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Benjamin and Berger (henceforth referred to as BB) pertinently advocated the users of NHST to convert p-values into Bayes factors according to a very simple, if not magic, formula: BFB=(−ep log p)−...

Microarrays allow researchers to measure the expression of thousands of genes in a single experiment. Before statistical comparisons can be made, the data must be assessed for quality and normalisation procedures must be applied, of which many have been proposed. Methods of comparing the normalised data are also abundant, and no clear consensus has yet been reached. The purpose of this paper was to compare those methods used by the EADGENE network on a very noisy simulated data set. With the a priori knowledge of which genes are differentially expressed, it is possible to compare the success of each approach quantitatively. Use of an intensity-dependent normalisation procedure was common, as was correction for multiple testing. Most variety in performance resulted from differing approaches to data quality and the use of different statistical tests. Very few of the methods used any kind of background correction. A number of approaches achieved a success rate of 95% or above, with relatively small numbers of false positives and negatives. Applying stringent spot selection criteria and elimination of data did not improve the false positive rate and greatly increased the false negative rate. However, most approaches performed well, and it is encouraging that widely available techniques can achieve such good results on a very noisy data set.

The importance of variance modelling is now widely known for the analysis of microarray data. In particular the power and accuracy of statistical tests for differential gene expressions are highly dependent on variance modelling. The aim of this paper is to use a structural model on the variances, which includes a condition effect and a random gene effect, and to propose a simple estimation procedure for these parameters by working on the empirical variances. The proposed variance model was compared with various methods on both real and simulated data. It proved to be more powerful than the gene-by-gene analysis and more robust to the number of false positives than the homogeneous variance model. It performed well compared with recently proposed approaches such as SAM and VarMixt even for a small number of replicates, and performed similarly to Limma. The main advantage of the structural model is that, thanks to the use of a linear mixed model on the logarithm of the variances, various factors of variation can easily be incorporated in the model, which is not the case for previously proposed empirical Bayes methods. It is also very fast to compute and is adapted to the comparison of more than two conditions.

The analysis of nonlinear function-valued characters is very important in genetic studies, especially for growth traits of agricultural and laboratory species. Inference in nonlinear mixed effects models is, however, quite complex and is usually based on likelihood approximations or Bayesian methods. The aim of this paper was to present an efficient stochastic EM procedure, namely the SAEM algorithm, which is much faster to converge than the classical Monte Carlo EM algorithm and Bayesian estimation procedures, does not require specification of prior distributions and is quite robust to the choice of starting values. The key idea is to recycle the simulated values from one iteration to the next in the EM algorithm, which considerably accelerates the convergence. A simulation study is presented which confirms the advantages of this estimation procedure in the case of a genetic analysis. The SAEM algorithm was applied to real data sets on growth measurements in beef cattle and in chickens. The proposed estimation procedure, as the classical Monte Carlo EM algorithm, provides significance tests on the parameters and likelihood based model comparison criteria to compare the nonlinear models with other longitudinal methods.

This article presents an extension of the methodology developed by Gilmour et al. [19], for ordered categorical data, taking into account the heterogeneity of residual variances of latent variables. Heterogeneity of residual variances is described via a structural linear model on log-variances. This method involves two main steps: i) a ’marginalization’ with respect to the random effects leading to quasi-score estimators; ii) an approximation of the variance-covariance matrix of the observations which leads to an analogue of the Henderson mixed model equations for continuous Gaussian data. This methodology is illustrated by a numerical example of footshape in sheep., Cet article présente une extension de la méthodologie développée par Gilmour et al. [19] dans le cas de variables qualitatives ordonnées, prenant en compte l’hétérogénéité des variances résiduelles des variables latentes. L’hétérogénéité des variances résiduelles est décrite par un modèle linéaire structurel sur les logarithmes des variances. Cette méthode comprend deux étapes principales : i) une « marginalisation » par rapport aux effets aléatoires qui conduit, grâce aux équations de quasi-score, à l’estimation des paramètres ; ii) une approximation de la matrice de variance-covariance des observations qui aboutit à un système analogue aux équations du modèle mixte d’Henderson dans le cas de variables continues gaussiennnes. Cette méthodologie est illustrée par un exemple sur la forme des pieds chez le mouton.

The diversity of a set of breeds or species is defined in the Weitzman approach by a recursion formula using the pairwise genetic distances between the elements of the set. The algorithm for computing the diversity function of Weitzman is described. It also provides a taxonomy of the set which is interpreted as the maximum likelihood phylogeny. The theory is illustrated by an application to 19 European cattle breeds. The possible uses of the method for defining optimal conservation strategies are briefly discussed., Un aperçu sur l’approche de la diversité selon Weitzman. La diversité d’un ensemble d’espèces, ou de races, est définie par Weitzman de façon récursive ; les données de départ sont les distances génétiques entre les éléments de l’ensemble pris deux à deux. L’algorithme de calcul de la diversité fournit, comme résultat intermédiaire, un arbre de classement des espèces en présence, qui est interprété comme une phylogénie du maximum de vraisemblance. La théorie est illustrée par un exemple d’application à 19 races bovines européennes, et les utilisations possibles de la méthode pour définir des stratégies optimales de conservation sont discutées brièvement.

This paper presents techniques of parameter estimation in heteroskedastic mixed models having i) heterogeneous log residual variances which are described by a linear model of explanatory covariates and ii) log residual and log u-components linearly related. This makes the intraclass correlation a monotonic function of the residual variance. Cases of a homogeneous variance ratio and of a homogeneous u-component of variance are also included in this parameterization. Estimation and testing procedures of the corresponding dispersion parameters are based on restricted maximum likelihood procedures. Estimating equations are derived using the standard and gradient EM. The analysis of a small example is outlined to illustrate the theory., Cet article présente des techniques d’estimation des paramètres intervenant dans des modèles mixtes caractérisés i) par des logvariances résiduelles décrites par un modèle linéaire de covariables explicatives et ii) par des composantes u et e liées par une fonction affine. Cela conduit à un coefficient de corrélation intraclasse qui varie comme une fonction monotone de la variance résiduelle. Le cas d’une corrélation constante et celui d’une composante u constante sont également inclus dans cette paramétrisation. L’estimation et les tests relatifs aux paramètres de dispersion correspondants sont basés sur les méthodes du maximum de vraisemblance restreint (REML). Les équations à résoudre pour obtenir ces estimations sont établies à partir de l’algorithme EM standard et gradient. La théorie est illustrée par l’analyse numérique d’un petit exemple.

