AN ANALYSIS OF VARIABILITY IN NUMBER OF DIGITS IN AN INBRED STRAIN OF GUINEA PIGS

Estimation of genetic parameters and prediction of breeding values for multivariate threshold and continuous data in a simulated horse population using Gibbs sampling and residual maximum likelihood

Simulated horse data were used to compare multivariate estimation of genetic parameters and prediction of breeding values (BV) for categorical, continuous and molecular genetic data using linear animal models via residual maximum likelihood (REML) and best linear unbiased prediction (BLUP) and mixed linear-threshold animal models via Gibbs sampling (GS). Simulation included additive genetic values, residuals and fixed effects for one continuous trait, liabilities of four binary traits, and quantitative trait locus (QTL) effects and genetic markers with different recombination rates and polymorphism information content for one of the liabilities. Analysed data sets differed in the number of animals with trait records and availability of genetic marker information. Consideration of genetic marker information in the model resulted in marked overestimation of the heritability of the QTL trait. If information on 10,000 or 5,000 animals was used, bias of heritabilities and additive genetic correlations was mostly smaller, correlation between true and predicted BV was always higher and identification of genetically superior and inferior animals was - with regard to the moderately heritable traits, in many cases - more reliable with GS than with REML/BLUP. If information on only 1,000 animals was used, neither GS nor REML/BLUP produced genetic parameter estimates with relative bias <or=25% and BV correlation >50% for all traits. Selection decisions for binary traits should rather be based on GS than on REML/BLUP breeding values.

Genetic Analysis of Male and Female Fertility After Artificial Insemination in Sheep: Comparison of Single-Trait and Joint Models

The outcome of an insemination depends on male and female fertility. Nevertheless, few studies have incorporated genetic evaluation of these 2 traits jointly. The aim of this work was to compare genetic parameter estimates of male and female fertility defined as success or failure to artificial insemination (AI), using 8 different models. The first 2 models were simple repeatability models studying fertility of one sex and ignoring any information of the other. Models 3 and 4 took into account the information of the other sex by the inclusion of its random permanent environmental effect, whereas models 5 and 6 included fixed effects of the other sex. Models 7 and 8 were joint genetic evaluation models of male and female fertility ignoring or considering genetic correlation. Data were composed of 147,018 AI of the Manech Tête Rousse breed recorded from 2000 to 2004 corresponding to 79,352 ewes and 963 rams. The pedigree file included 120,989 individuals. Variance component estimates from the different models were quite similar; heritabilities varied from 0.050 to 0.053 for female fertility and were near 0.003 for male fertility. Correlations among estimated breeding values in the same sex using different models were higher than 0.99. The genetic correlation between male and female fertility was not significantly different from 0. These results show that for French dairy sheep with extensive use of AI, estimation of breeding values for male and female fertility might be implemented with quite simple models.

Allelic penetrance approach as a tool to model two-locus interaction in complex binary traits

Many binary phenotypes do not follow a classical Mendelian inheritance pattern. Interaction between genetic and environmental factors is thought to contribute to the incomplete penetrance phenomena often observed in these complex binary traits. Several two-locus models for penetrance have been proposed to aid the genetic dissection of binary traits. Such models assume linear genetic effects of both loci in different mathematical scales of penetrance, resembling the analytical framework of quantitative traits. However, changes in phenotypic scale are difficult to envisage in binary traits and limited genetic interpretation is extractable from current modeling of penetrance. To overcome this limitation, we derived an allelic penetrance approach that attributes incomplete penetrance to the stochastic expression of the alleles controlling the phenotype, the genetic background and environmental factors. We applied this approach to formulate dominance and recessiveness in a single diallelic locus and to model different genetic mechanisms for the joint action of two diallelic loci. We fit the models to data on the genetic susceptibility of mice following infections with Listeria monocytogenes and Plasmodium berghei. These models gain in genetic interpretation, because they specify the alleles that are responsible for the genetic (inter)action and their genetic nature (dominant or recessive), and predict genotypic combinations determining the phenotype. Further, we show via computer simulations that the proposed models produce penetrance patterns not captured by traditional two-locus models. This approach provides a new analysis framework for dissecting mechanisms of interlocus joint action in binary traits using genetic crosses.

Technical note: How to use Winbugs to draw inferences in animal models1

This paper deals with Bayesian inferences of animal models using Gibbs sampling. First, we suggest a general and efficient method for updating additive genetic effects, in which the computational cost is independent of the pedigree depth and increases linearly only with the size of the pedigree. Second, we show how this approach can be used to draw inferences from a wide range of animal models using the computer package Winbugs. Finally, we illustrate the approach in a simulation study, in which the data are generated and analyzed using Winbugs according to a linear model with i.i.d errors having Student's t distributions. In conclusion, Winbugs can be used to make inferences in small-sized, quantitative, genetic data sets applying a wide range of animal models that are not yet standard in the animal breeding literature.

Bayesian mapping of genomewide interacting quantitative trait loci for ordinal traits

Development of statistical methods and software for mapping interacting QTL has been the focus of much recent research. We previously developed a Bayesian model selection framework, based on the composite model space approach, for mapping multiple epistatic QTL affecting continuous traits. In this study we extend the composite model space approach to complex ordinal traits in experimental crosses. We jointly model main and epistatic effects of QTL and environmental factors on the basis of the ordinal probit model (also called threshold model) that assumes a latent continuous trait underlies the generation of the ordinal phenotypes through a set of unknown thresholds. A data augmentation approach is developed to jointly generate the latent data and the thresholds. The proposed ordinal probit model, combined with the composite model space framework for continuous traits, offers a convenient way for genomewide interacting QTL analysis of ordinal traits. We illustrate the proposed method by detecting new QTL and epistatic effects for an ordinal trait, dead fetuses, in a F(2) intercross of mice. Utility and flexibility of the method are also demonstrated using a simulated data set. Our method has been implemented in the freely available package R/qtlbim, which greatly facilitates the general usage of the Bayesian methodology for genomewide interacting QTL analysis for continuous, binary, and ordinal traits in experimental crosses.

