Luis Alberto García-Cortés

A note on the rationale for estimating genealogical coancestry from molecular markers

BackgroundGenetic relatedness or similarity between individuals is a key concept in population, quantitative and conservation genetics. When the pedigree of a population is available and assuming a founder population from which the genealogical records start, genetic relatedness between individuals can be estimated by the coancestry coefficient. If pedigree data is lacking or incomplete, estimation of the genetic similarity between individuals relies on molecular markers, using either molecular coancestry or molecular covariance. Some relationships between genealogical and molecular coancestries and covariances have already been described in the literature.MethodsWe show how the expected values of the empirical measures of similarity based on molecular marker data are functions of the genealogical coancestry. From these formulas, it is easy to derive estimators of genealogical coancestry from molecular data. We include variation of allelic frequencies in the estimators.ResultsThe estimators are illustrated with simulated examples and with a real dataset from dairy cattle. In general, estimators are accurate and only slightly biased. From the real data set, estimators based on covariances are more compatible with genealogical coancestries than those based on molecular coancestries. A frequently used estimator based on the average of estimated coancestries produced inflated coancestries and numerical instability. The consequences of unknown gene frequencies in the founder population are briefly discussed, along with alternatives to overcome this limitation.ConclusionsEstimators of genealogical coancestry based on molecular data are easy to derive. Estimators based on molecular covariance are more accurate than those based on identity by state. A correction considering the random distribution of allelic frequencies improves accuracy of these estimators, especially for populations with very strong drift.

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 assignment 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.

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.

Combining Genomic and Genealogical Information in a Reproducing Kernel Hilbert Spaces Regression Model for Genome-Enabled Predictions in Dairy Cattle

Genome-enhanced genotypic evaluations are becoming popular in several livestock species. For this purpose, the combination of the pedigree-based relationship matrix with a genomic similarities matrix between individuals is a common approach. However, the weight placed on each matrix has been so far established with ad hoc procedures, without formal estimation thereof. In addition, when using marker- and pedigree-based relationship matrices together, the resulting combined relationship matrix needs to be adjusted to the same scale in reference to the base population. This study proposes a semi-parametric Bayesian method for combining marker- and pedigree-based information on genome-enabled predictions. A kernel matrix from a reproducing kernel Hilbert spaces regression model was used to combine genomic and genealogical information in a semi-parametric scenario, avoiding inversion and adjustment complications. In addition, the weights on marker- versus pedigree-based information were inferred from a Bayesian model with Markov chain Monte Carlo. The proposed method was assessed involving a large number of SNPs and a large reference population. Five phenotypes, including production and type traits of dairy cattle were evaluated. The reliability of the genome-based predictions was assessed using the correlation, regression coefficient and mean squared error between the predicted and observed values. The results indicated that when a larger weight was given to the pedigree-based relationship matrix the correlation coefficient was lower than in situations where more weight was given to genomic information. Importantly, the posterior means of the inferred weight were near the maximum of 1. The behavior of the regression coefficient and the mean squared error was similar to the performance of the correlation, that is, more weight to the genomic information provided a regression coefficient closer to one and a smaller mean squared error. Our results also indicated a greater accuracy of genomic predictions when using a large reference population.

Artificial selection with traditional or genomic relationships: consequences in coancestry and genetic diversity

Estimated breeding values (EBVs) are traditionally obtained from pedigree information. However, EBVs from high-density genotypes can have higher accuracy than EBVs from pedigree information. At the same time, it has been shown that EBVs from genomic data lead to lower increases in inbreeding compared with traditional selection based on genealogies. Here we evaluate the performance with BLUP selection based on genealogical coancestry with three different genome-based coancestry estimates: (1) an estimate based on shared segments of homozygosity, (2) an approach based on SNP-by-SNP count corrected by allelic frequencies, and (3) the identity by state methodology. We evaluate the effect of different population sizes, different number of genomic markers, and several heritability values for a quantitative trait. The performance of the different measures of coancestry in BLUP is evaluated in the true breeding values after truncation selection and also in terms of coancestry and diversity maintained. Accordingly, cross-performances were also carried out, that is, how prediction based on genealogical records impacts the three other measures of coancestry and inbreeding, and viceversa. Our results show that the genetic gains are very similar for all four coancestries, but the genomic-based methods are superior to using genealogical coancestries in terms of maintaining diversity measured as observed heterozygosity. Furthermore, the measure of coancestry based on shared segments of the genome seems to provide slightly better results on some scenarios, and the increase in inbreeding and loss in diversity is only slightly larger than the other genomic selection methods in those scenarios. Our results shed light on genomic selection vs. traditional genealogical-based BLUP and make the case to manage the population variability using genomic information to preserve the future success of selection programmes.

Variance and covariance of actual relationships between relatives at one locus.