This paper describes a further contribution to the problem of testing homogeneity of intra-class correlations among environments in the case of univariate linear models, without making any assumption about the genetic correlation between environments. An iterative generalized expectation-maximization (EM) algorithm, as described in Foulley and Quaas (1994), is presented for computing restricted maximum likelihood (REML) estimates of the residual and between-family components of variance and covariance. Three different parameterizations (cartesian, polar and spherical coordinates) are proposed to compute EM-REML estimators under the reduced (constant intra-class correlation between environments) model. This procedure is illustrated with the analysis of simulated data., Cet article décrit une approche permettant d’estimer les composantes de variance-covariance entre milieux dans le cas de corrélation intra-classe homogènes entre milieux, sans faire d’hypothèse sur les corrélations génétiques entre milieux pris 2 à 2. Un algorithme itératif d’espérance-maximisation (EM), comparable à celui décrit par Foulley et Quaas (1994), est proposé pour calculer les estimations du maximum de vraisemblance restreinte (REML) des composantes résiduelles et familiales de variance covariance. Trois paramétrisations différentes (coordonnées cartésiennes, polaires et sphériques) sont proposées pour calculer les estimateurs EM-REML sous le modèle réduit (les corrélations intra-classe sont supposées toutes égales à une même constante). Cette procédure est illustrée par l’analyse de données simulées.

Gene regulatory networks are collections of genes that interact with one other and with other substances in the cell. By measuring gene expression over time using high-throughput technologies, it may be possible to reverse engineer, or infer, the structure of the gene network involved in a particular cellular process. These gene expression data typically have a high dimensionality and a limited number of biological replicates and time points. Due to these issues and the complexity of biological systems, the problem of reverse engineering networks from gene expression data demands a specialized suite of statistical tools and methodologies. We propose a non-standard adaptation of a simulation-based approach known as Approximate Bayesian Computing based on Markov chain Monte Carlo sampling. This approach is particularly well suited for the inference of gene regulatory networks from longitudinal data. The performance of this approach is investigated via simulations and using longitudinal expression data from a genetic repair system in Escherichia coli., 16 pages, 11 figures

Structural Equation Models with latent variables (SEM) are multivariate models usedto model causality relationships in data (observed variables), such that the correlationstructure of the observed variables is transferred into the correlation structure of thelatent variables. A Bayesian approach of SEM is proposed based on the analysis of thecovariance matrix of latent variables using parameter expansion to overcome identifiabilityissues. This paper is applied to the estimation of a SEM modelling a measurement processof water pollutants., Les modèles à équations structurelles à variables latentes (SEM) sont utilisés pour représenter des relations de causalité dans les données, de telle façon que la structure de corrélation des variables observées est résumée dans la structure de corrélation de variables latentes construites à cet effet. Ce papier propose une analyse bayésienne des modèles SEM reposant sur l?analyse de la matrice de covariance des variables latentes utilisant l?expansion parametrique pour surmonter les problèmes d?identifiabilité. Ce papier est appliqué àl?estimation d?un modele structurel schématisant un processus de mesure de polluants de l'eau.

A statistical method for identifying meaningful sources of heterogeneity of residual and genetic variances in mixed linear Gaussian models is presented. The method is based on a structural linear model for log variances. Inference about dispersion parameters is based on the marginal likelihood after integrating out location parameters. A likelihood ratio test using the marginal likelihood is also proposed to test for hypotheses about sources of variation involved. A Bayesian extension of the estimation procedure of the dispersion parameters is presented which consists of determining the mode of their marginal posterior distribution using log inverted chi-square or Gaussian distributions as priors. Procedures presented in the paper are illustrated with the analysis of muscle development scores at weaning of 8575 progeny of 142 sires in the Maine-Anjou breed. In this analysis, heteroskedasticity is found, both for the sire and residual components of variance., Une méthode statistique est présentée, capable d’identifier les sources significatives d’hétérogénéité de variances résiduelles et génétiques dans un modèle linéaire mixte gaussien. La méthode est fondée sur un modèle structurel de décomposition du logarithme des variances. L’inférence concernant les paramètres de dispersion est basée sur la vraisemblance marginale obtenue après intégration des paramètres de position. Un test du rapport des vraisemblances utilisant la vraisemblance marginale est aussi proposé afin de tester des hypothèses sur différentes sources de variation. Une extension bayésienne de la procédure d’estimation des paramètres de dispersion est présentée; elle consiste en la maximisation de leur distribution marginale a posteriori, pour des distributions a priori log χ 2 inverse ou gaussienne. Les procédures présentées dans ce papier sont illustrées par l’analyse de notes de pointages sur le développement musculaire au sevrage de 8 575 jeunes veaux de race Maine-Anjou, issus de 142 pères. Dans cette analyse, une hétéroscédasticité a été trouvée sur les composantes père et résiduelle de la variance.

Background: The recent advent of high-throughput SNP genotyping technologies has opened new avenues of research for population genetics. In particular, a growing interest in the identification of footprints of selection, based on genome scans for adaptive differentiation, has emerged.[br/] Methodology/Principal Findings: The purpose of this study is to develop an efficient model-based approach to perform Bayesian exploratory analyses for adaptive differentiation in very large SNP data sets. The basic idea is to start with a very simple model for neutral loci that is easy to implement under a Bayesian framework and to identify selected loci as outliers via Posterior Predictive P-values (PPP-values). Applications of this strategy are considered using two different statistical models. The first one was initially interpreted in the context of populations evolving respectively under pure genetic drift from a common ancestral population while the second one relies on populations under migration-drift equilibrium. Robustness and power of the two resulting Bayesian model-based approaches to detect SNP under selection are further evaluated through extensive simulations. An application to a cattle data set is also provided.[br/] Conclusions/Significance: The procedure described turns out to be much faster than former Bayesian approaches and also reasonably efficient especially to detect loci under positive selection.

Genes are known to interact with one another through proteins by regulating the rate at which gene transcription takes place. As such, identifying these gene-to-gene interactions is essential to improving our knowledge of how complex biological systems work. In recent years, a growing body of work has focused on methods for reverse-engineering these so-called gene regula-tory networks from time-course gene expression data. However, reconstruction of these networks is often complicated by the large number of genes potentially involved in a given network and the limited number of time points and biological replicates typically measured. Bayesian methods are particularly well-suited for dealing with problems of this nature, as they provide a systematic way to deal with different sources of variation and allow for a measure of uncertainty in parameter esti-mates through posterior distributions, rather than point estimates. Our current work examines the application of approximate Bayesian methodology for the purpose of reverse engineering regula-tory networks from time-course gene expression data. We demonstrate the advantages of our pro-posed approximate Bayesian approaches by comparing their performance on a well-characterized pathway in [i]Escherichia coli[/i].

International audience; Growth curve data consist of repeated measurements of a continuous growth process over time among a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diusion process has no explicit form, we propose to approximate it using the Euler-Maruyama scheme. Finally, we suggest to validate the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chichen growth, the modeling being improved by the SDE approach.