Bayesian multiple quantitative trait loci mapping for complex traits using markers of the entire genome

A Bayesian methodology has been developed for multiple quantitative trait loci (QTL) mapping of complex binary traits that follow liability threshold models. Unlike most QTL mapping methods where only one or a few markers are used at a time, the proposed method utilizes all markers across the genome simultaneously. The outperformance of our Bayesian method over the traditional single-marker analysis and interval mapping has been illustrated via simulations and real data analysis to identify candidate loci associated with colorectal cancer.

Analysis of Fertility and Dystocia in Holsteins Using Recursive Models to Handle Censored and Categorical Data

A method based on the analysis of recursive multiple-trait models was used to 1) estimate genetic and phenotypic relationships of calving ease (CE) with fertility traits and 2) analyze whether dystocia negatively affects reproductive performance in the next reproductive cycle. Data were collected from 1995 through 2002, and contained 33,532 records of CE and reproductive data of 17,558 Holstein cows distributed across 560 herds in official milk recording from the Basque Country Autonomous Community (Spain). The following fertility traits were considered: days open (DO), days to first service, number of services per pregnancy (NINS), and outcome of first insemination (OFI). Four bivariate sire and sire-maternal grandsire models were used for the analyses. Censoring existed in DO (26.49% of the data) and NINS (12.22% of the data) because of cows having been sold or culled before reaching the next parturition. To avoid bias, a data augmentation technique was applied to censored data. Threshold models were used for CE and OFI. To consider that CE affects fertility and the genetic determination of CE and fertility traits, recursive models were applied, which simultaneously considered CE as a fixed effect on fertility performance and the existence of a genetic correlation between CE and fertility traits. The effects of CE score 3 (difficult birth) with respect to score 1 (no problem) for days to first service, DO, NINS, and OFI were 8 d, 31 d, 0.5 services, and -12% success at first insemination, respectively. These results showed poorer fertility after dystocia. Genetic correlations between genetic effects of fertility traits and CE were close to zero, except for the genetic correlations between direct effects of DO and CE, which were positive, moderate, and statistically different from 0 (0.47 +/- 0.24), showing that genes associated with difficult births also reduce reproductive success.

Analysis of litter size and days to lambing in the Ripollesa ewe. I. Comparison of models with linear and threshold approaches1

The analysis focused on model fitting of 2 ewe reproductive traits, litter size, and days to lambing (interval between the introduction of the ram into the flock and the subsequent parturition of the ewes). The experimental data set of the Universitat Autònoma of Barcelona flock was used, including 1,598 records of litter size and 1,699 records of days to lambing from 376 Ripollesa ewes between 1986 and 2005. Univariate and bivariate models were considered as beginning points with linear or threshold approximation for litter size. Model fitting was evaluated in terms of goodness-of-fit and predictive ability, using the mean square error and the correlation between phenotypic and predicted records (rho(y,ŷ)) as reference parameters. The bivariate model was preferable for both variables, minimizing mean square error and maximizing rho(y,ŷ). A threshold approximation for litter size was preferable over a linear approximation. Models were also compared with a simulation study, comparing the correlation coefficient between simulated and predicted breeding values (rho(a,â)). The bivariate threshold model was favored, with a rho(y,ŷ) of 0.677 and 0.834 for litter size and days to lambing, respectively. Correlation coefficients between simulated and predicted breeding values in the bivariate linear model were reduced slightly to 0.651 and 0.831, respectively, and they were lowest with linear univariate models (0.642 and 0.802). Although the bivariate models for ewe litter size and days to lambing were more accurate than the univariate models, the threshold approaches showed a greater advantage under the bivariate model. For the purpose of genetic evaluation of litter size in sheep, use of the threshold-linear model seems justified. In the Ripollesa breed, the evaluation of litter size can benefit from recording birth weight.

Association analyses between the prion protein locus and reproductive and lamb weight traits in Ripollesa sheep1

The objective of this study was to analyze the association between the haplotypes of the prion protein (PrP) locus and several reproductive and lamb weight traits in Ripollesa sheep. Prion protein genotypes were available for a total of 310 sheep (7 rams, 114 ewes, and 189 lambs), all of them belonging to the purebred Ripollesa flock of the Universitat Autònoma of Barcelona, for which all sheep had a known pedigree. In addition, the genotype of 24 historical descendants of the previously genotyped adult individuals was reconstructed, provided that both parents were homozygous for PrP haplotypes. Only 3 haplotypes (ARR, ARQ, and ARH) were observed in the PrP locus of the sheep sampled. Reproductive traits included conception rate and litter size, whereas birth BW and 90-d BW were the lamb weight traits studied. The additive effect of PrP haplotypes was analyzed through Bayesian animal threshold and linear models, for reproduction and weight traits, respectively. Ewe reproductive data belonged to 89 ewes that gave 492 conception rate records and 440 litter size records. Analyses of BW at birth and at 90 d of age were made on 323 and 164 lamb records, respectively. No associations between PrP haplotypes and conception rate and BW traits were observed. For litter size, the effect of the ARH haplotype was greater than that of the ARQ haplotype. Differences between ARH and ARR haplotypes also suggested an advantage for the ARH. As a whole, our results indicated that the selection favorable to increase litter size in Ripollesa ewes may also increase the ARH haplotype frequency, which contradicts the recommendations of the current European Union legislation aiming to increase the genetic resistance to scrapie. As a consequence, scrapie genotyping needs to be included as a new selection criterion in the breed.

