Variance Component and Breeding Value Estimation for Reproductive Traits in Laying Hens Using a Bayesian Threshold Model

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1 Variance Component and Breeding Value Estimation for Reproductive Traits in Laying Hens Using a Bayesian Threshold Model J. Bennewitz,* 1 O. Morgades,* R. Preisinger, G. Thaller,* and E. Kalm* *Institut für Tierzucht und Tierhaltung, Christian-Albrechts-Universität, D Kiel, Germany; and Lohmann Tierzucht GmbH, D Cuxhaven, Germany ABSTRACT Variance components and breeding values applied that considered the record of each egg set as a were estimated for 3 reproductive traits in a pure line of repeated observation of the hen. The estimated heritability was 0.067, 0.126, and for the traits proportion of White Leghorn laying hens. The traits were proportion of fertile eggs of eggs set, proportion of first quality chicks fertile eggs of eggs set, proportion of first quality chicks of eggs set, and proportion of first-quality chicks of fertile of eggs set, and proportion of first-quality chicks of fertile eggs, respectively, and the SE were small. It was shown eggs. A total of 3,020 hens were tested up to 3 times over that the heritability estimates were substantially higher a period of 7 d. For the definition of the traits, each egg from their expected values based on linear models. This was scored for each trait either as 0 or 1. To account for results in a higher genetic progress and consequently the binomial distribution of the data, a Bayesian animal threshold model implemented in a Gibbs sampler was favors the applied Bayesian threshold model for a genetic evaluation of binomial distributed reproductive traits. Key words: variance component, breeding value, reproductive trait, laying hen, Bayesian threshold model 2007 Poultry Science 86: INTRODUCTION The most economically important traits for commercial poultry production are laying intensity, egg size, eggshell strength, and feed conversion ratio (Preisinger, 1998). Good fertility is the prerequisite for a genetic improvement in these production traits, because fertile animals allow a higher selection intensity in the nucleus tie of a breeding scheme and, in addition, speed up the transfer of genetic progress from the nucleus into the production tie by the multipliers. However, breeding for reproductive traits is a difficult task, because the heritability of fitness-related traits is generally low (Falconer and MacKay, 1996). Hence, especially for the breeding of these traits, the choice of a suitable statistical model is important. In poultry breeding, important reproductive traits are the proportion of fertile and hatchable eggs (Gowe et al., 1993). Common to these traits are their binomial distribution. In some genetic evaluations, these traits are treated as normal distributed traits (Förster, 1993; Szwaczkowski et al., 2000), which is based on the approximation of the binomial distribution by the normal distribution, if the number of observations is large (Collett, 1991). This has, however, disadvantages. First, the approximation might 2007 Poultry Science Association Inc. Received September 8, Accepted January 21, Corresponding author: jbennewitz@tierzucht.uni-kiel.de be of unequal quality for the hens, because they show a different number of observations. Second, estimated genetic parameters (e.g., heritabilities) are difficult to interpret, because they depend on the mean of the trait, known as the problem of the mean dependent variance (Lynch and Walsh, 1998). To overcome the shortcoming of the nonconstant variance, the authors mentioned above (Förster, 1993; Szwaczkowski et al., 2000) applied a variance stabilization transformation to the data (Collett, 1991), which assumed that the number of observations was more or less equal for all hens. An alternative way of modeling binomial distributed traits is to apply threshold models, which assume a continuous but unobservable, normally distributed variable underlying the phenotypic expression of a binary scored trait (Sorensen and Gianola, 2002). If the unobservable variable value exceeds a fixed threshold, the respective binary variable takes value 1 and 0 otherwise. Threshold models are frequently applied in the genetic analysis of disease traits in dairy cattle (Heringstad et al., 2004; Hinrichs et al., 2005). The aim of the present study was the application of a Bayesian threshold animal model for the estimation of variance components and, subsequently, breeding values for 3 reproductive traits in a pure line of a laying stock. MATERIALS AND METHODS Data and Traits Records from 3,020 hens from a full-pedigreed pure line of White Leghorn were used. Average number of 823