Dairy cattle breeds have been subjected over the last fifty years to intense artificial selection towards improvement of milk production traits. In this study, we performed a whole genome scan for differentiation using 42,486 SNPs in the three major French dairy cattle breeds (Holstein, Normande and Montbéliarde) to identify the main physiological pathways and regions which were affected by this selection. After analyzing the population structure, we estimated F(ST) within and across the three breeds for each SNP under a pure drift model. We further considered two different strategies to evaluate the effect of selection at the genome level. First, smoothing F(ST) values over each chromosome with a local variable bandwidth kernel estimator allowed identifying 13 highly significant regions subjected to strong and/or recent positive selection. Some of them contained genes within which causal variants with strong effect on milk production traits (GHR) or coloration (MC1R) have already been reported. To go further in the interpretation of the observed signatures of selection we subsequently concentrated on the annotation of differentiated genes defined according to the F(ST) value of SNPs localized close or within them. To that end we performed a comprehensive network analysis which suggested a central role of somatotropic and gonadotropic axes in the response to selection. Altogether, these observations shed light on the antagonism, at the genome level, between milk production and reproduction traits in highly producing dairy cows.

Background The recent settlement of cattle in West Africa after several waves of migration from remote centres of domestication has imposed dramatic changes in their environmental conditions, in particular through exposure to new pathogens. West African cattle populations thus represent an appealing model to unravel the genome response to adaptation to tropical conditions. The purpose of this study was to identify footprints of adaptive selection at the whole genome level in a newly collected data set comprising 36,320 SNPs genotyped in 9 West African cattle populations. Results After a detailed analysis of population structure, we performed a scan for SNP differentiation via a previously proposed Bayesian procedure including extensions to improve the detection of loci under selection. Based on these results we identified 53 genomic regions and 42 strong candidate genes. Their physiological functions were mainly related to immune response (MHC region which was found under strong balancing selection, CD79A, CXCR4, DLK1, RFX3, SEMA4A, TICAM1 and TRIM21), nervous system (NEUROD6, OLFM2, MAGI1, SEMA4A and HTR4) and skin and hair properties (EDNRB, TRSP1 and KRTAP8-1). Conclusion The main possible underlying selective pressures may be related to climatic conditions but also to the host response to pathogens such as Trypanosoma(sp). Overall, these results might open the way towards the identification of important variants involved in adaptation to tropical conditions and in particular to resistance to tropical infectious diseases.

Motivation: With the proliferation of microarray experiments and their availability in the public domain, the use of meta-analysis methods to combine results from different studies increases. In microarray experiments, where the sample size is often limited, meta-analysis offers the possibility to considerably increase the statistical power and give more accurate results. Results: A moderated effect size combination method was proposed and compared with other meta-analysis approaches. All methods were applied to real publicly available datasets on prostate cancer, and were compared in an extensive simulation study for various amounts of inter-study variability. Although the proposed moderated effect size combination improved already existing effect size approaches, the P-value combination was found to provide a better sensitivity and a better gene ranking than the other meta-analysis methods, while effect size methods were more conservative. Availability: An R package metaMA is available on the CRAN. Contact: guillemette.marot@jouy.inra.fr

International audience; Time-course microarray studies require a particular modelling of covariance matrices when measures are repeated on the same individuals. Taking into account the within-subject correlation in the test statistics for differential gene expression, however, requires a large number of parameters when a gene-specific approach is used, which often results in a lack of power due to the small number of individuals usually considered in microarray experiments. Shrinkage approaches can improve this detection power in differential gene expression studies by reducing the number of parameters, while offering a good flexibility and a small rate of false positives. A natural extension of the shrinkage approach based on a structural mixed model to variance–covariance matrices is proposed. The structural model was used in three configurations to shrink (i) the eigenvalues in an eigenvalue/eigenvector decomposition, (ii) the innovation variances in a Cholesky decomposition, (iii) both the variances and correlation parameters of a gene-by-gene covariance matrix using a Fisher transformation. The proposed methods were applied both to a publicly available data set and to simulated data. They were found to perform well, compared to previously proposed empirical Bayesian approaches, and outperformed the gene-specific or common-covariance methods in many cases.

International audience; Generalized linear mixed models (GLMM) form a very general class of random effects models for discrete and continuous responses in the exponential family. They are useful in a variety of applications. The traditional likelihood approach for GLMM usually involves high dimensional integrations which are computationally intensive. In this work, we investigate the case of binary outcomes analyzed under a two stage probit normal model with random effects. First, it is shown how ML estimates of the fixed effects and variance components can be computed using a stochastic approximation of the EM algorithm (SAEM). The SAEM algorithm can be applied directly, or in conjunction with a parameter expansion version of EM to speed up the convergence. A procedure is also proposed to obtain REML estimates of variance components and REML-based estimates of fixed effects. Finally an application to a real data set involving a clinical trial is presented, in which these techniques are compared to other procedures (penalized quasi-likelihood, maximum likelihood, Bayesian inference) already available in classical softwares (SAS Glimmix, SAS Nlmixed, WinBUGS), as well as to a Monte Carlo EM (MCEM) algorithm.

A large variety of methods has been proposed in the literature for microarray data analysis. The aim of this paper was to present techniques used by the EADGENE (European Animal Disease Genomics Network of Excellence) WP1.4 participants for data quality control, normalisation and statistical methods for the detection of differentially expressed genes in order to provide some more general data analysis guidelines. All the workshop participants were given a real data set obtained in an EADGENE funded microarray study looking at the gene expression changes following artificial infection with two different mastitis causing bacteria: Escherichia coli and Staphylococcus aureus. It was reassuring to see that most of the teams found the same main biological results. In fact, most of the differentially expressed genes were found for infection by E. coli between uninfected and 24 h challenged udder quarters. Very little transcriptional variation was observed for the bacteria S. aureus. Lists of differentially expressed genes found by the different research teams were, however, quite dependent on the method used, especially concerning the data quality control step. These analyses also emphasised a biological problem of cross-talk between infected and uninfected quarters which will have to be dealt with for further microarray studies.

DNA markers are commonly used for large-scale evaluation of genetic diversity in farm animals, as a component of the management of animal genetic resources. AFLP markers are useful for such studies as they can be generated relatively simply; however, challenges in analysis arise from their dominant scoring and the low level of polymorphism of some markers. This paper describes the results obtained with a set of AFLP markers in a study of 59 pig breeds. AFLP fingerprints were generated using four primer combinations (PC), yielding a total of 148 marker loci, and average harmonic mean of breed sample size was 37.3. The average proportion of monomorphic populations was 63% (range across loci: 3%-98%). The moment-based method of Hill and Weir (2004, Mol Ecol 13:895-908) was applied to estimate gene frequencies, gene diversity (F(ST)), and Reynolds genetic distances. A highly significant average F(ST) of 0.11 was estimated, together with highly significant PC effects on gene diversity. The variance of F(ST) across loci also significantly exceeded the variance expected under the hypothesis of AFLP neutrality, strongly suggesting the sensitivity of AFLP to selection or other forces. Moment estimates were compared to estimates derived from the square root estimation of gene frequency, as currently applied for dominant markers, and the biases incurred in the latter method were evaluated. The paper discusses the hypotheses underlying the moment estimations and various issues relating to the biallelic, dominant, and lowly polymorphic nature of this set of AFLP markers and to their use as compared to microsatellites for measuring genetic diversity.

