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Poult Sci 2007. 86:823-828
© 2007 Poultry Science Association
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GENETICS

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

J. Bennewitz*,1, O. Morgades*, R. Preisinger{dagger}, G. Thaller* and E. Kalm*

* Institut für Tierzucht und Tierhaltung, Christian-Albrechts-Universität, D-24098 Kiel, Germany; and {dagger} Lohmann Tierzucht GmbH, D-27454 Cuxhaven, Germany

1 Corresponding author: jbennewitz{at}tierzucht.uni-kiel.de


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 REFERENCES
 
Variance components and breeding values were estimated for 3 reproductive traits in a pure line of White Leghorn laying hens. The traits were proportion of fertile eggs of eggs set, proportion of first quality chicks of eggs set, and proportion of first-quality chicks of fertile eggs. A total of 3,020 hens were tested up to 3 times over a period of 7 d. For the definition of the traits, each egg was scored for each trait either as 0 or 1. To account for the binomial distribution of the data, a Bayesian animal threshold model implemented in a Gibbs sampler was applied that considered the record of each egg set as a repeated observation of the hen. The estimated heritability was 0.067, 0.126, and 0.136 for the traits proportion of fertile eggs of eggs set, proportion of first quality chicks of eggs set, and proportion of first-quality chicks of fertile eggs, respectively, and the SE were small. It was shown that the heritability estimates were substantially higher from their expected values based on linear models. This results in a higher genetic progress and consequently 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


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 REFERENCES
 
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 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
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 REFERENCES
 
Data and Traits

Records from 3,020 hens from a full-pedigreed pure line of White Leghorn were used. Average number of 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 1Go and 2Go.


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  		Table 1. Phenotypic mean, SD, and minimum and maximum number of eggs set, of fertile eggs, and of first-quality chicks per hen (n = 8,466 hen and test combinations)
 

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  		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)
 
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:


Formula

where {lambda}= a vector of unobservable variables of the hens with phenotypic information, subsequently denoted as liabilities; ß = a 23 x 1 vector of the effects of the multicode specified above; a = a g x 1 vector of additive animal effects, with g being the number of animals, pe = a h x 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|A{sigma}2a) ~N(0,A {sigma}2a), where A = the numerator relationship matrix of the animals and {sigma}2a = 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|{sigma}2pe) ~N(0,I {sigma}2pe), where I = an identity matrix and {sigma}2pe = 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 ßi was sampled from:


Formula

where ßi = the ith component of ß; xi = the ith column vector extracted from X; x'i{lambda}* = the sum of corrected liabilities pertaining to the ith level of ß; {lambda}* = {lambda} corrected for all fixed and random effects except the ith component of ß, and dxx was the ith diagonal element of X' X. With binary data, the threshold and the residual variance ({sigma}2e) were not identifiable. Therefore these parameters were set to 0 and 1, respectively. The ith permanent environmental pei was sampled from:


Formula

where pi = the ith component of p; wi = the ith column vector extracted from W; wi {lambda} ** = the sum of corrected observations pertaining to the ith level of pe; {lambda}** = {lambda} corrected for all fixed and random effects except the ith component of pe, and dwwi = the ith diagonal element of W'W + Ia1, where I = an identity matrix and a1 = {sigma}2e/{sigma}2e. The ith animal effect ai was sampled from:


Formula

where ai = the ith component of a, {lambda}*** = {lambda}corrected for all fixed and random effects except the ith animal effect; zi = the ith column vector extracted from Z; zi ' {lambda}*** = the sum of corrected liabilities pertaining to the ith level of a; dzzi = the ith diagonal element of the matrix Z ' Z + A– 1a2 and a2 = {sigma}2e/{sigma}a. The {sigma}pe was sampled from an inverted {chi}2 distribution with h – 2 df. The inverted {chi}2 distribution was scaled by pe pe. The {chi}2a was sampled from an inverted {chi}2 distribution with g – 2 df. Here, the inverted {chi}2 distribution was scaled by a'A–1a.

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 (burn-in plus safety margin). The effective sample size was estimated using time series methods as described by Sorensen et al. (1995), applying the SAS procedure AUTOREG (SAS Institute, 2002). It was >250 for the variance components for all traits. The mean of the respective posterior 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:


Formula

where pi = the expected trait value of animal I; {Phi} (·) = the cumulative probability function of the standard normal distribution; µ = the probit function corresponding to the mean liability of the respective trait; and EBVi = 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
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
REFERENCES
 
The mean and SD of posterior distribution of additive genetic variance, permanent environmental variance, heritability, and repeatability for the liabilities to the 3 traits are shown in Table 3Go. 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 3Go. 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.