Analysis of litter size and days to lambing in the Ripollesa ewe. II. Estimation of variance components and response to phenotypic selection on litter size1

A performance data set from 376 Ripollesa purebred ewes of the experimental flock of the Universitat Autònoma of Barcelona was analyzed using a bivariate Bayesian threshold-linear model. The data set contained 1,598 litter size records and 1,699 days-to-lambing records. The model included the additive genetic effect of each animal and 3 nongenetic sources of variation: ewe age, year of lambing, and the permanent environmental effect characterized by the ewe. The flock was phenotypically selected for litter size since 1986, and replacement ewes and rams were selected from the progeny of the more prolific ewes, which had at least 3 deliveries recorded. The phenotypic trend for litter size was positive, whereas days to lambing followed an unclear pattern. Both traits had low heritabilities; 0.13 for litter size and 0.11 for days to lambing. Response to selection was evaluated through (a) the average breeding value of the ewe lambs chosen annually, and (b) the average breeding value of the overall flock. The first measurement suggested a positive trend for litter size, although it showed important oscillations. On the other hand, the average breeding value for the overall flock showed a stable positive tendency after yr 4 of selection, with estimates clearly different from zero after yr 11 of selection. A significant increase in the incidence of multiple births was observed, with a mode of approximately 10%. The correlated response in days to lambing did not show a significant trend. The effect of year of lambing also positively influenced both litter size and days to lambing, although important oscillations were observed between years. Results indicated that litter size in sheep can be effectively improved through phenotypic selection, even in small flocks; moreover, days to lambing could also be genetically improved, given the estimate obtained for its heritability.

Effect of the threshold nature of traits on heritability estimates obtained by linear model

In this research we used stochastic method to simulate litter size data under assumption that the phenotype of the trait of interest is categorically distributed with a liability of normally distributed background effect. For the phenotypic categorical measurements, normal distribution of liability was assumed. Analysis was conducted by DFREML for phenotypic categorical data set. In the simulation process, we applied different true background heritabilities in combination with different threshold points. The points with the lowest heritability had the lowest variation in estimated heritability and those with the highest heritability had the highest variation. Results clearly show that by moving threshold points from the population mean towards the two tails of the distribution and also by increasing the true heritability, the differences between estimated heritability and true heritability increases. The linear model has always underestimated true heritabilities. It is possible to use our estimates to obtain a confidence range for true heritability when we have information about approximate location of threshold and estimates of heritability obtained by linear model.

Control of canalization and evolvability by Hsp90

Partial reduction of Hsp90 increases expression of morphological novelty in qualitative traits of Drosophila and Arabidopsis, but the extent to which the Hsp90 chaperone also controls smaller and more likely adaptive changes in natural quantitative traits has been unclear. To determine the effect of Hsp90 on quantitative trait variability we deconstructed genetic, stochastic and environmental components of variation in Drosophila wing and bristle traits of genetically matched flies, differing only by Hsp90 loss-of-function or wild-type alleles. Unexpectedly, Hsp90 buffering was remarkably specific to certain normally invariant and highly discrete quantitative traits. Like the qualitative trait phenotypes controlled by Hsp90, highly discrete quantitative traits such as scutellor and thoracic bristle number are threshold traits. When tested across genotypes sampled from a wild population or in laboratory strains, the sensitivity of these traits to many types of variation was coordinately controlled, while continuously variable bristle types and wing size, and critically invariant left-right wing asymmetry, remained relatively unaffected. Although increased environmental variation and developmental noise would impede many types of selection response, in replicate populations in which Hsp90 was specifically impaired, heritability and ‘extrinsic evolvability’, the expected response to selection, were also markedly increased. However, despite the overall buffering effect of Hsp90 on variation in populations, for any particular individual or genotype in which Hsp90 was impaired, the size and direction of its effects were unpredictable. The trait and genetic-background dependence of Hsp90 effects and its remarkable bias toward invariant or canalized traits support the idea that traits evolve independent and trait-specific mechanisms of canalization and evolvability through their evolution of non-linearity and thresholds. Highly non-linear responses would buffer variation in Hsp90-dependent signaling over a wide range, while over a narrow range of signaling near trait thresholds become more variable with increasing probability of triggering all-or-none developmental responses.

Genetic parameters for calving rate and calf survival from linear, threshold, and logistic models in a multibreed beef cattle population1

Generalized mixed linear, threshold, and logistic sire models and Markov chain, Monte Carlo simulation procedures were used to estimate genetic parameters for calving rate and calf survival in a multibreed beef cattle population. Data were obtained from a 5-generation rotational crossbreeding study involving Angus, Brahman, Charolais, and Hereford (1969 to 1995). Gelbvieh and Simmental bulls sired terminal-cross calves from a sample of generation 5 cows. A total of 1,458 cows sired by 158 bulls had a mean calving rate of 78% based on 4,808 calving records. Ninety-one percent of 5,015 calves sired by 260 bulls survived to weaning. Mean heritability estimates and standard deviations for daughter calving rate from posterior distributions were 0.063 +/- 0.024, 0.150 +/- 0.049, and 0.130 +/- 0.047 for linear, threshold, and logistic models, respectively. For calf survival, mean heritability estimates and standard deviations from posterior distributions were 0.049 +/- 0.022, 0.160 +/- 0.058, and 0.190 +/- 0.078 from linear, threshold, and logistic models, respectively. When transformed to an underlying normal scale, linear sire, mixed model, heritability estimates were similar to threshold and logistic sire mixed model estimates. Posterior density distributions of estimated heritabilities from all models were normal. Spearman rank correlations between sire EPD across statistical models were greater than 0.97 for daughter calving rate and for calf survival. Sire EPD had similar ranges across statistical models for daughter calving rate and for calf survival.