Genetic diversity within and between breeds (and lines) of pigs was investigated. The sample comprised 68 European domestic breeds (and lines), including 29 local breeds, 18 varieties of major international breeds, namely Duroc, Hampshire, Landrace, Large White and Pietrain, and 21 commercial lines either purebred or synthetic, to which the Chinese Meishan and a sample of European wild pig were added. On average 46 animals per breed were sampled (range 12–68). The genetic markers were microsatellites (50 loci) and AFLP (amplified fragment length polymorphism, 148 loci). The analysis of diversity showed that the local breeds accounted for 56% of the total European between-breed microsatellite diversity, and slightly less for AFLP, followed by commercial lines and international breeds. Conversely, the group of international breeds contributed most to within-breed diversity, followed by commercial lines and local breeds. Individual breed contributions to the overall European between- and within-breed diversity were estimated. The range in between-breed diversity contributions among the 68 breeds was 0.04–3.94% for microsatellites and 0.24–2.94% for AFLP. The within-breed diversity contributions varied very little for both types of markers, but microsatellite contributions were negatively correlated with the between-breed contributions, so care is needed in balancing the two types of contribution when making conservation decisions. By taking into account the risks of extinction of the 29 local breeds, a cryopreservation potential (priority) was estimated for each of them.

International audience; The aim of this study was to analyze somatic cell counts which is an indirect criterion. to assess susceptibility to mastitis. Data analyzed were weekly records (6448 somatic cell scores) out of 159 primiparous Holsteins and Holsteins x Normande cows raised at the INRA" Le Pin au Haras" experimental farm, France. Given the longitudinal structure of this data set, the analysis consists of modeling both the mean and the individual profiles. This was achieved via the use of mixed models including fixed effects for the average profiles and random effects for the adjusted individual profiles. As far as fixed effects are concerned, the main issue is to fit a time trend to the average profiles. For this, we employed the technique of fractional polynomials described in Royston and Altman (Appl. Stat. 43 (1994) 429-467) under several variance-covariance structures. The best second degree polynomial involved an intercept plus the time at the power (-1/3) (i.e., t-1/3) plus the latter times the logarithm of the time (i.e., t-1/3 × log (t)). Regarding random effects, model comparisons involved random coefficient models, exponential stationary stochastic processes and heterogeneous variances. The models that simultaneously included all these three structures turned out to be the best. For instance, random coefficient models did not fit the variance function well, even when the degree of the polynomial was high. This phenomenon partly justified the introduction of heteroskedastic models.; Analyse statistique des scores de cellules somatiques par la méthodologie des modèles mixtes appliquée à des données longitudinales. L'objectif de cette étude est d'analyser les comptages de cellules somatiques, qui constituent un critère indirect d'appréciation de la sensibilité aux mammites. Les données analysées étaient relatives à 6448 contrôles hebdomadaires de cellules somatiques du lait effectués sur 159 génisses de race Holstein et croisées Holstein x Normande, entretenues au domaine expérimental INRA-Le pin au Haras. Eu égard à la structure longitudinale des données, l'analyse a consisté à modéliser les profils moyens et individuels de réponse. À ce propos, ont été mis en œuvre des modèles mixtes dont les effets fixes décrivent les profils moyens et les effets aléatoires les profils individuels. Concernant les effets fixes, on a utilisé la technique des polynômes fractionnaires, décrite par Royston et Altman [29] assortie de différentes structures de variance-covariance. Le meilleur ajustement est fourni par un polynôme de second degré comportant un terme constant, le temps à la puissance moins un tiers (i.e., t-1/3) et ce même terme multiplié par le logarithme du temps (i.e., t-1/3 × log(t)). Concernant la partie aléatoire du modèle, les modèles mis en comparaison faisaient appel à une régression sur le temps à coefficients aléatoires, à des processus stochastiques stationnaires et à des variances hétérogènes. Ce sont les modèles qui incluaient simultanément ces trois structures qui se sont avérés les meilleurs. Ainsi, les modèles polynomiaux à degré élevé ne rendaient pas compte de l'évolution réelle de la variance avec le temps d'où le recours à des modèles hétéroscédastiques.

International audience; A heteroskedastic random coefficients model was described for analyzing weight performances between the 100th and the 650th days of age of Maine-Anjou beef cattle. This model contained both fixed effects, random linear regression and heterogeneous variance components. The objective of this study was to analyze the difference of growth curves between animals born as twin and single bull calves. The method was based on log-linear models for residual and individual variances expressed as functions of explanatory variables. An expectation-maximization (EM) algorithm was proposed for calculating restricted maximum likelihood (REML) estimates of the residual and individual components of variances and covariances. Likelihood ratio tests were used to assess hypotheses about parameters of this model. Growth of Maine-Anjou cattle was described by a third order regression on age for a mean growth curve, two correlated random effects for the individual variability and independent errors. Three sources of heterogeneity of residual variances were detected. The difference of weight performance between bulls born as single and twin bull calves was estimated to be equal to about 15 kg for the growth period considered.

International audience; This paper presents procedures for implementing the PX-EM algorithm of Liu, Rubin and Wu to compute REML estimates of variance covariance components in Henderson's linear mixed models. The class of models considered encompasses several correlated random factors having the same vector length e.g., as in random regression models for longitudinal data analysis and in sire-maternal grandsire models for genetic evaluation. Numerical examples are presented to illustrate the procedures. Much better results in terms of convergence characteristics (number of iterations and time required for convergence) are obtained for PX-EM relative to the basic EM algorithm in the random regression.; L'algorithme PX-EM dans le contexte de la méthodologie du modèle mixte d'Henderson. Cet article présente des procédés permettant de mettre en œuvre l'algorithme PX-EM de Liu, Rubin et Wu à des modèles linéaires mixtes d'Henderson. La classe de modèles considérée concerne plusieurs facteurs aléatoires corrélés ayant la même dimension vectorielle comme c'est le cas avec les modèles de régression aléatoire dans l'analyse des données longitudinales ou avec les modèles père-grand-père maternel en évaluation génétique. Des exemples numériques sont présentés pour illustrer ces techniques. L'algorithme PX-EM présente de nettement meilleurs résultats en terme de caractéristiques de convergence (nombre d'itérations et temps de calcul) que l'EM de base sur les exemples ayant trait à des modèles de régression aléatoire.