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  		Table 3. Mean and SD in parenthesis of posterior distribution of additive genetic variance ({sigma}2a), permanent environmental variance ({sigma}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
 
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 3Go. 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 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 1Go to 3GoGo. The posterior distributions were more or less sharp for all traits leading to the small SE of the estimated variance components (Table 3Go). 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.


Figure 1
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  		Figure 1. Posterior distribution of additive genetic variance of liability to fertile eggs of eggs set ({diamondsuit}), liability to first-quality chicks of eggs set ({blacktriangleup}), and liability to first-quality chicks of fertile eggs set ({blacksquare}).

 

Figure 2
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  		Figure 2. Posterior distribution of permanent environmental variance of liability to fertile eggs of eggs set ({diamondsuit}), liability to first-quality chicks of eggs set ({blacktriangleup}), and liability to first-quality chicks of fertile eggs set ({blacksquare}).

 

Figure 3
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  		Figure 3. Posterior distribution of heritability of liability to fertile eggs of eggs set ({diamondsuit}), liability to first-quality chicks of eggs set ({blacktriangleup}), and liability to first-quality chicks of fertile eggs set ({blacksquare}).

 
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 4Go 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.


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  		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
 
The permanent environmental variance and, subsequently, the repeatability, were considerable for all 3 traits (Table 3Go). 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.

Received for publication September 8, 2006. Accepted for publication January 21, 2007.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 REFERENCES
 
Collett, D. 1991. Modelling Binary Data. Chapman and Hall, London, UK.

Dempster, E. R., and I. M. Lerner. 1950. Heritability of threshold characters. Genetics 35:212–236.[Free Full Text]

Falconer, D. S., and T. F. C. MacKay. 1996. Introduction to Quantitative Genetics. 4th ed. Longman Sci. Tech., New York, NY.

Förster, A. 1993. Züchterische Möglichkeiten einer Verbesserung der Schlupfrate in Reinzuchtlinien eines Zuchtprogrammes für braune Legehybriden. PhD Thesis. Schriftenr. Inst. Tierzucht und Tierhaltung, Christian-Albrechts-Univ., Kiel, Germany.

Gowe, R. S., R. W. Fairfull, I. Mcmillan, and G. S. Schmidt. 1993. A strategy for maintaining high fertility and hatchability in a multiple-trait egg stock selection program. Poult. Sci. 72:1433–1448.[Web of Science]

Heringstad, B., Y. M. Chang, D. Gianola, and G. Klemetsdal. 2004. Multivariate threshold model analysis of clinical mastitis in multiparous Norwegian dairy cattle. J. Dairy Sci. 87:3038–3046.[Abstract/Free Full Text]

Hinrichs, D., E. Stamer, W. Junge, and E. Kalm. 2005. Genetic analyses of mastitis data using animal threshold models and genetic correlation with production traits. J. Dairy Sci. 88:2260–2268.[Abstract/Free Full Text]

Ledur, M. C., R. W. Fairfull, I. Mcmillan, and L. Asseltine. 2000. Genetic effects of aging on egg production traits in the first laying cycle of White Leghorn strain crosses. Poult. Sci. 79:296–304.[Abstract/Free Full Text]

Lynch, M., and B. Walsh. 1998. Genetics and Analysis of Quantitative Traits. Sinauer Assoc. Inc., Sunderland, MA.

Misztal, I., L. Varona, M. Culbertson, J. K. Bertrand, J. Mabry, T. J. Lawlor, C. P. van Tassel, and N. Gengler. 1998. Studies on the value of incorporating the effect of dominance in genetic evaluations of dairy cattle, beef cattle and swine. Biotechnol. Agron. Environ. 2:227–233.

Preisinger, R. 1998. Internationalisation of breeding programmes—Breeding egg-type chickens for a global market. Pages 135–142 in Proc. 6th World Congr. Genet. Appl. Livest. Prod., Armidale, Australia.

Reinsch, N. 1996. Two fortran programs for the Gibbs sampler in univariate linear mixed models. Arch. Tierz. 39:203–209.

Sapp, R. L., R. Rekaya, I. Misztal, and T. Wing. 2004. Male and female fertility and hatchability in chickens: A longitudinal mixed model approach. Poult. Sci. 83:1253–1259.[Abstract/Free Full Text]

SAS Institute. 2002. SAS/STAT User’s Guide. Version 8. SAS Inst. Inc., Cary, NC.

Schaeffer, L. R. 2004. Application of random regression models in animal breeding. Livest. Prod. Sci. 86:35–45.

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