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1 Danish Institute of Agricultural Sciences, Dept. of Animal Breeding and Genetics, Research Centre Foulum, P.O. Box 50, DK-8830 Tjele, Denmark
2 The Danish Agricultural Advisory Centre, Udkærsvej 15, Skejby, 8200 Aarhus N, Denmark
3 The Royal Veterinary and Agricultural University, Dept. of Animal Science and Animal Health, Grønnegårdsvej 2, DK-1870, Frederiksberg C, Denmark
Corresponding author: M. Hansen; e-mail: mxh{at}lr.dk.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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Key Words: stillbirth • Holstein-Friesian • genetic parameter • threshold model
Abbreviation key: CG = contemporary group, ECP = extreme category problem, HF = Holstein-Friesian, MGS = maternal grandsire, ODBW = original Danish black and white, S-MGS = sire-maternal grandsire
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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Many studies of genetic parameters for stillbirth have applied linear models and thereby ignored the categorical nature of stillbirth data (e.g., Niskanen and Juga, 1998; Luo et al., 1999; Harbers et al., 2000; Meyer et al., 2001b). From these studies, the direct and maternal heritabilities ranged from 0.01 to 0.06. It is not straightforward to compare these heritabilities, because they will depend on the frequency of stillbirth in the population analyzed. To compare heritabilities obtained with linear models, they can be transformed to a scale of underlying liabilities using the formula from Robertson in Dempster and Lerner (1950). However, this transformation has been shown to give biased results when the frequency is low (Van Vleck, 1972). It is desirable to analyze stillbirth data with a model that directly takes account of the categorical nature of these data, such as the threshold model (Wright, 1934). The threshold model assumes an unobservable underlying continuous variable (liability), with one or more thresholds deciding the observed categorical outcome. An empirical Bayesian method to draw inference about variance components in a threshold model was proposed by Harville and Mee (1983). This method relies on asymptotic properties and has been found to give biased inferences (Hoeschele and Gianola, 1989; Hoeschele and Tier, 1995). Sorensen et al. (1995) showed that exact and full Bayesian inference about the marginal (and joint) posterior distributions of all parameters in a threshold model could be accomplished by use of Gibbs sampling (Geman and Geman, 1984). Sorensen et al. (1995) implemented the Gibbs sampler with augmentation of liabilities as described in Albert and Chip (1993). Only a few studies of stillbirth have used this approach. Luo (1999) used this approach for a threshold sire-maternal grandsire (S-MGS) model [a model containing correlated genetic effects of sires and maternal grandsires (MGS)] but did not succeed with the analysis as the sire variance was trapped at zero. Steinbock et al. (2003) used a similar approach and found marginal posterior means of direct and maternal heritabilities on 0.12 and 0.08 for stillbirth in first calving Swedish Holstein cows.
The proportion of genes in the Danish Holstein population descending from the original Danish black and white population in the 1960s (ODBW) has decreased since an upgrading with Holstein-Friesian (HF) began in the 1970s. In Danish Holstein calves today, the proportion of genes originating from HF is higher than 90%. During this upgrading period, the stillbirth rate at first calving has increased in the Danish Holstein breed. A similar pattern has been observed in the Swedish Holstein breed (Steinbock et al., 2003). Therefore, it is possible that genes descending from HF have influenced the stillbirth rate. The current model used for the official Danish breeding value evaluation of stillbirth includes covariates of the proportion of ODBW and HF and the heterozygosity between these 2 breeds for both the calf and the dam (Danish Cattle Federation, 2002) (the way heterozygosity is defined will be explained later). Results from this evaluation indicated an unfavorable breed effect of HF and a small favorable effect of heterozygosity for both the direct and the maternal effect (Pedersen, 2002, personal communication). But this model is a linear model and does not account for the categorical nature of stillbirth. Inference about effects of breed and heterozygosity can be different in a threshold model.
The objectives with this study were: 1) to estimate the direct and maternal heritability for stillbirth at first calving in the Danish Holstein population using a Bayesian threshold model with Gibbs sampling; 2) to compare these heritabilities with heritabilities from a linear model transformed to the liability scale with the formula by Robertson in Dempster and Lerner (1950); 3) to test the hypotheses that the additive genetic effect of HF relative to the additive genetic effect of ODBW is unfavorable, and the effect of heterozygosity between HF and ODBW is favorable.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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Editing.
The editing procedure is summarized in Table 1
. The main editing rules were that the sire and the MGS of the calf had to be a known AI-bull with 10 records as sire or as MGS. Twins were omitted for most analyses, but they were kept for one analysis. Records were clustered in subclasses of contemporary groups (CG) using the method described by Schmitz et al. (1991) with a maximum interval on 2 yr and a minimum desired size of clusters on 20. Due to this method, a few CG subclasses were small, and records in CG subclasses with 5 or fewer records were omitted.