International audience; This paper presents procedures for implementing the EM algorithm to compute REML estimates of variance covariance components in Gaussian mixed models for longitudinal data analysis. The class of models considered includes random coefficient factors, stationary time processes and measurement errors. The EM algorithm allows separation of the computations pertaining to parameters involved in the random coefficient factors from those pertaining to the time processes and errors. The procedures are illustrated with Pothoff and Roy's data example on growth measurements taken on 11 girls and 16 boys at four ages. Several variants and extensions are discussed.; Estimation EM-REML des paramètres de covariance en modèles mixtes gaussiens en vue de l'analyse de données longitudinales. Cet article présente des procédés permettant de mettre en œuvre l'algorithme EM en vue du calcul d'estimations REML des composantes de variance covariance en modèles mixtes gaussiens d'analyse de données longitudinales. La classe de modèles considérée concerne les coefficients aléatoires, les processus temporels stationnaires et les erreurs de mesure. L'algorithme EM permet de dissocier formellement les calculs relatifs aux paramètres des coefficients aléatoires de ceux impliqués dans les processus et la résiduelle. Ces méthodes sont illustrées par un exemple provenant de Pothoff et Roy sur des mesures de croissance prises sur 11 filles et 16 garçons à quatre âges différents. On discute enfin plusieurs variantes et extensions de cette méthode.

International audience; This paper presents simulation formulae of two-sided truncated normal random variables using a completion distribution and its two corresponding conditionals generated via a Gibbs sampler. This procedure extends formulae given by Robert and Casella for the one-sided case.; Un simulateur de "completion "pour la loi normale tronquée aux deux extrémités. Cet article présente des formules de simulation de variables aléatoires normales réduites tronquées aux deux extrémités à partir d'une distribution de "completion "et ses deux lois conditionnelles générées par un algorithme de Gibbs. Ces formules généralisent celles de Robert et Casella établies pour le cas d'une seule troncature.

La réussite de la reproduction est primordiale pour la rentabilité de l’élevage, elle constitue un préalable indispensable à toute production. L’aptitude à la reproduction d’un animal au cours d’une carrière dépend des caractères de précocité sexuelle, de fertilité et de prolificité, cette dernière pouvant être décomposée en termes de taux d’ovulation et de mortalité embryonnaire. Ces différents aspects font l’objet de recherches en génétique animale et sont intégrés à des degrés divers dans les schémas de sélection, en race pure ou en croisement. La précocité sexuelle et la prolificité doivent être améliorées quand elles sont limitantes, mais certains seuils ne doivent pas être dépassés sous peine de voir se réduire le gain économique espéré. S’intéresser aux composantes de la prolificité devrait en permettre un meilleur contrôle. Une fertilité maximale est toujours recherchée mais elle revêt différentes facettes qui en compliquent son amélioration. Elle est à la fois dépendante du sexe (la fertilité femelle reste souvent la plus limitante) de la technique de reproduction employée (naturelle ou artificielle) et du moment choisi pour la mise à la reproduction (intervalle mise bas - reproduction, saison). Les recherches sur la génétique des caractères de reproduction doivent être poursuivies tout en intégrant les résultats de cartographie génétique., Reproduction is of prime importance for the profitability of a herd, it is a prerequisite of every animal production. The breeding ability during the productive life of an animal depends on sexual maturity, fertility and prolificacy the components of which are the ovulation and the embryonic mortality rates. Those traits are studied in animal breeding and are variously taken into account in the purebred and crossbred selection schemes. Sexual maturity and prolificacy have to be improved when they are limiting factors, but for economic reasons they have to stay beneath optimal thresholds. Studying the components of prolificacy would allow a better control. Maximal fertility is always expected, but its improvement is made difficult because it depends on numerous factors : sex (female fertility is often the more limiting), reproduction technique used (natural or artificial) and the choice of the breeding period (time between littering and breeding, season). Research on genetics of the breeding traits has to be pursued, while including the recent advances in genetic cartography.

This paper reviews basic theory and features of the multiplicative model of gene action. A formal decomposition of the mean and of the genotypic variance is presented. Connections between the statistical parameters of this model and those of the factorial decomposition into additive, dominance and epistatic effects are also emphasized. General formulae for the genotypic covariance among inbred relatives are given in the case of linkage equilibrium. It is shown that neglecting the epistatic components of variation makes the multiplicative model a pseudo-additive one, since this approximation does not break the strong dependency between mean and variance effects. Similarities and differences between the classical polygenic ’additive-dominance’ and the multiplicative gene action approaches are outlined and discussed. Numerical examples for the biallelic case are produced to illustrate that comparison., Cet article présente la théorie et les principales caractéristiques du modèle multiplicatif d’action des gènes. Une décomposition formelle de la moyenne et de la variance génotypique permet d’établir les relations entre les paramètres statistiques de ce modèle et ceux issus de la décomposition factorielle de l’effet des gènes en effets additifs, de dominance et d’épistasie. Une formule générale de la covariance entre apparentés dans une population consanguine en équilibre de liaison est proposée. On montre que les composantes épistatiques de la variabilité génétique peuvent être négligées ; le modèle multiplicatif devient alors un modèle pseudo-additif, l’approximation ne supprimant pas la forte liaison entre moyenne et variance. Les similitudes et les différences entre le modèle polygénique « additif-dominance » classique et le modèle multiplicatif d’action des gènes sont discutées et illustrées par des exemples dans le cas biallélique.

Data from the Spanish Holstein-Friesian milk recording system were used to investigate sources of heterogeneity of phenotypic herd-period variances for milk yield and to evaluate the impact of data standardization in the genetic evaluation. Factors such as region, year of calving and management level of the herd were found to be associated to heterogeneity of herd-period variances in a correspondence analysis. Applying clustering techniques, initial 36781 herd-periods were stratified into 38 subclasses defined by combinations of levels of those factors. A statistical analysis that used a structural model on log-variances showed that a significant amount of the observed variability among herd-period variances could be explained by the defined subclasses. Variability between subclasses was 62.8% of the total between and within true variability. The herd-period stratification was used to obtain estimates of herd-period milk yield variances in a Bayesian analysis. These estimates were then applied to produce standardized milk yield records. Genetic evaluations were run on standardized and unstandardized data. Genetic evaluation of cows was noticeably affected by standardization. Twenty percent of cows in the 1000 best cows list were excluded from the list when standardization was performed. Estimated correlation coefficient between the two evaluations for these cows was 0.82. Bulls evaluation was affected to a less extent. Only 9% of the top 100 bulls were out of the 100 top list after standardization. A slight tendency to use North American bulls from the most variable herds was observed. However, this fact did not imply significant changes in ranking of North American bulls compared to domestic bulls when standardization was performed.