Genetic and environmental interactions determine seizure susceptibility in epileptic EL mice

Gene identification has progressed rapidly for monogenic epilepsies, but complex gene-environmental interactions have hindered progress in gene identification for multifactorial epilepsies. We analyzed the role of environmental risk factors in the inheritance of multifactorial idiopathic generalized epilepsy in the EL mouse. Seizure susceptibility was evaluated in the EL (E) and seizure-resistant ABP/LeJ (A) parental mouse strains and in their AEF1 and AEF2 hybrid offspring using a handling-induced seizure test. The seizure test was administered in three environments (environments I, II and III) that differed with respect to the number of seizure tests administered (one test or four tests) and the age of the mice when tested (young or old). The inheritance of seizure susceptibility appeared dominant after repetitive seizure testing in young or old mice, but recessive after a single test in old mice. Heritability was high (0.67-0.77) in each environment. Significant quantitative trait loci (QTL) that were associated with environments I and III (repetitive testing) were found on chromosomes 2 and 9 and colocalized with previously mapped El2 and El4, respectively. The El2 QTL found in environment I associated only with female susceptibility. A novel QTL, El-N, for age-dependent predisposition to seizures was found on proximal chromosome 9 only in environment II. The findings indicate that environmental risk factors determine the genetic architecture of seizure susceptibility in EL mice and suggest that QTL for complex epilepsies should be defined in terms of the environment in which they are expressed.

Bayes factor for testing the genetic background of quantitative threshold traits

The recent development of the Bayes factor methodology to test between nested competing models has allowed to evaluate the genetic component of continuous traits through a methodology based on the reparameterization of a variance component model in terms of intra-class correlation. In this study, we present a modification of that procedure to compare between threshold models that only differ in a bounded variable. An extra data-augmentation step is required to sample liability, an underlying continuous variable related with phenotypic records through an adequate number of thresholds. In this context, we describe a procedure to sample from a truncated multivariate normal distribution with non-null covariances between records, taking as starting point the Cholesky factorization and the inclusion of linear restrictions. The method was tested by computer simulations with satisfactory results.

A comparison between multivariate Slash, Student's t and probit threshold models for analysis of clinical mastitis in first lactation cows

Robust threshold models with multivariate Student's t or multivariate Slash link functions were employed to infer genetic parameters of clinical mastitis at different stages of lactation, with each cow defining a cluster of records. The robust fits were compared with that from a multivariate probit model via a pseudo-Bayes factor and an analysis of residuals. Clinical mastitis records on 36 178 first-lactation Norwegian Red cows from 5286 herds, daughters of 245 sires, were analysed. The opportunity for infection interval, going from 30 days pre-calving to 300 days postpartum, was divided into four periods: (i) -30 to 0 days pre-calving; (ii) 1-30 days; (iii) 31-120 days; and (iv) 121-300 days of lactation. Within each period, absence or presence of clinical mastitis was scored as 0 or 1 respectively. Markov chain Monte Carlo methods were used to draw samples from posterior distributions of interest. Pseudo-Bayes factors strongly favoured the multivariate Slash and Student's t models over the probit model. The posterior mean of the degrees of freedom parameter for the Slash model was 2.2, indicating heavy tails of the liability distribution. The posterior mean of the degrees of freedom for the Student's t model was 8.5, also pointing away from a normal liability for clinical mastitis. A residual was the observed phenotype (0 or 1) minus the posterior mean of the probability of mastitis. The Slash and Student's t models tended to have smaller residuals than the probit model in cows that contracted mastitis. Heritability of liability to clinical mastitis was 0.13-0.14 before calving, and ranged from 0.05 to 0.08 after calving in the robust models. Genetic correlations were between 0.50 and 0.73, suggesting that clinical mastitis resistance is not the same trait across periods, corroborating earlier findings with probit models.

A Thurstonian model for quantitative genetic analysis of ranks: a Bayesian approach

A fully Bayesian method for quantitative genetic analysis of data consisting of ranks of, e.g., genotypes, scored at a series of events or experiments is presented. The model postulates a latent structure, with an underlying variable realized for each genotype or individual involved in the event. The rank observed is assumed to reflect the order of the values of the unobserved variables, i.e., the classical Thurstonian model of psychometrics. Parameters driving the Bayesian hierarchical model include effects of covariates, additive genetic effects, permanent environmental deviations, and components of variance. A Markov chain Monte Carlo implementation based on the Gibbs sampler is described, and procedures for inferring the probability of yet to be observed future rankings are outlined. Part of the model is rendered nonparametric by introducing a Dirichlet process prior for the distribution of permanent environmental effects. This can lead to potential identification of clusters of such effects, which, in some competitions such as horse races, may reflect forms of undeclared preferential treatment.

Major gene detection for fusiform rust resistance using Bayesian complex segregation analysis in loblolly pine

The presence of major genes affecting rust resistance of loblolly pine was investigated in a progeny population that was generated with a half-diallel mating of six parents. A Bayesian complex segregation analysis was used to make inference about a mixed inheritance model (MIM) that included polygenic effects and a single major gene effect. Marginalizations were achieved by using Gibbs sampler. A parent block sampling by which genotypes of a parent and its offspring were sampled jointly was implemented to improve mixing. The MIM was compared with a pure polygenic model (PM) using Bayes factor. Results showed that the MIM was a better model to explain the inheritance of rust resistance than the pure PM in the diallel population. A large major gene variance component estimate (> 50% of total variance), indicated the existence of major genes for rust resistance in the studied loblolly pine population. Based on estimations of parental genotypes, it appears that there may be two or more major genes affecting disease phenotypes in this diallel population.