2 824 BENNEWITZ ET AL. Table 1. Phenotypic mean, SD, and minimum and maximum number of eggs set, of fertile eggs, and of firstquality chicks per hen (n = 8,466 hen and test combinations) Trait Mean SD Minimum Maximum Number of eggs set per hen Number of fertile eggs per hen Number of first-quality chicks per hen daughters per sire and per dam was 36 and 6, respectively. The number of animals in the pedigree was 4,108. The hens were artificially inseminated using pooled semen from different males. This was done to eliminate the effect of each individual male in terms of fertility rate and to give each hen a similar chance when producing fertile eggs. Each hen was tested up to 3 times within a period of 4 wk, resulting in 8,466 hen and test combinations, with an average of 2.65 tests per hen. Age at first test was 32 wk. For each test and each hen, hatching eggs were collected over a period of 7 d, stored for 7 d, and subsequently incubated for 504 h. The incubation time was shorter than usual to reduce early chick mortality due to dehydration during the hatch. If the incubation time takes a long time for hatching every chick, the very first chicks will suffer from high ambient temperature in the hatcher. If all chicks hatch in a shorter time frame, the challenge for the chicks in the hatcher will be reduced, and early chick liveability will be improved. Fertility was measured on the 18th day of incubation by candling the eggs during transfer from the incubator to the hatcher. The number of eggs set, fertile eggs, and first-quality chicks were recorded for each animal and each test. From these figures, the traits fertile eggs of eggs set (FE), first-quality chicks of eggs set (CE), and first-quality chicks of fertile eggs (CFE) were calculated as follows. For FE, each fertile (unfertile) egg set was coded with a 1 (0). For CE, each egg that resulted (not) in a first-quality chick obtained a 1 (0). Similarly, for CFE, each fertile egg that resulted (not) in a first-quality chick obtained a 1 (0). For description of the data, see Tables 1 and 2. Bayesian Statistical Analysis A multicode was created using the age of the hen at the beginning of the respective test (in weeks), house, row, and tier where the animals were kept, and the number of tests. The effect of this multicode was tested with the procedure GENMOD of the statistical program package SAS (SAS Institute, 2002) and was highly significant for all 3 traits (not shown). Variance components were estimated univariately applying the following repeatability Bayesian threshold animal model: λ = Xβ + Za + Wpe + e where λ = a vector of unobservable variables of the hens with phenotypic information, subsequently denoted as liabilities; β =a23 1 vector of the effects of the multicode specified above; a =ag 1 vector of additive animal effects, with g being the number of animals, pe =ah 1 vector with permanent environmental effects common to all observations of a hen and h being the number of permanent environmental effects; and e = a vector with residuals. Additionally, X, Z, and W were known incidence matrices. Improper uniform priors were assumed for the effect of the multicode. A normal distribution was used as prior for the effect of the animals as p(a Aσ 2 a ) N(0,Aσ 2 a), where A = the numerator relationship matrix of the animals and σ 2 a = the unknown additive genetic variance with an improper uniform prior. Similarly, a normal distribution for the permanent environmental effects was used as prior as p(pe σ 2 pe) N(0,Iσ 2 pe), where I = an identity matrix and σ 2 pe = the unknown permanent environmental variance with an improper uniform prior. The marginal posterior distributions of all unknowns in the model were obtained using Gibbs sampling. The liabilities were created by data augmentation, as described by Sorensen et al. (1995), drawing random variables from truncated normal distributions, which are conditional upon the other fixed and random effects in the model. The effect β i was sampled from: β i N(x iλ* dxx 1 i, σ 2 e dxx 1 i ) where β i = the ith component of β; x i = the ith column vector extracted from X; x i λ* = the sum of corrected liabilities pertaining to the ith level of β; λ* = λ corrected for all fixed and random effects except the ith component of β, and dxx was the ith diagonal element of X X. With binary data, the threshold and the residual variance (σ 2 e) were not identifiable. Therefore these parameters were set to 0 and 1, respectively. The ith permanent environmental pe i was sampled from: pe i N(w iλ** dww 1 i, σ 2 e dww 1 i ) where p i = the ith component of p; w i = the ith column vector extracted from W; w iλ** = the sum of corrected observations pertaining to the ith level of pe; λ** = λ Table 2. Phenotypic mean and SD for the traits liability to fertile eggs of eggs set (FE), liability to first-quality chicks of eggs set (CE), and liability to first-quality chicks of fertile eggs set (CFE) Trait n Mean SD FE 55, CE 55, CFE 47,