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). A sire or a MGS was regarded as unproven when it was less than 5 yr old when the calf was born or when the dam was born, respectively. The final data set included 983 bulls as sires and 1055 bulls as MGS (Table 3). In total, 823 bulls had records both as a sire and as a MGS, and 1205 different bulls were included as a sire or as a MGS. Pedigree for these bulls was traced on both their sire and their dam paths as far back as parents were known. This resulted in an additive relationship matrix (A) consisting of 4445 animals. Most of the bulls with records were Danish AI-bulls, but some were foreign AI-bulls. Danish AI-bulls were regarded as unproven if they had progeny included that were born less than 5 yr after the sire, or else they were regarded as proven. In total, 100 sires and 118 MGS were regarded as Danish proven AI-bulls. These bulls could have been selected for fewer stillbirths based on data not included in the analyses. Selection of Danish proven AI-bulls for mating with heifers seemed to exist as 9.9% of their calves were stillborn compared with 12.2% for unproven sires. Also, calves with a proven Danish MGS had a lower incidence of stillbirth than calves from an unproven MGS. When some sires used for estimation of genetic parameters were selected for the trait of interest, Van Vleck (1985) suggested treating them as nongenetic effects to avoid selection bias. A disadvantage of this approach is that the genetic and environmental effects become less connected. This procedure would provide greater uncertainty for the genetic parameters and was not done in this study.
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Statistical Procedures
Threshold model.
We assumed that the observed outcome (yi) for the i’th calf (i = 1,2,...,n) followed a Bernoulli distribution (stillborn = 1, alive = 0), where n is the total number of records included. We assumed that each calf had an unknown liability for stillbirth (Ui). All calves with liabilities exceeding the threshold (t) were observed as stillborn. The model for the liabilities was in matrix notation:
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where U was a vector of liabilities, b was a vector of nongenetic effects plus effects of breed and heterosis, h was a vector of CG effects, s was a vector of sire effects, mgs was a vector of MGS effects, and e was a vector of residuals. X, W, Z1, and Z2 were incidence matrices relating the effects to liabilities.
Three different analyses were carried out with the threshold model: TM, TM-TWIN, and TM-HET. Effects of breed and heterosis were only included in TM-HET and records of twins were only included in TM-TWIN. The vector b contained cross-classified effects of: sex of calf (male, female), month of calving (1,2,...,12), calving age in months (22,23,...,35), year of calving (1988,...,2002), twin (single, twin) (only included for TM-TWIN). In TM-HET, b also contained the coefficients of the regressions bHF-calf and bHF-dam for the proportion of HF in the calf and in the dam and bHet-calf and bHet-dam for the degree of heterozygosity between HF and ODBW in the calf and in the dam.
It was assumed that the liabilities conditional on all effects were independent and normally distributed:
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To obtain identifiable parameters, the residual variance 
was set to one, and the threshold was set to zero (Sorensen et al., 1995). The conditional probability that record i (yi) falls into the category "stillborn" given b, h, s, and mgs was:
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where 
is the cumulative distribution function of a standard normal distribution, and xi, wi, zli, and z2i are row i of each of the respective incidence matrices.
Priors for the location parameters were:
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where 
and A is the additive genetic relationship matrix consisting of bulls, which are sires or MGS of calves, and all their known ancestors.
Conjugate proper prior distributions were used for the dispersion parameters. The prior for 
was an inverted chi-square distribution with 4 degrees of freedom and with an expectation = 0.11511. The prior for G0 was a two-dimensional inverted Wishart distribution with 4 degrees of freedom and with an expectation of the (co)variance matrix = 
. The expectations of these priors were chosen such that, after transformation to the derived variables (described in the following), they were equivalent to a direct heritability of 0.10, a maternal heritability of 0.10, a correlation between direct and maternal effects of zero, and a ratio of the variance of CG effects to the phenotypic variance of 0.10. These prior expectations were based on estimates from Luo (1999) and Steinbock et al. (2003).