Multiple-interval mapping for ordinal traits

Many statistical methods have been developed to map multiple quantitative trait loci (QTL) in experimental cross populations. Among these methods, multiple-interval mapping (MIM) can map QTL with epistasis simultaneously. However, the previous implementation of MIM is for continuously distributed traits. In this study we extend MIM to ordinal traits on the basis of a threshold model. The method inherits the properties and advantages of MIM and can fit a model of multiple QTL effects and epistasis on the underlying liability score. We study a number of statistical issues associated with the method, such as the efficiency and stability of maximization and model selection. We also use computer simulation to study the performance of the method and compare it to other alternative approaches. The method has been implemented in QTL Cartographer to facilitate its general usage for QTL mapping data analysis on binary and ordinal traits.

Genetic Associations Between Clinical Mastitis and Somatic Cell Score in Early First-Lactation Cows

The objectives of this study were to examine genetic associations between clinical mastitis and somatic cell score (SCS) in early first-lactation cows, to estimate genetic correlations between SCS of cows with and without clinical mastitis, and to compare genetic evaluations of sires based on SCS or clinical mastitis. Clinical mastitis records from 15 d before to 30 d after calving and first test-day SCS records (from 6 to 30 d after calving) from 499,878 first-lactation daughters of 2,043 sires were analyzed. Results from a bivariate linear sire model analysis of SCS in cows with and without clinical mastitis suggest that SCS is a heterogeneous trait. Heritability of SCS was 0.03 for mastitic cows and 0.08 for healthy cows, and the genetic correlation between the 2 traits was 0.78. The difference in rank between sire evaluations based on SCS of cows with and without clinical mastitis varied from -994 to 1,125, with mean 0. A bivariate analysis with a threshold-liability model for clinical mastitis and a linear Gaussian model for SCS indicated that heritability of liability to clinical mastitis is at least as large as that of SCS in early lactation. The mean (standard deviation) of the posterior distribution of heritability was 0.085 (0.006) for liability to clinical mastitis and 0.070 (0.003) for SCS. The posterior mean (standard deviation) of the genetic correlation between liability to clinical mastitis and SCS was 0.62 (0.03). A comparison of sire evaluations showed that genetic evaluation based on SCS was not able to identify the best sires for liability to clinical mastitis. The association between sire posterior means for liability to clinical mastitis and sire predicted transmitting ability for SCS was far from perfect.

Recombinant inbreeding in mice reveals thresholds in embryonic corpus callosum development

The inbred strains BALB/cWah1 and 129P1/ReJ both show incomplete penetrance for absent corpus callosum (CC); about 14% of adult mice have no CC at all. Their F(1) hybrid offspring are normal, which proves that the strains differ at two or more loci pertinent to absent CC. Twenty-three recombinant inbred lines were bred from the F(2) cross of BALB/c and 129, and several of these expressed a novel and severe phenotype after only three or four generations of inbreeding - total absence of the CC and severe reduction of the hippocampal commissure (HC) in every adult animal. As inbreeding progressed, intermediate sizes of the CC and the HC remained quite rare. This striking phenotypic distribution in adults arose from developmental thresholds in the embryo. CC axons normally cross to the opposite hemisphere via a tissue bridge in the septal region at midline, where the HC forms before CC axons arrive. The primary defect in callosal agenesis in the BALB/c and 129 strains is severe retardation of fusion of the hemispheres in the septal region, and failure to form a CC is secondary to this defect. The putative CC axons arrive at midline at the correct time and place in all groups, but in certain genotypes, the bridge is not yet present. The relative timing of axon growth and delay of the septal bridge create a narrow critical period for forming a normal brain.

A Bayesian threshold-normal mixture model for analysis of a continuous mastitis-related trait

Mastitis is associated with elevated somatic cell count in milk, inducing a positive correlation between milk somatic cell score (SCS) and the absence or presence of the disease. In most countries, selection against mastitis has focused on selecting parents with genetic evaluations that have low SCS. Univariate or multivariate mixed linear models have been used for statistical description of SCS. However, an observation of SCS can be regarded as drawn from a 2- (or more) component mixture defined by the (usually) unknown health status of a cow at the test-day on which SCS is recorded. A hierarchical 2-component mixture model was developed, assuming that the health status affecting the recorded test-day SCS is completely specified by an underlying liability variable. Based on the observed SCS, inferences can be drawn about disease status and parameters of both SCS and liability to mastitis. The prior probability of putative mastitis was allowed to vary between subgroups (e.g., herds, families), by specifying fixed and random effects affecting both SCS and liability. Using simulation, it was found that a Bayesian model fitted to the data yielded parameter estimates close to their true values. The model provides selection criteria that are more appealing than selection for lower SCS. The proposed model can be extended to handle a wide range of problems related to genetic analyses of mixture traits.

Using the quantitative genetic threshold model for inferences between and within species

Sewall Wright's threshold model has been used in modelling discrete traits that may have a continuous trait underlying them, but it has proven difficult to make efficient statistical inferences with it. The availability of Markov chain Monte Carlo (MCMC) methods makes possible likelihood and Bayesian inference using this model. This paper discusses prospects for the use of the threshold model in morphological systematics to model the evolution of discrete all-or-none traits. There the threshold model has the advantage over 0/1 Markov process models in that it not only accommodates polymorphism within species, but can also allow for correlated evolution of traits with far fewer parameters that need to be inferred. The MCMC importance sampling methods needed to evaluate likelihood ratios for the threshold model are introduced and described in some detail.