3 GENETIC PARAMETERS FOR REPRODUCTIVE TRAITS IN LAYING HENS 825 Table 3. Mean and SD in parenthesis of posterior distribution of additive genetic variance (σ 2 a), permanent environmental variance (σ 2 pe), heritability, and repeatability of liability to fertile eggs of eggs set (FE), liability to first-quality chicks of eggs set (CE), and liability to first-quality chicks of fertile eggs set (CFE) 1 Trait σ 2 a σ 2 pe Heritability Repeatability FE (0.019) (0.016) (0.014) CE (0.029) (0.021) (0.019) CFE (0.029) (0.021) (0.020) The error variance (σ 2 e) was set to 1. corrected for all fixed and random effects except the ith component of pe, and dww i = the ith diagonal element of W W + Ia 1, where I = an identity matrix and a 1 = σ 2 e/σ 2 pe. The ith animal effect a i was sampled from: a i N (z i λ*** dzz 1 i, σ 2 e dzz 1 i ) where a i = the ith component of a, λ*** = λ corrected for all fixed and random effects except the ith animal effect; z i = the ith column vector extracted from Z; z iλ*** = the sum of corrected liabilities pertaining to the ith level of a; dzz i = the ith diagonal element of the matrix Z Z + A 1 a 2 and a 2 = σ 2 e/σ 2 a. The σ 2 pe was sampled from an inverted χ 2 distribution with h 2 df. The inverted χ 2 distribution was scaled by pe pe. The σ 2 a was sampled from an inverted χ 2 distribution with g 2 df. Here, the inverted χ 2 distribution was scaled by a A 1 a. The Gibbs sampler was run in a single long-chain scheme. For all traits, the sampler ran 120,000 rounds. Convergence was determined by visual inspection of the trace plots. The first 20,000 iterations were deleted (burnin plus safety margin). The effective sample size was estimated using time series methods as described by Sorensen et al. (1995), applying the SAS procedure AUT- OREG (SAS Institute, 2002). It was >250 for the variance components for all traits. The mean of the respective pos- terior distribution provided an estimate for the additive genetic variance and the permanent environmental variance for the liabilities to the traits, respectively. The estimation of the heritability and repeatability from the estimated variance components was straightforward. For the estimation of best linear unbiased prediction (BLUP) breeding values, the same Gibbs sampling algorithm was used a second time, keeping the variance components fixed at their estimated values. The posterior mean of the animal effects provided estimates of BLUP breeding values on the liability scale, and they were transformed to the phenotypic scale using: p i = Φ ( + EBV i ) where p i = the expected trait value of animal I; Φ ( ) = the cumulative probability function of the standard normal distribution; = the probit function corresponding to the mean liability of the respective trait; and EBV i = the breeding value estimated on the liability scale. Again, the sampler ran 120,000 rounds, and the results of the first 20,000 rounds were deleted. The Gibbs sampler implemented in the program LMMG (Reinsch, 1996) was used throughout. RESULTS AND DISCUSSION The mean and SD of posterior distribution of additive genetic variance, permanent environmental variance, her- Figure 1. Posterior distribution of additive genetic variance of liability to fertile eggs of eggs set ( ), liability to first-quality chicks of eggs set ( ), and liability to first-quality chicks of fertile eggs set ( ).

4 826 BENNEWITZ ET AL. Figure 2. Posterior distribution of permanent environmental variance of liability to fertile eggs of eggs set ( ), liability to first-quality chicks of eggs set ( ), and liability to first-quality chicks of fertile eggs set ( ). itability, and repeatability for the liabilities to the 3 traits are shown in Table 3. The heritability was lower for the fertility trait FE compared with the 2 hatchability traits CF and CFE. This was also found by Förster (1993), Gowe et al. (1993), and Szwaczkowski et al. (2000). On the contrary, Sapp et al. (2004) reported almost identical heritabilities for both types of traits, which were, in addition, substantially lower compared with those in Table 3. However, it should be kept in mind that a comparison of heritabilities across literature reports is difficult due to different trait definitions and different models applied and the genetic origin of the chicken line used in the studies. The ignorance of the binary nature of observations in genetic analysis leads to a so-called heritability on the phenotypic scale, which is lower than the heritability obtained from threshold models (Dempster and Lerner, 1950; Lynch and Walsh, 1998). Indeed, higher heritability estimates, as well as consequently more reliable estimated breeding values (EBV), which are less regressed back to the mean, are the advantages of applying threshold models, if appropriate. Applying the Dempster and Lerner (1950) formula for conversion of the heritability on the liability scale to the phenotypic scale to the current data, revealed a heritability on the phenotypic scale for FE, CE, and CFE of 0.027, 0.079, and 0.086, respectively (ignoring errors in the phenotypic means), which are substantially lower compared with the respective estimates shown in Table 3. The full benefit of the threshold models is utilized when the breeding value estimation is done with the same models. In the present study, breeding values were estimated as the mean of the posterior distribution of the respective animal effects obtained from the Gibbs sampler keeping the variance components fixed. These are BLUP EBV, because they consider the uncertainty of the nuisance effect multicode. Alternatively, one Figure 3. Posterior distribution of heritability of liability to fertile eggs of eggs set ( ), liability to first-quality chicks of eggs set ( ), and liability to first-quality chicks of fertile eggs set ( ).