The objectives of this work were to examine the usefulness of measuring ovulation rate (OR) in order to improve genetic progress of litter size (LS) in sheep and to study different selection criteria combining OR and prenatal survival (ES) performance. Responses to selection for 5 generations within a population of 20 male and 600 female parents were compared using Monte-Carlo simulation techniques with 50 replicates per selection method. Two breeds with low (Merino) and medium (Lacaune) prolificacy were considered. Records were generated according to a bivariate threshold model for OR and ES. Heritabilities of OR and ES in the underlying scale were assumed constant over breeds and equal to 0.35 and 0.11, respectively, with a genetic correlation of -0.40 between these traits. Four methods of genetic evaluation were compared: univariate best linear unbiased prediction (BLUP) using LS records only (b-LS); univariate BLUP on OR records (b-OR); bivariate BLUP using OR and LS records (b-ORLS); and a maximum a posteriori predictor of a generalised linear model whereby OR was analysed as a continuous trait and ES as a binary threshold trait (t-ORES). Response in LS was very similar with b-LS, b-ORLS and t-ORES, whereas it was significantly lower with b-OR. Response in OR was maximum with b-OR and minimum with b-LS. In contrast, response in ES was maximum with b-LS. This study raised the question as to why selection based on indices combining information from both OR and ES did not perform better than selection using LS only., Cet article discute l’intérêt du taux d’ovulation (OR) pour accroître le progrès génétique sur la taille de portée (LS) et étudie à cet effet divers critères de sélection combinant OR et le taux de survie embryonnaire (ES). On a examiné par simulation les réponses à la sélection en 5 générations dans une population de 20 et 600 reproducteurs mâles et femelles avec 50 réplications par méthode. On a considéré 2 races, de prolificité faible (Mérinos) et moyenne (Lacaune). Les performances ont été générées à partir d’un modèle bicaractère à seuils. Les héritabilitiés d’OR et ES ont été supposées constantes sur l’échelle sous-jacente dans les 2 races et prises égales respectivement à 0,35 et 0,11 avec une corrélation génétique entre ces 2 caractères de - 0,40. Quatre méthodes d’évaluation génétiques ont été comparées : i) Blup unicaractère basé sur LS (b-LS) ; ii) Blup unicaractère basé sur OR (b-OR) ; iii) Blup bicaractère basé sur OR et LS (b-ORLS) ; iv) Prédicteur du maximum a posteriori bicaractère d’un modèle linéaire généralisé où OR est traité comme un caractère continu et ES comme un caractère à seuils (t-ORES). Les réponses observées étaient très voisines avec b-LS, b-ORLS et t-ORES, alors que b-OR donne une réponse significativement inférieure. La réponse sur OR était maximum avec b-OR et minimum avec b-LS, tandis que la réponse sur ES était maximum avec b-LS. Cette étude pose la question de savoir pourquoi la sélection basée sur des indices combinant OR et ES ne donne pas de résultats significativement supérieurs à la sélection sur LS.

This paper reviews some problems encountered in estimating heterogeneous variances in Gaussian linear mixed models. The one-way and multiple classification cases are considered. EM-REML algorithms and Bayesian procedures are derived. A structural mixed linear model on log-variance components is also presented, which allows identification of meaningful sources of variation of heterogeneous residual and genetic components of variance and assessment of their magnitude and mode of action., Cet article fait le point sur un certain nombre de problèmes qui surviennent lors de l’estimation de variances hétérogènes dans des modèles linéaires mixtes gaussiens. On considère le cas d’un ou plusieurs facteurs d’hétéroscédasticité. On développe des algorithmes EM-REML et bayésiens. On propose également un modèle linéaire mixte structurel des logarithmes des variances qui permet de mettre en évidence des sources significatives de variation des variances résiduelles et génétiques et d’appréhender leur importance et leur mode d’action.

Breeding values and genetic parameters for dystocia were estimated in Normande and Holstein breeds. Dystocia scores were related to an underlying continuous variable via a threshold model. The underlying linear model included the effects of calving season, sex of calf by parity of dam, sire of calf, grandsire of calf, dam within maternal grandsire and herd-year effects. Typical results were found for the environmental effects, with a strong influence of dam parity on dystocia, a strong influence of sex of calf, and a small effect of calving season. Herd-year variances were 32 and 40% of the residual variance in the Normande and Holstein breeds, respectively. Heritabilities for the Normande (Holstein) breed were .08 (.07) for direct effects and .11 (.07) for maternal effects. Correlations between sire and grandsire effects were .51 and .36 for the Normande and Holstein breeds, respectively. The corresponding correlations between direct and maternal effects were .15 and -.09. The results of this study show that a complete model for dystocia including the threshold concept and maternal effects can be applied for routine evaluation of dairy AI bulls. Maternal effects are important, and they should be considered in dystocia analysis, especially if nonrandom mating is present. Selection for reducing dystocia in calf and cow effects are not antagonistic.

Summary - Genetic parameters and bull transmitting abilities were estimated for twinning in the Maine-Anjou breed. Twin calving performance was analyzed as a threshold binary trait assuming direct and fetal effects and polygenic inheritance. The statistical model included the effects of parity, year-season, herd, sire of the fetus, sire of the cow and cow within sire. Heritabilities were 0.13 and 0.02 for direct and fetal effects, respectively, with a correlation between both effects of 0.36. Transmitting abilities of bulls were expressed on the underlying and observed scales; the bull ranked first had an observed twinning rate of 13.7% among its 259 female progeny, corresponding to an estimated breeding value of 2.6 units of underlying standard deviation or 13.9% as the probability for a future daughter to have a twin calving in her second parity. It is concluded that there is considerable place for twinning selection among Maine-Anjou bulls. cattle / twinning / genetic parameter / sire evaluation / threshold model

A statistical method is presented for identifying sources of heterogeneity of residual variance in mixed linear models. The method is based on a log-linear model for the residual variances from which parameters can be estimated and hypotheses tested using the marginal likelihood function. It can be viewed as a generalization of Bartlett's test. Computing procedures are presented and an illustration is given. The method can be used with experimental or field data, e.g., to detect possible preferential treatment of daughters of sires or to investigate the possible segregation of major genes.