The relationship between pairs of individuals is an important topic in many areas of population and quantitative genetics. It is usually measured as the proportion of thegenome identical by descent shared by the pair and it can be inferred from pedigree information. But there is a variance in actual relationships as a consequence of Mendelian sampling, whose general formula has not been developed. The goal of this work is to develop this general formula for the one-locus situation,. We provide simple expressions for the variances and covariances of all actual relationships in an arbitrary complex pedigree. The proposed method relies on the use of the nine identity coefficients and the generalized relationship coefficients; formulas have been checked by computer simulation. Finally two examples for a short pedigree of dogs and a long pedigree of sheep are given.

Model determination in a case of heterogeneity of variance using sampling techniques

SUMMARY A sampling determination procedure has been described in a case of heterogeneity of variance. The procedure makes use of the predictive distributions of each data given the rest of the data and the structure of the assumed model. The computation of these predictive distributions is carried out using a Gibbs Sampling procedure. The final criterion to compare between models is the Mean Square Error between the expectation of predictive distributions and real data. The procedure has been applied to a data set of weight at 210 days in the Spanish Pirenaica beef cattle breed. Three proposed models have been compared: (a) Single Trait Animal Model; (b) Heterogeneous Variance Animal Model; and (c) Multiple Trait Animal Model. After applying the procedure, the most adjusted model was the Heterogeneous Variance Animal Model. This result is probably due to a compromise between the complexity of the model and the amount of available information. The estimated heritabilities under the preferred model have been 0.489 ± 0.076 for males and 0.331 ± 0.082 for females. RESUMEN: Contraste de modelos en un caso de heterogeneidad de varianzas usando métodos de muestreo Se ha descrito un método de contraste de modelos mediante técnicas de muestreo en un caso de heterogeneidad de varianza entre sexos. El procedimiento utiliza las distribucviones predictivas de cada dato, dado el resto de datos y la estructura del modelo. El criterio para coparar modelos es el error cuadrático medio entre la esperanza de las distribuciones predictivas y los datos reales. El procedimiento se ha aplicado en datos de peso a los 210 días en la raza bovina Pirenaica. Se han propuesto tres posibles modelos: (a) Modelo Animal Unicaracter; (b) Modelo Animal con Varianzas Heterogéneas; (c) Modelo Animal Multicaracter. El modelo mejor ajustado fue el Modelo Animal con Varianzas Heterogéneas. Este resultado es probablemente debido a un compromiso entre la complejidad del modelo y la cantidad de datos disponibles. Las heredabilidades estimadas bajo el modelo preferido han sido 0,489 ± 0,076 en los machos y 0,331 ± 0,082 en las hembras.

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.

Bayes factor between Student t and Gaussian mixed models within an animal breeding context

The implementation of Student t mixed models in animal breeding has been suggested as a useful statistical tool to effectively mute the impact of preferential treatment or other sources of outliers in field data. Nevertheless, these additional sources of variation are undeclared and we do not know whether a Student t mixed model is required or if a standard, and less parameterized, Gaussian mixed model would be sufficient to serve the intended purpose. Within this context, our aim was to develop the Bayes factor between two nested models that only differed in a bounded variable in order to easily compare a Student t and a Gaussian mixed model. It is important to highlight that the Student t density converges to a Gaussian process when degrees of freedom tend to infinity. The twomodels can then be viewed as nested models that differ in terms of degrees of freedom. The Bayes factor can be easily calculated from the output of a Markov chain Monte Carlo sampling of the complex model (Student t mixed model). The performance of this Bayes factor was tested under simulation and on a real dataset, using the deviation information criterion (DIC) as the standard reference criterion. The two statistical tools showed similar trends along the parameter space, although the Bayes factor appeared to be the more conservative. There was considerable evidence favoring the Student t mixed model for data sets simulated under Student t processes with limited degrees of freedom, and moderate advantages associated with using the Gaussian mixed model when working with datasets simulated with 50 or more degrees of freedom. For the analysis of real data (weight of Pietrain pigs at six months), both the Bayes factor and DIC slightly favored the Student t mixed model, with there being a reduced incidence of outlier individuals in this population.

A novel recursive algorithm for the calculation of the detailed identity coefficients

BackgroundA recursive algorithm to calculate the fifteen detailed coefficients of identity is introduced. Previous recursive procedures based on the generalized coefficients of kinship provided the detailed coefficients of identity under the assumption that the two individuals were not an ancestor of each other.FindingsBy using gametic relationships to include three, four or two pairs of gametes, we can obtain these coefficients for any pair of individuals. We have developed a novel linear transformation that allows for the calculation of pairwise detailed identity coefficients for any pedigree given the gametic relationships. We illustrate the procedure using the well-known pedigree of Julio and Mencha, which contains 20 Jicaque Indians of Honduras, to calculate their detailed coefficients.ConclusionsThe proposed algorithm can be used to calculate the detailed identity coefficients of two or more individuals with any pedigree relationship.