The coefficient of dominance is not (always) estimable with biallelic markers

The genetic relationship among individuals at one locus is characterized by nine coefficients of identity. The coefficients of inbreeding, coancestry and dominance (or fraternity) are just linear functions of them. Here, it is shown how they can be estimated using biallelic and triallelic markers using the method of moments, and comparisons are made with other methods based on molecular coancestry or molecular covariance. It is concluded that in the general case of dominance and inbreeding with biallelic markers, only the coefficients of inbreeding and coancestry can be estimated, but neither the single coefficients of identity nor the coefficient of dominance can be estimated. More than two alleles are required for a full estimation as illustrated with the triallelic situation.

Accuracy of genome-wide evaluation for disease resistance in aquaculture breeding programs

Current aquaculture breeding programs aimed at improving resistance to diseases are based on challenge tests, where performance is recorded on sibs of candidates to selection, and on selection between families. Genome-wide evaluation (GWE) of breeding values offers new opportunities for using variation within families when dealing with such traits. However, up-to-date studies on GWE in aquaculture programs have only considered continuous traits. The objectives of this study were to extend GWE methodology, in particular the Bayes B method, to analyze dichotomous traits such as resistance to disease, and to quantify, through computer simulation, the accuracy of GWE for disease resistance in aquaculture sib-based programs, using the methodology developed. Two heritabilities (0.1 and 0.3) and 2 disease prevalences (0.1 and 0.5) were assumed in the simulations. We followed the threshold liability model, which assumes that there is an underlying variable (liability) with a continuous distribution and assumed a BayesB model for the liabilities. It was shown that the threshold liability model used fits very well with the BayesB model of GWE. The advantage of using the threshold model was clear when dealing with disease resistance dichotomous phenotypes, particularly under the conditions where linear models are less appropriate (low heritability and disease prevalence). In the testing set (where individuals are genotyped but not measured), the increase in accuracy for the simulated schemes when using the threshold model ranged from 4 (for heritability equal to 0.3 and prevalence equal to 0.5) to 16% (for heritability and prevalence equal to 0.1) when compared with the linear model.

Sex ratio variation in Iberian pigs

Within the area of sex allocation, one of the topics that has attracted a lot of attention is the sex ratio problem. Fisher (1930) proposed that equal numbers of males and females have been promoted by natural selection and it has an adaptive significance. But the empirical success of Fisher's theory remains doubtful because a sex ratio of 0.50 is also expected from the chromosomal mechanism of sex determination. Another way of approaching the subject is to consider that Fisher's argument relies on the underlying assumption that offspring inherit their parent's tendency in biased sex ratio and therefore that genetic variance for this trait exists. Here, we analyzed sex ratio data of 56,807 piglets coming from 550 boars and 1893 dams. In addition to classical analysis of heterogeneity we performed analyses fitting linear and threshold animal models in a Bayesian framework using Gibbs sampling techniques. The marginal posterior mean of heritability was 2.63 x 10(-4) under the sire linear model and 9.17 x 10(-4) under the sire threshold model. The probability of the hypothesis p(h(2) = 0) fitting the last model was 0.996. Also, we did not detect any trend in sex ratio related to maternal age. From an evolutionary point of view, the chromosomal sex determination acts as a constraint that precludes control of offspring sex ratio in vertebrates and it should be included in the general theory of sex allocation. From a practical view that means that the sex ratio in domestic species is hardly susceptible to modification by artificial selection.

Bayes factors for detection of quantitative trait loci

A fundamental issue in quantitative trait locus (QTL) mapping is to determine the plausibility of the presence of a QTL at a given genome location. Bayesian analysis offers an attractive way of testing alternative models (here, QTL vs. no-QTL) via the Bayes factor. There have been several numerical approaches to computing the Bayes factor, mostly based on Markov Chain Monte Carlo (MCMC), but these strategies are subject to numerical or stability problems. We propose a simple and stable approach to calculating the Bayes factor between nested models. The procedure is based on a reparameterization of a variance component model in terms of intra-class correlation. The Bayes factor can then be easily calculated from the output of a MCMC scheme by averaging conditional densities at the null intra-class correlation. We studied the performance of the method using simulation. We applied this approach to QTL analysis in an outbred population. We also compared it with the Likelihood Ratio Test and we analyzed its stability. Simulation results were very similar to the simulated parameters. The posterior probability of the QTL model increases as the QTL effect does. The location of the QTL was also correctly obtained. The use of meta-analysis is suggested from the properties of the Bayes factor.

Hypothesis testing for the genetic background of quantitative traits

The testing of Bayesian point null hypotheses on variance component models have resulted in a tough assigmment for which no clear and generally accepted method exists. In this work we present what we believe is a succeeding approach to such a task. It is based on a simple reparameterization of the model in terms of the total variance and the proportion of the additive genetic variance with respect to it, as well as on the explicit inclusion on the prior probability of a discrete component at origin. The reparameterization was used to bypass an arbitrariness related to the impropriety of uninformative priors onto unbounded variables while the discrete component was necessary to overcome the zero probability assigned to sets of null measure by the usual continuous variable models. The method was tested against computer simulations with appealing results.

Using coupling with the Gibbs sampler to assess convergence in animal models

The coupling method was investigated as a method to assess the convergence of the Gibbs sampler when drawing marginal inferences with an animal model. This method is based on the output of two Markov chains with different starting values but the same conditional deviates. The coupling method shows that the Gibbs sampler has an exponential convergence when the variance components are assumed to be known. All the variables in the model have the same rate of convergence, and it is closely related with the largest eigenvalue of a matrix derived from the coefficient matrix of the mixed model equations. Models including the variance components as unknowns showed a rate of convergence equal for all the variables, and this rate of convergence was approximately exponential. The coupling method provides an estimation of the convergence at the current iteration, and it does not require a post-Gibbs analysis as do the single chain-based methods. The coupling method is less computationally demanding than multiple chain methods, because only two chains are required to assess convergence.

Bayesian inference of variance components for litter size in Rasa Aragonesa sheep

Variance components were estimated for litter size in Rasa Aragonesa sheep, a meat breed from northern Spain, to determine whether selective breeding for litter size is a reasonable strategy to improve reproductive performance. We assumed an animal mixed effect threshold model with a binary response variable. Marginal estimates of the genetic parameters were obtained in the underlying scale using Bayesian inference, implemented via the Gibbs sampling procedure and a data augmentation approach. Posterior marginal means of heritability and repeatability were .077 and .141, respectively. Moreover, the 95% highest marginal posterior density region of heritability ranged from .051 to .101. Therefore, we conclude that litter size is a trait that could be selected for in breeding programs. The effect of the loss of pedigree information, a common feature of sheep production, on the estimation of the genetic parameters was also studied using simulation. The results indicate that the lack of pedigree information has little effect on our estimates of heritability.