens & Chagunda, 2005), the effect of sex selection in the commercial production of herds (Hohenboken, 1999), and bioeconomic performance of the multiple ovulation and embryo transfer (MOET) system in select herds (Ruvuna et al., 1992a,b,c).However, models describing the production of embryos in superovulated females have yet to be developed.
Most reproduction models use the average value of the number of embryos for bioeconomic analysis (Ruvuna et al., 1992a,b,c;Friggens & Chagunda, 1993;Cardoso et al., 1999;Hohenboken, 1999).How-Sci.Agric.(Piracicaba, Braz.), v.66, n.4, p.447-450, July/August 2009 ever, Beltrame et al. (2007) showed that economic evaluations based on average responses (deterministic simulations) of embryo production yield biased results.Therefore, in practical situations, decisions related to synchronization of the recipient females should be based on the expected distribution of the results.
Recently, some models have included stochasticity in the number generated (Ostergaard et al., 2005;Barioni et al., 2007;Beltrame et al., 2007).Most of them have assumed normal distribution, since the probability distribution of that variable seemed undetermined in the literature (Barioni et al., 2007;Beltrame et al., 2007).In this context, this study has the objective of identifying and fitting an adequate probability density function for the number of embryos generated by superovulated Nelore females, aiming to provide more reliable stochastic simulation models.

Determination of the frequency distribution
Datasets were supplied by the Associação Brasileira dos Criadores de Zebu (Brazilian Zebu Cattle Breeders Association -ABCZ) and by a Controlmax Consultoria e Sistemas Ltda.(Controlmax Consulting anda Systems, Ltd.).The ABCZ dataset (ABCZ DS ), includes all the communications of embryos collected from Nelore breed animals in Brazil from 1991 to 2005; the Controlmax dataset (CM DS ), contained data collected from two embryo transfer centers and two farms.A total three hundred and fifty nine collections of embryos were analyzed.
Datasets were checked for consistency.Data of 254 collections were discarded on the grounds of incoherent date of collection, null fields or number of embryos higher than 35.Records of 61928 collections from 26767 donors remained in the dataset and were used to fit the models.The frequency of collections with no embryo produced was calculated through Equation 1.
where: F 0 is the frequency of superovulated donors with no embryo produced; O 0,CM is the number of superovulated donors with no embryo produced and Nobs CM DS is the total number of collections in the CM DS dataset.
The frequencies of collections with number of embryos other than zero were defined using Equation 2, which corrects for the absence of collection with null observations in the ABCZ DS dataset.
where: F n is the frequency of collections with a given number (n) of embryos collected; O n,ABCZ DS is the number of observations of collection with n embryos collected in the ABCZ dataset and Nobs ABCZ DS is the total number of collections in the CM dataset.

Model fitting
To analyze data collected for using on stochastic analysis in a simulator, a theoretical distribution of the probabilities has to be identified.This distribution must represent, as reliable as possible, the behavior of the variable under study (Freitas Filho, 2001).A graphical comparison of the observed and estimated frequencies for different candidate models was carried out to select the most promising models (Figure 1).
The behavior of the studied data suggested that the negative exponential distribution [EX(λ)] and the Gamma distribution [G(k, λ)] (Benjamin & Cornell., 1970) would result in good data fit .A numerical optimization method was then used to identify the best value of the parameters.
Parameter estimation of the probability distributions were made through the least squares method using a maximum likelihood estimator for normally distributed data (Kostina, 2004).As there is no algorithm with undisputed efficacy for nonlinear estimation of parameters (Dantas et al., 2007), three algorithms were applied: (1) GRG2 (Lasdon et al. 1978); (2) Quasi-Newton (Press et al., 1990); and (3) the Downhill Simplex (Nelder & Mead, 1965;Walsh, 1979;Press et al., 1990).Because the method is not functional for the adjustment of only one parameter, the third algorithm was applied only for fitting the Gamma model.The GRG2 method was applied using Microsoft Excel 2003 ® Solver.The last two methods were applied using Matlab 1 .

RESULTS AND DISCUSSION
The average number embryos per collection (6.1) was found to be higher than that reported by Slenning et al. (1989), based on 39 published studies (4.4 ± 2.8), but lower than that obtained by Baruselli et al. (2006) (8.7 ± 0.9), and close to values reported by Reichenbach2 (2003), Merton et al. (2003) and Visintin et al. (1999) (5~6).The estimate of frequency of null collections, based on the CM DS was 16% (58 null collections in 359 records) which is intermediate to the values reported by Slenning et al. (1989) and by Baruselli et al. (2006), 8.2%, and 20%, respectively.
Both Gamma and Exponential distributions adjustments were significant (p < 0.0001; Tables 1 and 2).The coefficient of determination was high for both models (R 2 = 0.96 for both the Exponential and the Gamma distributions), therefore distributions explained most of the variance in the data.The standard errors of regression -0.0072 and 0.0071 -for the Exponential and Gamma distributions, respectively, are acceptable for simulation purposes.Parameter estimates were λ = 0.156 for the Exponential distribution and k= 0.947, λ = 6.775 for the Gamma distribution.
Although model fitting was completed with a large dataset, it was necessary to include an independent estimate of frequency of unsuccessful collections.There-fore, the greater residual variation (Figure 2) for the frequencies of low number of embryos in the probability distribution may have resulted from the use of a different sample (the CM DS ) to originate the estimated frequencies of unsuccessful collections.Better parameter estimation would require a more comprehensive database reporting unsuccessful embryo collections.
Adjustments herein presented are deemed adequate for the purpose of giving a reference probability distribution function from which adjustments for specific situations may derive.The number of viable embryos per collection in the Nelore breed can be estimated by either the gamma or the exponential distributions, that should, instead of the normal distribution be used for simulations of that trait.

Figure 1 -
Figure 1 -Comparison between the observed and estimated frequencies obtained by the Exponential and Gama distributions.

FV=
Figure 2 -Residual variation (estimated frequency -observed frequency) vs the number of embryos adjusted with the Gamma and Exponential distributions.
of variance for the Gamma distribution fitting FV = Source of variation; DF = Degrees of freedom; SQ = Sum of errors; QM = Average Squared; F = Test of Significance; ** = p < 0.01 -high significance

Table 1 -
Analysis of variance for the Exponential distribution fitting.