This article gives a formal proof of a formula for the precision of estimated genetic distances proposed by Barker et al. which can be used in designing experimental sampling programmes. The derivation is given in the general multi- allelic case using the Sanghvi distance. Two sources of sampling are considered, i.e. i) among individuals (or gametes) within locus and ii) among loci within populations. Distribution assumptions about gene frequencies are discussed, especially the normal used in Barker et al. versus the Dirichlet via simulation. © Inra/Elsevier, Paris

A method of genetic evaluation for multiple binary responses is presented. An underlying multivariate normal distribution is rendered discrete, in m dimensions, via a set of m fixed thresholds. There are 2m categories of response and the probability of response in a given category is modeled with an m-dimensional multivariate normal integral. The argument of this integral follows a multivariate mixed linear model. The randomness of some elements in the model is taken into account using a Bayesian argument. Assuming that the variance-covariance structure is known, the mode of the joint posterior distribution of the fixed and random effects is taken as a point estimator. The problem is non-linear and iteration is required. The resulting equations indicate that the approach falls in the class of generalized linear models, with additional generalization stemming from the accommodation of random effects. A remarkable similarity with multiple trait evaluation via mixed linear models is observed. Important numerical issues arise in the implementation of the procedure and these are discussed in detail. An application of the method to data on calving preparation, calving difficulties and calf viability is presented., Cet article présente une méthode d’évaluation génétique multidimensionnelle de caractères binaires. La distribution multinormale sous-jacente est discrétisée en m dimensions par le biais de m seuils. On considère les 2m catégories de réponse et la probabilité de réponse dans une catégorie est modélisée par une intégrale d’une densité multinormale de dimension m. L’argument de cette intégrale est décomposé suivant un modèle mixte multidimensionnel. Le caractère aléatoire de certains éléments du modèle est pris en compte par une approche bayésienne. Le mode de la distribution conjointe a posteriori est choisi comme estimateur de position des effets fixes et aléatoires sachant la structure de variances-covariances connue. Le système obtenu est non linéaire et résolu par itérations. La forme des équations montre que cette approche fait partie de la classe des modèles linéraires généralisés avec l’extension supplémentaire au traitement de facteurs aléatoires. Le système présente en outre une analogie remarquable avec celui de l’évaluation multi-caractère par modèle mixte linéraire. La résolution soulève l’importantes difficultés numériques. La méthode est illustrée par une application numérique à des données de préparation au vêlage, de difficulté de celui-ci et de viabilité des veaux.

A sire evaluation procedure is proposed for situations in which there is uncertainty with respect to the assignment of progeny to sires. The method requires the specification of the prior probabilities Pij that progeny i is out of sire j. Inferences about location parameters (« fixed » environmental and group effects and transmitting abilities of sires) are based on Bayesian statistical procedures. Modal values of the posterior distribution of these parameters are taken as point estimators. Finding this mode entails solving a nonlinear system of equations and several algorithms are suggested. The methodology is described for univariate evaluations obtained from normal or binary traits. Estimation of unknown variances is also addressed. A small numerical example is presented to illustrate the procedure. Potential applications to livestock breeding are discussed., Une méthode d’évaluation des pères est proposée en situation d’incertitude vis-à-vis de l’assignation des descendants à leurs pères. La méthode requiert la spécification des probabilités a priori Pij que le descendant i provienne du père j. L’inférence des paramètres de position (effets « groupe » et de milieu, considérés comme fixes et valeurs génétiques transmises des pères) est basée sur des procédures statistiques bayésiennes. Les valeurs modales de la distribution a posteriori de ces paramètres ont été prises comme estimateurs ponctuels. La recherche du mode nécessite la résolution d’un système d’équations non linéaire pour lequel plusieurs algorithmes sont proposés. La méthodologie est développée dans le cadre univariate pour des caractères normaux et binaires. Le cas de variances inconnues est également abordé. Un petit exemple numérique est présenté à titre d’illustration. Enfin, les applications possibles aux espèces domestiques sont discutées.

The threshold model equations are shown to correspond to a pseudo-linear model. A modification of Jacobi iteration "on data" is presented that does not require setting up the system of equations explicitly. Two iterative threshold model programs were developed; these involve absorption-inversion and iteration "on data". The programs were applied to a set of simulated data. Normal termination did not occur when the data contained fixed effects with observations only in an extreme category, and methods for solving such problems are described. The absorption-inversion program behaved erratically when the model was not of full rank; numerical accuracy problems precluded detection of dependencies between thresholds and other effects. The iteration "on data" version was fastest when only a single round of modified Jacobi was performed for each Newton-Raphson round. A round of iteration for the absorption-inversion program consumed 837, 11.8, and 1.24s of central processing unit time on a Personal Computer AT, an IBM 3081 mainframe, and a CRAY X-MP/48 supercomputer, respectively. Corresponding times for the iteration "on data" program were 55.2, 2.6, and .57s. The cost of a threshold model analysis would be three to five times larger than that of a linear model.

Cette note présente une méthode de prise en compte de la consanguinité dans la simulation de valeurs génétiques à partir de valeurs parentales. Par rapport aux formules négligeant la consanguinité adoptées par WILLHAM & THOMSON (1970) dans le cow-game et sous l’hypothèse d’un déterminisme génétique additif, cette méthode revient à changer l’écart type résiduel (formule,voir document attaché ) étant les coefficients de consanguinité du père et de la mère respectivement et σg. l’écart-type génétique. La même modification s’applique à la simulation d’un caractère corrélé., This paper presents a method for taking into account inbreeding when genetic values of individuals are simulated on the basis of the parental values. Compared to formulae used by WILLHAM and THOMSON (1970) in the cow game and, under the assumption of an additive genetic determinism, this method leads to change the residual standard deviation being the inbreeding coefficients of the sire and dam respectively and σg the genetic standard deviation. The same modification has to be applied when simulating a correlated trait.

A procedure of sire evaluation for multiple binary responses when information is missing for some traits is described. The genetic model assumes a conceptual underlying multivariate normal distribution rendered discrete by abrupt thresholds. Statistical inferences are made from a posterior distribution consisting of several conditionally independent likelihood functions and a multivariate normal prior distribution. Each likelihood function corresponds to a particular class of information available. Point estimators and predictors of fixed and random effects are the values that maximize the posterior distribution conditionally on heritabilities and genetic correlations. The procedure involves nonlinear maximization. An example involving joint selection for calving ease and skeletal development illustrates the principles. Application of the methodology as a potential means of removing bias due to selection for categorical traits is discussed.

International audience

Cet article généralise la méthode donnée par OLLIVIER (1974) pour estimer la régression parent-descendant au cas où plusieurs mesures du même caractère, en nombre variable, sont prises par parent. La méthode proposée repose sur un changement de variable parentale. Une application numérique est présentée.