Estimation of genetic parameters for the prevalence of osseous fragments in limb joints of Hanoverian Warmblood horses

Genetic parameters were estimated for the prevalence of osseous fragments in distal (DIJ) and proximal interphalangeal (PIJ), fetlock (FJ) and hock joints (HJ) of Hanoverian Warmblood horses by using residual maximum likelihood (REML) with linear animal models. The analyses were based on the results of 10 standardized radiographs of all four limbs of 3725 young riding horses selected for sale at auction. Transformation factors onto the underlying liability scale were verified by a simulation study. The heritability estimates of osseous fragments on the liability scale were in the range of h2 = 0.19-0.60. Further analyses of osseous fragments in FJ and HJ were performed separately in males and females. In both sexes, the heritabilities of osseous fragments in HJ were higher (h2 = 0.41 in males, h2 = 0.25 in females) than those of osseous fragments in FJ (h2 = 0.21 in males, h2 = 0.23 in females). Osseous fragments in the phalangeal joints (DIJ, PIJ, FJ) were genetically correlated moderately positive (r(g) = 0.19-0.41). The genetic correlations between osseous fragments in the phalangeal joints and in HJ were negative (r(g) = -0.27 to -0.67). Particularly, this applied to osseous fragments in FJ in both sexes, to those in front FJ in males and to osseous fragments in front and hind FJ of females (up to r(g) = -1). The heritability of height at withers was estimated at h2 = 0.27-0.28. Genetic correlations between height at withers and osseous fragments in equine limb joints were mostly moderately positive (up to r(g) = 0.75). We conclude from our results that osseous fragments in phalangeal and hock joints are genetically different traits but sex-specific expression of osseous fragments was unlikely.

Evidence of a major gene from Bayesian segregation analyses of liability to osteochondral diseases in pigs

Bayesian segregation analyses were used to investigate the mode of inheritance of osteochondral lesions (osteochondrosis, OC) in pigs. Data consisted of 1163 animals with OC and their pedigrees included 2891 animals. Mixed-inheritance threshold models (MITM) and several variants of MITM, in conjunction with Markov chain Monte Carlo methods, were developed for the analysis of these (categorical) data. Results showed major genes with significant and substantially higher variances (range 1.384-37.81), compared to the polygenic variance (sigmau2). Consequently, heritabilities for a mixed inheritance (range 0.65-0.90) were much higher than the heritabilities from the polygenes. Disease allele frequencies range was 0.38-0.88. Additional analyses estimating the transmission probabilities of the major gene showed clear evidence for Mendelian segregation of a major gene affecting osteochondrosis. The variants, MITM with informative prior on sigmau2, showed significant improvement in marginal distributions and accuracy of parameters. MITM with a "reduced polygenic model" for parameterization of polygenic effects avoided convergence problems and poor mixing encountered in an "individual polygenic model." In all cases, "shrinkage estimators" for fixed effects avoided unidentifiability for these parameters. The mixed-inheritance linear model (MILM) was also applied to all OC lesions and compared with the MITM. This is the first study to report evidence of major genes for osteochondral lesions in pigs; these results may also form a basis for underpinning the genetic inheritance of this disease in other animals as well as in humans.

Reflections on the pathogenesis of Down syndrome

Present efforts to identify, isolate, and characterize in molecular terms the "consensus" segment of 21q sufficient to cause most of the major and some of the most characteristic minor manifestations of Down syndrome will soon provide answers to many questions. However, we think that a reductionist approach to explain the Down syndrome phenotype in a "linear" manner from the DNA sequence of the segment will be doomed to failure from the outset because of the open, complex, nonlinear, hierarchical nature of morphogenetic systems. Neo-Darwinism is under strong attack; most genetic changes accumulated over time may very well be of neutral effect, and detailed studies in several related groups of vertebrate species has shown that molecular and organismal evolution are largely independent of one another. It has been pointed out recently that biology lacks a theory of ontogenetic and phylogenetic development, and that a purely "genocentric" view of biology at the expense of the complexly hierarchical intrinsic epigenetic attributes of developmental systems is "out of focus with respect to ... biological organization and morphogenesis," and may be "a residue of nineteenth century romantic idealism." Down syndrome impresses us as a paradigm of increased developmental variability due to a deceleration of the rate of development (neoteny) with many anomalies of incomplete morphogenesis (vestigia), atavisms, increased morphometric variability with many decreased means, increased variances, and increased fluctuating asymmetry. These abnormalities, together with highly increased risk of prenatal death and postnatal morbidity, impaired growth, and abnormal CNS and gonadal structure and function characteristic of most aneuploidy syndromes, suggest to us that the pathogenesis of Down syndrome is best viewed in terms of the mechanisms of speciation. Transgenic experiment involving sequential or overlapping pieces of "the consensus segment" on distal 21q22.1-22.3 may help decide to what extent the Down syndrome phenotype can be resolved into the additive effect of several pleiotropic oligogenes with epistatic interaction or the indirect secondary "mass" effect of a specific segment of 21q with epistatic interaction involving multiple loci on 21q and other chromosomes.