To obtain marginal posterior distributions of derived variables, they were generated from each saved sample of dispersion parameters. Direct and maternal genetic (co)variances were generated following derivations from Willham (1972):
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The genetic correlation between direct and maternal effects was generated as:
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The expectation of the phenotypic variance for this model was:
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where aS,MGS was the genetic relationship between the sire and the MGS. When we calculated the phenotypic variance we assumed that the sire and the MGS were unrelated. Therefore, the term aS,MGS2
S,MGS dropped out. Using this phenotypic variance, the direct and the maternal heritabilities were generated as: 
and 
, and a CG ratio was generated as:
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Draws from the posterior distribution of the parameters in the model were obtained by Gibbs sampling (Geman and Geman, 1984). The Gibbs sampling approach was carried out by software developed by Korsgaard et al. (2003), which is implemented in the DMU-package (Madsen and Jensen, 2000). In this software two different methods to sample location parameters are implemented: univariate and joint sampling. In the joint sampling, all location parameters are sampled simultaneously based on the method by García-Cortés and Sorensen (1996), which do not need inversion of the coefficient matrix of the mixed model equations. Joint sampling can often reduce the lag correlation between subsequent samples (Liu et al., 1994), but the computing time for one round in the Gibbs sampler is 2 to 3 times longer than univariate sampling (for the current analyses). For each analysis, 2 chains (A and B) consisting of 60,000 to 86,000 rounds were carried out using univariate sampling of location parameters. Additionally, two chains (C and D) each on 21,000 rounds were carried out for TM-HET where joint sampling of location parameters was used.
The length of the burn-in period was assessed by visual examination of trace plots of all effects in b, all (co)variance components, and all derived variables. In the chains with univariate sampling of location parameters, samples from the first 10,000 rounds were discarded as burn-in period for TM-TWIN and TM, but in TM-HET 40000 rounds were discarded. In the chains with joint sampling, only 2000 rounds were discarded. The samples after burn-in were divided into 20 sequential batches. The means of the batches should converge to independent and identically distributed normal random variables. This was assessed by calculating lag correlations between batch means and performing one-way analyses of variance of the batch means. Effective sample size was calculated for each parameter by dividing the variance within batch with the variance between batches (e.g., Sorensen and Gianola, 2002).
Summary statistics of nongenetic effects were given as the predicted probabilities of stillbirth for a calf conditional on marginal posterior means of all nongenetic effects. The predicted probability of stillbirth in TM-TWIN for, e.g., a single born male calf with a 22-mo-old dam born in January 2002 was the sum of the marginal posterior means of these nongenetic effects evaluated in the cumulative distribution function of a standard normal distribution.
Linear model.
The linear models included the same effects as the threshold model, but the response variable was the actual observation (1 or 0). Estimates of (co)variance components were obtained with AI-REML (Jensen et al., 1997) using the DMU-package (Madsen and Jensen, 2000). Two analyses were carried out: LM and LM-HET, where LM-HET included effects of breed and heterosis.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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) were both relatively symmetric. For all the analysis, the two chains (A and B) gave very similar inference (Table 4). The marginal posterior mean of the direct heritability from TM-TWIN was 0.10 and the standard deviation was 0.014. The marginal posterior mean of the maternal heritability from the same analysis was 0.13, and the standard deviation was 0.015. Only small differences in these estimates were observed when twins were discarded and when effects of breed and heterosis were not included in the model (Table 4).
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). The transformation of these heritabilities from LM to the liability scale gave a heritability of 0.15 for direct effects and 0.17 for maternal effects.
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), with means from 0.03 to 0.06 and standard deviations from 0.10 to 0.11 (Table 4). The estimates of the correlation from the linear models were both -0.04, with standard errors on 0.09 (Table 5).
Breed and Heterosis
The marginal posterior distributions for effects of breed and heterosis were very inaccurate for the two chains, which used univariate sampling of location parameters (A and B). In contrast, the two chains with joint sampling (C and D) gave much more accurate inferences. For a computing time on 20 to 25 d, only 20 to 43 effective samples were obtained with the univariate sampling, but more than 2750 were obtained with the joint sampling. The reason for these differences was a considerable difference in mixing properties for the methods as illustrated in Figure 2. With univariate sampling, the lag 300 correlations for all effects of breed and heterosis were from 0.45 to 0.95, but with the joint sampling the lag 10 correlation was below 0.05.
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and Figure 3). The largest and most significant effect was the effect of HF in the dam, which had a marginal posterior mean of 0.50. This effect can be interpreted as the difference between a dam with 100% HF genes compared with a dam with 100% ODBW genes. When the posterior mean of this effect was transformed to the observable scale, the probability of stillbirth was 13.9% for dams with 100% HF genes, given that the stillbirth rate was 7% for dams with 100% ODBW genes. The estimates of breed and heterosis from the linear model were generally in accordance with the effects from the threshold model (Table 6).
