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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)−...

Gene diversity and allelic diversity are both recognized as useful criteria for establishing priorities of conservation. Similarly to gene diversity partitioning, based on variation of gene frequencies both within and between subpopulations, allelic diversity can be partitioned using variation in allele numbers. Allelic diversity components have to meet a number of desirable properties, among which orthogonality (i.e. independence of the within-subpopulation and between-subpopulation components), and concavity (i.e. no within-subpopulation diversity larger than total diversity). In this note it is shown how the partitioning recommended in species diversity studies should be transposed to allelic diversity. Attention is drawn to some methods of allelic diversity partitioning currently proposed, essentially the widely used rarefaction method of Petit et al. (Conserv Biol 12:844–855, 1998) and a more recently proposed method by Caballero and Rodriguez-Ramilo (Conserv Genet 11:2219–2229, 2010). Their contrasting properties and the potentially contrasting conclusions to be expected from their application in conservation are emphasized. Some simple examples are used to show how an improper measure of allelic diversity may lead to misleading conclusions when defining priorities of conservation.

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.

We investigated the genetic variability within and between cattle breeds. The polymorphisms of 17 microsatellites were studied in 220 unrelated animals belonging to four Italian beef cattle breeds (Chianina, Marchigiana, Romagnola, and Piemontese). Variations of allelic frequencies were examined to characterize the breeds and their relationships. Wahlund coefficients, Polymorphism Information Content values, and Haldane exact test for Hardy-Weinberg proportions were calculated. The results show that the Hardy-Weinberg equilibrium is not always maintained. Moreover, in addition to the classical genetic distances, a new method, based on the consideration of a multilocus genotype of each animal, was set up to measure the genetic similarity between animals or within groups of animals. All the results showed that, whereas Chianina occupies an intermediate position and Piemontese is the most distinct of all four breeds, Marchigiana and Romagnola display the strongest similarity. The new method also provides evidence that average similarities are always higher within breeds than between breeds. By comparing pairwise the multilocus genotypes, it was also possible to discriminate the individuals with higher or lower genetic similarities so that each breed could be subdivided into two groups of animals in relation to their similarity to the average breed multilocus genotype. High similarities between breeds were detected, somewhat surprisingly, when the most homogeneous groups of each breed were compared. The microsatellite multilocus genotype is particularly efficient in evaluating the between- and within-breeds genetic similarities and for subgrouping genetically more homogeneous animals.

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.

A bivariate analysis is described for one continuous and one discrete trait to estimate sire effects in a progeny test. Unequal design matrices, in terms of both missing data as well as different fixed effects are allowed for. Three independent sub data sets are formed, for which the corresponding log-likelihoods are expressed. The parameters are estimated as their maxima a posteriori by maximizing the log-posterior density using a Newton Raphson algorithm; equations are given in a mixed model form. A simulation study is presented showing the benefits of this bivariate analysis for the evaluation of beef bulls for calving difficulty, using birth weight as a correlated trait. Standard errors obtained from the information matrix were close to empirical standard errors, and it was shown that bivariate analysis with unequal design matrices can correct for a bias which may occur when birth weight records are not missing at random.

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

Genetic parameters for dystocia in the Main-Anjou breed were estimated. Data consisted of 28 178 birth records collected between 1978 and 1989 in 995 herds, with 161, 71 and 12 415 sires, maternal grandsires and dams, respectively, represented. Original scores (1 through 5) were collapsed in order to set two dystocia definitions: dystocia 1 (scores 1+2v.3+4+5) and dystocia 2 (scores 1v.2+3+4+5). Four models were proposed for genetic parameter estimation: (1) fixed effects plus sire effects; (2) model 1 plus maternal grandsire effect; (3) model 2 plus dam within maternal grandsire effects; (4) same as model 3 but a random effect ‘herds’ replaced a fixed effect ‘regions’. Two methods of fitting models were applied: marginal maximum likelihood and the ‘tilde-hat’ approach. Estimates of genetic parameters by the two methods were similar. Models ignoring maternal effects overestimated the heritability of direct effects especially in the case of dystocia 2. Dystocia definition was responsible for the greatest difference among estimated genetic parameters. Possible reasons for this are discussed. When analysing large data sets, it is recommended judiciously to collapse dystocia categories and to apply approximate statistical procedures to complete models including maternal effects.