A method of prediction of genetic merit from jointly distributed quanta] and quantitative responses is described. The probability of response in one of two mutually exclusive and exhaustive categories is modeled as a non-linear function of classification and « risk » variables. Inferences are made from the mode of a posterior distribution resulting from the combination of a multivariate normal density, a priori, and a product binomial likelihood function. Parameter estimates are obtained with the Newton-Raphson algorithm, which yields a system similar to the mixed model equations. « Nested » Gauss-Seidel and conjugate gradient procedures are suggested to proceed from one iterate to the next in large problems. A possible method for estimating multivariate variance (covariance) components involving, jointly, the categorical and quantitative variates is presented. The method was applied to prediction of calving difficulty as a binary variable with birth weight and pelvic opening as « risk » variables in a Blonde d’Aquitaine population., Cet article présente une méthode de prédiction de la valeur génétique à partir d’observations quantitatives et qualitatives. La probabilité de réponse selon l’une des deux modalités exclusives et exhaustives envisagées est exprimée comme une fonction non linéaire d’effets de facteurs d’incidence et de variables de risque. L’inférence statistique repose sur le mode de la distribution a posteriori qui combine une densité multinormale a priori et une fonction de vraisemblance produit de binomiales. Les estimations sont calculées à partir de l’algorithme de Newton-Raphson qui conduit à un système d’équations similaires à celles du modèle mixte. Pour les gros fichiers, on suggère des méthodes itératives de résolution telles que celles de Gauss-Seidel et du gradient conjugué. On propose également une méthode d’estimation des composantes de variances et covariances relatives aux variables discrètes et continues. Enfin, la méthodologie présentée est illustrée par une application numérique qui a trait à la prédiction des difficultés de vêlage en race bovine Blonde d’Aquitaine utilisant d’une part, l’appréciation tout-ou-rien du caractère, et d’autre part, le poids à la naissance du veau et l’ouverture pelvienne de la mère comme des variables de risque.

Chapitre 12; Structural Equation Models with latent variables (SEM) are hypothetical constructsused to represent causality relationships in data, where the observedcorrelation structure is transferred into the correlation structure of latent variables.In this paper a Bayesian analysis of SEM is proposed using parameterexpansion to overcome identi fiability issues. An original use of posterior drawsfrom latent variables is proposed to model expert knowledge in uncertaintyanalysis.

International audience; A set of eleven pig breeds originating from six European countries, and including a small sample of wild pigs, was chosen for this study of genetic diversity. Diversity was evaluated on the basis of 18 microsatellite markers typed over a total of 483 DNA samples collected. Average breed heterozygosity varied from 0.35 to 0.60. Genotypic frequencies generally agreed with Hardy-Weinberg expectations, apart from the German Landrace and Schwäbisch-Hällisches breeds, which showed significantly reduced heterozygosity. Breed differentiation was significant as shown by the high among-breed fixation index (overall $F_{\rm ST} = 0.27$), and confirmed by the clustering based on the genetic distances between individuals, which grouped essentially all individuals in 11 clusters corresponding to the 11 breeds. The genetic distances between breeds were first used to construct phylogenetic trees. The trees indicated that a genetic drift model might explain the divergence of the two German breeds, but no reliable phylogeny could be inferred among the remaining breeds. The same distances were also used to measure the global diversity of the set of breeds considered, and to evaluate the marginal loss of diversity attached to each breed. In that respect, the French Basque breed appeared to be the most "unique" in the set considered. This study, which remains to be extended to a larger set of European breeds, indicates that using genetic distances between breeds of farm animals in a classical taxonomic approach may not give clear resolution, but points to their usefulness in a prospective evaluation of diversity.; Diversité génétique de onze races porcines européennes. Un ensemble de onze races porcines en provenance de six pays européens, et incluant un petit échantillon de sangliers, a été choisi pour une étude de diversité génétique. Cette diversité a été évaluée sur la base de 18 marqueurs microsatellites typés sur un total de 483 échantillons d'ADN. Les races étudiées manifestent un taux d'hétérozygotie allant de 0,35 à 0,60. Les locus sont en équililibre de Hardy-Weinberg à l'exception du cas des races allemandes Landrace et Schwäbisch-Hällisches, qui manifestent un déficit d'hétérozygotes. L'indice de différenciation entre races est élevé ($F_{\rm ST}$ global de 0,27) et les distances génétiques entre individus permettent de les regrouper pratiquement en 11 ensembles distincts, correspondant aux 11 races considérées. Les distances génétiques entre races ont d'abord été utilisées pour construire des arbres phylogénétiques. Ces arbres suggèrent qu'un modèle de dérive génétique pourrait expliquer la divergence des deux races allemandes, mais aucune phylogénie fiable n'a pu être établie entre les races restantes. Les mêmes distances ont ensuite été utilisées pour mesurer la diversité génétique globale de l'ensemble et évaluer la perte marginale de diversité associée à chacune des races étudiées. De ce point de vue, la race française Basque apparaît comme la plus originale dans l'ensemble considéré. Cette étude, qui reste à étendre à un plus grand nombre de races européennes, indique que l'utilisation des distances entre races animales domestiques dans une approche taxonomique classique risque d'avoir un faible pouvoir de résolution, mais elle souligne l'intérêt de les utiliser plutôt pour des évaluations prospectives de diversité.

The joint distribution of breeding values and of records usually depends on unknown parameters such as means, variances and covariances in the case of the multivariate normal distribution. If the objective of the analysis is to make selection decisions, these parameters should be considered as « nuisances ». If the values of the parameters are unknown, the state of uncertainty can be represented by a prior probability distribution. This is then combined with the information contributed by the data to form a posterior distribution from which the needed predictors are calculated after integrating out the « nuisances ». Prediction under alternative states of knowledge is discussed in this paper and the corresponding solutions presented. It is shown that when the dispersion structure is unknown, modal estimators of variance parameters should be considered. Because a Bayesian framework is adopted, the estimates so obtained are necessarily non-negative. If prior knowledge about means and variances is completely vague and the distribution is multivariate normal, the « optimal predictors in the sense of maximizing the expected merit of the selected candidates can be approximated by using the « mixed model equations » with the unknown variances replaced by restricted maximum likelihood estimates. This leads to empirical Bayes predictors of breeding values., La distribution conjointe des valeurs génétiques et des performances dépend habituellement de paramètres inconnus tels que les espérances, les variances et covariances dans le cas de la distribution multinormale. Quand l’analyse statistique vise à des décisions de sélection, ces paramètres devraient être considérés comme des paramètres « parasites ». L’état d’incertitude sur les paramètres peut être représenté par une distribution a priori. Celle-ci, combinée à l’information procurée par les données, permet d’aboutir à une distribution a posteriori des paramètres d’intérêt après intégration des paramètres « parasites ». Cet article envisage la prédiction des valeurs génétiques sous différentes hypothèses de connaissance des paramètres et présente les solutions correspondantes. Lorsque les paramètres de dispersion sont inconnus, des estimateurs de la variance basés sur le mode a posteriori sont suggérés. Du fait du mode d’inférence, de type bayésien, ces estimateurs s’avèrent nécessairement non négatifs. Avec une distribution a priori des moyennes et des variances uniforme et sous l’hypothèse de normalité, les prédicteurs optimum (au sens de la maximisation du mérite espéré des individus sélectionnés) sont ceux obtenus à partir des équations du modèle mixte dans lesquelles les variances sont remplacées par leurs estimées du maximum de vraisemblance restreint. Cela conduit à des prédicteurs des valeurs génétiques de type bayésien empirique.