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). For a calf born in January 2002 from a dam with a calving age at 22 mo, the probability was 0.29 for a male and 0.21 for a female. For dams with a calving age of 28 mo or older, the probability was around 0.15 for male calves and 0.10 for female calves. The probability of stillbirth for calves born in July, August, and September was about 0.010 to 0.015 lower than for calves born in autumn and early winter (results not shown).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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The transformation of heritabilities obtained from linear models to the liability scale with the formula by Robertson in Dempster and Lerner (1950) gave an upward bias. The same upward bias was seen in Weller et al. (1988) and Steinbock et al. (2003). These results confirm the simulation study of Van Vleck (1972), where the transformation gave upward biased heritabilities when the frequency was low. Therefore, inference about the heritabilities on the liability scale ought to be from models accounting for the categorical nature of the data.
Genetic Correlations
The genetic association between direct and maternal effects was found to be small as the marginal posterior mean of the correlation was 0.03 with twins included in the data and 0.06 without twins. A slightly different estimate of the genetic correlation (-0.04) was obtained from the linear models. A correlation close to zero implies that selection on, e.g., the direct effect of stillbirth would not influence the maternal effect of stillbirth. The present results are in agreement with other studies using threshold models, where Luo (1999) found a correlation on 0.03 and Steinbock et al. (2003) found a correlation on -0.11. However, it contradicts the correlation on -0.24 found by Luo et al. (1999), where a linear model was used.
Breed and Heterosis
The indication of an unfavorable effect of HF on stillbirth in the Danish Holstein population (however, not significant) were confirmed in a larger study of 1.8 million calves with a threshold model assuming known (co)variances (Hansen et al. April 2003, unpublished). However, both the analyzed data sets were not well suited for inference about effects of breed and heterosis effects because only a small part of the animals had less than 50% HF genes and because the proportion of HF genes increased through the whole period. This would result in some confounding between the effects of breed, heterosis, and year. A balanced data set is needed to avoid this confounding. But an important conclusion of this study was that fitting effects of breed and heterosis or not, did not change the inference about the additive genetic (co)variances.
The exact marginal posterior distributions for effects of breed and heterosis were very inaccurate with univariate sampling of location parameters due to slow mixing. The mixing problems were due to a high correlation between samples of the effects of breed, heterosis, and sex (Table 7). However, sampling location parameters jointly improved the mixing properties enormously. Therefore it seems to be very advantageous to use joint sampling when posterior samples of effects are highly correlated.
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Complex Vertebral Malformation
The models used in this study were limited to assume that the genetic effects (additive and heterosis) were a sum of infinitely many, small independent gene effects. Complex vertebral malformation, which leads to mortality of fetuses and calves probably caused by an autosomal inherited recessive defect (Agerholm et al., 2001), was not accounted for in the model. Nielsen et al. (2003) showed that 77% of complex vertebral malformation homozygous fetuses were aborted before 260 d after insemination. As stillbirth is recorded for calvings 260 d after insemination, the impact of complex vertebral malformation on the inference of quantitative genetic parameters of stillbirth is believed to be limited.
Choice of Model
In this study we chose to model direct and maternal effects by fitting a model with correlated random effects of sires and MGS. The S-MGS model can be justified as (artificial) selection operates mainly on bulls in dairy cattle. Theoretically, an animal threshold model would be more appropriate than a S-MGS model, but it has been shown to cause problems, which are related to the extreme category problem (ECP) (Hoeschele and Tier, 1995; Luo et al., 2001). The ECP named by Misztal et al. (1989) can arise in threshold models, when all observations in a subclass for a location parameter are in one of the categories. In a comparison of all combinations of linear, threshold, animal, and S-MGS models with fixed or random CG effects, Luo et al. (2001) found that the threshold S-MGS model with random CG effects performed best for categorical data when Gibbs sampling was used. In a Bayesian analysis of a sire threshold model using Gibbs sampling, Hoeschele and Tier (1995) detected problems when CG effects were given a flat uniform prior. In this case individual solutions for CG subclasses with ECP tended towards plus or minus infinity. Even though 45% of subclasses for CG effects had ECP, these problems disappeared when a normal prior with an unknown variance was used for the CG effects. Then the posterior means of all the variance components were in agreement with simulated values. However, this approach has been shown to give biased inference in some extreme cases (Moreno et al., 1997; Rekaya et al., 2000). Moreno et al. (1997) found bias when 80% of the subclasses for CG effects had ECP, but no bias was detected when 60% of the subclasses for CG effects had ECP. In the present study, the effects of CG were given a normal prior, and only 11% of the subclasses for CG effects had ECP. Therefore, no bias was expected to occur.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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Received for publication January 26, 2003. Accepted for publication July 14, 2003.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
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), the maternal heritability (
), and the genetic correlation between direct and maternal effects (rDM).
), CG ratio (cg2), direct and maternal heritability (