Comparison Between Bivariate Models for 56-Day Nonreturn and Interval from Calving to First Insemination in Norwegian Red

A bivariate threshold-linear (TL) and a bivariate linear-linear (LL) model were assessed for the genetic analysis of 56-d nonreturn (NR56) and interval from calving to first insemination (CFI) in first-lactation Norwegian Red (former Norwegian Dairy Cattle) (NRF). Three different datasets were used to infer genetic parameters and to predict transmitting abilities for NRF sires. Mean progeny group sizes were 147.8, 102.7, and 56.5 daughters, and the corresponding number of sires were 746, 743, and 742 in the 3 datasets. Otherwise, the structures of the 3 datasets were similar. When the TL model was used, heritability of liability to NR56 was 2.8% in the 2 larger datasets and 3.8% in the smallest dataset. In the LL model, the heritability of NR56 in the largest dataset and in the 2 smaller datasets was 1.2 and 0.9%, respectively. For CFI, the heritability was similar in TL and LL models, ranging from 2.4 to 2.7%. The small heritability of the 2 reproductive traits implies that most of the variation is environmental and that large progeny groups are required to get accurate sire PTA. The point estimates of the genetic correlation between NR56 and CFI were near zero in both models. The 2 bivariate models were compared in terms of predictive ability using logistic regression and a chi2 statistic based on differences between observed and predicted outcomes for NR56 in a separate dataset. Comparison was also with respect to ranking of sires and correlations between sire posterior means (TL model) and PTA (LL model). We found very small differences in ability to predict NR56 between the 2 bivariate models, regardless of the dataset used. Correlations between sire posterior means (TL) and sire PTA (LL) and rank correlations between sire evaluations were all >0.98 in the 3 datasets. At present, the LL model is preferred for sire evaluations of NR56 and CFI in NRF. This is because the LL model is less computationally demanding and more robust with respect to the structure of the data than TL.

Genetic Analyses of Mastitis Data Using Animal Threshold Models and Genetic Correlation with Production Traits

In the present study, 6 different mastitis data sets of 3 dairy herds with an overall herd size of 3200 German Holstein cows were analyzed. Data collection periods included the first 50, 100, or 300 d of lactation. The 3 data collection periods were analyzed with a lactation model and a test-day model. All models were animal threshold models. Mastitis frequencies in the lactation model data sets varied between 29 and 45%, and varied between 3 and 6% in the test-day model data sets. Depending on the period of data collection, heritabilities of liability to mastitis in the lactation models were 0.05 (50 d), 0.06 (100 d), and 0.07 (300 d). In the test-day models, heritabilities were slightly higher with values of 0.09 (50 and 100 d), and 0.06 (300 d). Between lactation models, the rank correlations between the relative breeding values were high and varied between 0.86 and 0.94. Rank correlations between the relative breeding values of the test-day models ranged from 0.68 to 0.87. The rank correlations between the relative breeding values of lactation models and test-day models varied from 0.51 and 0.80. Genetic correlations between mastitis and milk production traits were estimated with a linear animal test-day model. The correlations with mastitis were 0.29 (milk yield), 0.30 (fat yield), 0.20 (fat content), 0.34 (protein yield), and 0.20 (protein content). The estimated genetic correlation between mastitis and somatic cell score was 0.84.

Power of variance component linkage analysis-II. Discrete traits

We determine the power of variance component linkage analysis in the case of discrete, dichotomous traits analyzed under a classical liability threshold model. For simplicity we consider randomly ascertained samples and an additive model of variation incorporating a qtl, residual additive genetic factors, and individual-specific random environmental effects. We derive an expression for the power of variance component linkage analysis in arbitrary relative pairs, and compare the power of discrete and quantitative trait linkage analysis in the specific case of sibpairs. The predicted sample sizes required in linkage analysis of sibpairs are confirmed by analysis of simulated data. Unlike the affected-sibpair method, the power of discrete trait variance component analysis increases with trait prevalence. The relative efficiency of a discrete trait for linkage analysis increases with population trait prevalence, but does not exceed about 40% and is typically much less.

The ecological genetics of conditional strategies

We develop a quantitative genetic model for conditional strategies that incorporates the ecological realism of previous strategic models. Similar to strategic models, the results show that environmental heterogeneity, cue reliability, and environment-dependent fitness trade-offs for the alternative tactics of the conditional strategy interact to determine when conditional strategies will be favored and that conditional strategies should be a common form of adaptive variation in nature. The results also show that conditional and unconditional development can be maintained in one of two ways: by frequency-dependent selection or by the maintenance of genetic variation that exceeds the threshold for induction. We then modified the model to take into account variance in exposures to the environmental cue as well as variance in response to the cue, which allows a derivation of a dose-response curve. Here the results showed that increasing the genetic variance for response both flattens and shifts the dose-response curve. Finally, we modify the model to derive the dose-response curve for a population polymorphic for a gene that blocks expression of the conditional strategy. We illustrate the utility of the model by application to predator-induced defense in an intertidal barnacle and compare the results with phenotypic models of selection.

Genetic Parameters for Stillbirth in Danish Holstein Cows Using a Bayesian Threshold Model

The objective of this study was to make an inference about the direct and maternal genetic variation of stillbirth for first-calving Holstein cows and to estimate the effect of breed and heterosis for original Danish black and white and Holstein-Friesian. A Bayesian threshold model, which included correlated genetic effects of sires and maternal grandsires was used. Marginal posterior distributions of effects were obtained using Gibbs sampling. Point estimates were compared with results from a linear model using REML. Data with and without twins were analyzed and models with and without effects of breed and heterosis were fitted, but estimates of genetic parameters were almost identical. In all the analyses with threshold models, the marginal posterior mean (and standard deviation) was 0.10 (0.014) for the direct heritability, 0.13 (0.015) for the maternal heritability, and 0.05 (0.10) for the genetic correlation between direct and maternal effects. The stillbirth rate tended to increase with a higher proportion of Holstein-Friesian in the calf and in the dam, but no effects of breed and heterosis were significant. Joint sampling of all location parameters was found superior to univariate sampling in terms of much better mixing properties of the fixed effects. Based on the results showing genetic variation for stillbirth at first calving, both the direct and the maternal effect could be included in the breeding program.