5 GENETIC PARAMETERS FOR REPRODUCTIVE TRAITS IN LAYING HENS 827 Table 4. Means, SD, and minimum and maximum estimated breeding values of the worst and the best 100 birds of liability to fertile eggs of eggs set (FE), liability to first-quality chicks of eggs set (CE), and liability to first-quality chicks of fertile eggs set (CFE) 1 Trait Mean SD Minimum Maximum FE Worst Best CE Worst Best CFE Worst Best Only birds with a reliability above 0.5 were considered. could estimate Bayesian EBV as the posterior mean of the respective animal effect obtained from the full Gibbs sampler that also samples the variance components, which would consider the uncertainty of all unknowns in the model. The posterior distributions of the additive genetic variance, permanent environmental variance, and heritability of the liability to FE, CE, and CFE are shown in Figures 1 to 3. The posterior distributions were more or less sharp for all traits leading to the small SE of the estimated variance components (Table 3). Intuitively, this is somewhat surprising, because the pedigree is of small to medium size compared with, for instance, dairy cattle pedigrees (Heringstad et al., 2004). The reason is that each egg was treated as a repeated observation of the hen with a binary outcome. Subsequently, the number of observations was much higher compared with a model that would use summarized observations as proportion of fertile eggs, for instance. The assumption of the applied modeling is that, for a defined trait, the repeated observations show a genetic correlation close to 1 and subsequently contribute to the same trait. This might be true, because eggs were collected over a relatively short period. If, however, data collection would be expanded, for instance over the whole laying period, random regression longitudinal models (Schaeffer, 2004) are probably more appropriate, because they do account for a putative change of the covariance structure of observations that are collected over a life time span of individuals and relax the assumption that the observations belong to the same trait. The transformation of the EBV from the liability scale to the phenotypic scale simplifies their interpretation, because they reflect the probability for each animal to produce an observation that falls into the category 1. In the present study, this would be the probability to set a fertile egg, to produce a first-quality chick from an egg set, and to produce a first-quality chick from a fertile egg set, respectively. Additionally, the comparison of the worst and the best EBV gives an idea of the scope for genetic improvement of the respective traits. This is not directly observable from estimated genetic parameters because of the binary nature of the trait. The mean EBV of the 100 worst and best birds (only birds with a reliability above 0.5 are considered) are shown in Table 4 for all 3 traits. Note that for CE and CFE, the differences between the best and the worst group were also affected by the challenge conditions during incubation as mentioned above and hence cannot be interpreted concerning this. The permanent environmental variance and, subsequently, the repeatability, were considerable for all 3 traits (Table 3). Similar results were reported by Sapp et al. (2004). Ledur et al. (2000) found significant heterosis effects for fertility traits in White Leghorns. Following this, it can be expected that dominance might explain a part of the permanent environmental effects, because dominance is one of the bases for heterosis effects (Falconer and MacKay, 1996). A model that includes, aside from the additive genetic effects, dominance would provide information as to how much permanent environmental variance can be attributed to dominance. However, estimation of dominance needs a substantial amount of data (Misztal et al., 1998), preventing the application of such a model to the present data set. In summary, a Bayesian threshold model was introduced that estimates variance components for binary data from 3 reproductive traits in laying hens. It was shown that the obtained heritability estimates were higher compared with their expected values obtained from linear models, which results in a higher expected genetic progress. This is especially the case if selection is based on BLUP EBV obtained from animal models that consider pedigree information. REFERENCES Collett, D Modelling Binary Data. Chapman and Hall, London, UK. Dempster, E. R., and I. M. Lerner Heritability of threshold characters. Genetics 35: Falconer, D. S., and T. F. C. 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