Bayesian Estimates of Covariance Components Between Lactation Curve Parameters and Disease Liability in Danish Holstein Cows

In the present work, covariance components for milk yield and disease liability were estimated with bivariate random regression test-day sire models using a Bayesian approach and implemented via the Gibbs sampler. The data consist of 8075 first-parity Danish Holstein (DH) cows, from 1259 sires, performing in 57 herds from 1992 to 1997. Treatments associated with five different type of diseases were pooled into a single general disease liability for each cow. Two models were fitted to the data. First, using a bivariate model, milk yield is modeled via a random regression, and disease liability via a repeatablility model. Second, using a bivariate model, both milk yield and disease liability are modeled using random regressions. A comparison based on a Bayes factor provides very strong support for the bivariate random regression model. Posterior means of heritabilities for each of the traits were estimated for five different points in time throughout lactation. Across models, heritabilities for milk yield are lowest in the beginning of the lactation (0.19) and highest at the end of the lactation (0.35). Posterior means of heritabilities of disease liability range from 0.04 to 0.10 for test days, and is equal to 0.20 for the whole lactation. Heritability of persistency measures estimated from the two models are 0.20 and 0.21. Estimates of posterior means of genetic correlations between single test-day milk yield and single test-day disease liability are in the range of 0.31 to 0.57. The estimates of posterior mean and of the 95% posterior interval of the genetic correlation between persistency and (total) disease liability using the model with the highest posterior probability are -0.12 and (-0.44; 0.20), respectively. Even though the largest proportion of the posterior probability mass is spread along negative values of the correlation (indicating that individuals with a flatter lactation curve tend to have lower disease liability), a value of zero of the genetic correlation falls comfortably within the 95% posterior interval. Thus the prospects of reducing incidence of disease by manipulating persistency as defined in this work remain inconclusive.

Genetic change for clinical mastitis in Norwegian cattle: a threshold model analysis

Records of clinical mastitis on 1.6 million first-lactation daughters of 2,411 Norwegian Cattle sires that were progeny tested from 1978 through 1998 were analyzed with a threshold model. The main objective was to infer genetic change for the disease in the population. A Bayesian approach via Gibbs sampling was used. The model for the underlying liability had age at first calving, month x year of calving, herd x 3-year-period, and sire of the cow as explanatory variables. Posterior mean (SD) of heritability of liability to clinical mastitis was 0.066 (0.003). Genetic evaluations (posterior means) of sires both in the liability and observable scales were computed. Annual genetic change of liability to clinical mastitis for progeny tested bulls born from 1973 to 1993 was assessed. The linear regression of mean sire effect on year of birth had a posterior mean (SD) of -0.00018 (0.0004), suggesting a nearly constant genetic level for clinical mastitis. However, an analysis of sire posterior means by birth-year of daughters indicated an approximately constant genetic level in the cow population from 1976 to 1990 (-0.02%/yr), and a genetic improvement thereafter (-0.27%/yr). This reflects more emphasis on mastitis in selection of bulls in recent years. Corresponding results obtained with a standard linear model analysis were -0.01% and -0.23% per year, respectively (regression of sire predicted transmitting ability on birth-year of daughters). Genetic change seems to be slightly understated with the linear model, assuming the threshold model holds true.

Finding genes influencing susceptibility to complex diseases in the post-genome era

During the last decade, hundreds of genes that harbor mutations causing simple Mendelian disorders have been identified using a combination of linkage analysis and positional cloning techniques. Traditional approaches to gene mapping have been largely unsuccessful in mapping genes influencing so-called 'complex' genetic diseases, however, because of low power and other factors. Complex genetic diseases do not display simple Mendelian patterns of inheritance, although genes do have an influence and close relatives of probands consequently have an increased risk. These disorders are thought to be due to the combined effects of variation at multiple interacting genes and the environment. Complex diseases have a significant impact on human health because of their high population incidence (unlike simple Mendelian disorders, which tend to be rare). New techniques are being developed aimed specifically at mapping genes conferring susceptibility to complex diseases. A project aimed at mapping genes influencing susceptibility to a complex disease may be undertaken in several stages: establishing a genetic basis for the disease in one or more populations; measuring the distribution of gene effects; studying statistical power using models; carrying out marker-based mapping studies using linkage or association. Quantitative genetic models can be used to estimate the heritability of a complex (polygenic) disease, as well as to predict the distribution of gene effects and to test whether one or more quantitative trait loci (QTLs) exist. Such models can be used to predict the power of different mapping approaches, but are often unrealistic and therefore provide only approximate predictions. Linkage analyses, association studies and family-based association tests are all hindered by low power and other specific problems. Association studies tend to be more powerful but can generate spurious associations due to population admixture. Alternative strategies for association mapping include the use of recent founder populations or unique isolated populations that are genetically homogeneous, and the use of unlinked markers (so-called genomic controls) to assign different regions of the genome of an admixed individual to particular source populations. Linkage disequilibrium observed in a sample of unrelated affected and normal individuals can also be used to fine-map a disease susceptibility locus in a candidate region. New Bayesian strategies make use of an annotated human genome sequence to further refine the position of a candidate disease susceptibility locus.

Generalized character process models: estimating the genetic basis of traits that cannot be observed and that change with age or environmental conditions

The genetic analysis of characters that change as a function of some independent and continuous variable has received increasing attention in the biological and statistical literature. Previous work in this area has focused on the analysis of normally distributed characters that are directly observed. We propose a framework for the development and specification of models for a quantitative genetic analysis of function-valued characters that are not directly observed, such as genetic variation in age-specific mortality rates or complex threshold characters. We employ a hybrid Markov chain Monte Carlo algorithm involving a Monte Carlo EM algorithm coupled with a Markov chain approximation to the likelihood, which is quite robust and provides accurate estimates of the parameters in our models. The methods are investigated using simulated data and are applied to a large data set measuring mortality rates in the fruit fly, Drosophila melanogaster.