Starting from the approach of Wright (1931), some methods were developed for predicting the selection limit for a quantitative trait. These methods, however, do not take into account the effects of selection on the genetic variance, which are mainly (i) the induction of linkage disequilibrium and (ii) the change in family structure. Some algorithms intended to predict the evolution of the genetic variance and mean under the infinitesimal model are presented and discussed. Simulation results are compared with the expected values. They clearly show the irrelevance of equations taking into account drift or selection separately. On the other hand, taking into account both effects leads to prediction of responses to selection close to those observed. However, no algorithm provides a strictly exact prediction of the genetic variance, and the most critical point lies in the prediction of the effect of selection on drift. More complex situations are also discussed: (1) the consequences of selection when an animal model and BLUP techniques are used; (2) the case of a finite number of loci, with more or less linkage, (3) the impact of non-random mating or mutation; (4) trying to make predictions under practical situations such as overlapping generations. Practical applications in breeding programmes are outlined.

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.

Accurate prediction of the cumulated genetic gain requires predicting genetic variance over time under the joint effects of selection and limited population size. An algorithm is proposed to quantify at each generation the effects of these factors on average coefficient of inbreeding, genetic variance, and genetic mean, under a purely additive polygenic model, with no mutation, and under the assumption of absence of inbreeding depression on viability affecting selection differentials. This algorithm is relevant to populations where mating is at random and generations do not overlap. It was tested via Monte Carlo simulation on a population of 3 males and 25 females mass selected out of 50 candidates of each sex, over 30 generations. For two values of the initial heritability of the selected trait, 0.5 and 0.9 (to represent high accuracy in index selection), predicted values of the genetic variance are in agreement with observed results up to the 12th and 19th generations, respectively. Beyond these generations, the variance is overestimated, due to an underestimation of the effect of selection on the rate of inbreeding. Finally, the algorithm provides predictions of the cumulated responses close to the observed values in both selected populations. It is concluded that, as regards the hypotheses of the study, the proposed algorithm is satisfactory, and could be used to optimize selection methods with respect to the cumulated genetic gain in the mid- or long-term. Possible extensions of the algorithm to more realistic situations are discussed.

Nonlinear mixed effects models are now widely used in biometrical studies, especially in pharmacokinetic research or for the analysis of growth traits for agricultural and laboratory species. Most of these studies, however, are often based on ML estimation procedures, which are known to be biased downwards. A few REML extensions have been proposed, but only for approximated methods. The aim of this paper is to present a REML implementation for nonlinear mixed effects models within an exact estimation scheme, based on an integration of the fixed effects and a stochastic estimation procedure. This method was implemented via a stochastic EM, namely the SAEM algorithm. The simulation study showed that the proposed REML estimation procedure considerably reduced the bias observed with the ML estimation, as well as the residual mean squared error of the variance parameter estimations, especially in the unbalanced cases. ML and REML based estimators of fixed effects were also compared via simulation. Although the two kinds of estimates were very close in terms of bias and mean square error, predictions of individual profiles were clearly improved when using REML vs. ML. An application of this estimation procedure is presented for the modelling of growth in lines of chicken.

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.

In this review, we first analyse the objectives to consider in preserving diversity, fitness and adaptability of farm animal genetic resources (AnGR), given the links between genetic diversity and fitness-adaptedness (FA) traits. Ways to measure diversity are then presented, and tools available for managing genetic diversity within given economic constraints are described, under a theoretical framework proposed by Martin Weitzman. The emphasis is on molecular diversity and the exploitation of genetic marker information for use in conservation management of AnGR, through the use of a diversity function which may be applied either to gene diversity (based on allele frequencies) or to allelic diversity (based on allele counts). The question is raised of the possible relationships between molecular diversity of AnGR and FA traits. Methods for investigating those relationships based on within-breed heterozygosity or between-breed molecular divergence are reviewed. The concepts are illustrated on data from a European study of pig diversity.

Assessing allelic richness in a set of populations requires that variations of sample size be taken into account. One way of doing this is to estimate the number of alleles expected in samples of specified size, using the rarefaction method applied in ecology. An alternative method, based on extrapolation, consists of adding to the number of alleles actually seen in a population the expected number of alleles missing, given the number of genes examined in the population and the allelic frequencies observed over the whole set of populations. Heterogeneity of allelic richness across populations and across loci can also be tested in this framework by numerical re-sampling. Both methods provide a measure of “private” allelic richness, a useful criterion in genetic diversity preservation, by allowing evaluation of the uniqueness of each population in terms of allele numbers. The two methods are compared on isozyme loci in the argan tree of Morocco and on microsatellite genotypes in the European pig. In both species, allelic richness and gene diversity behave quasi-independently over the populations compared and a higher differentiation is observed in allelic richness compared to gene diversity. In general, the rarefaction technique is sensitive to the sample size of reference and may lack sensitivity to rare alleles when the sample size of reference is small. Extrapolation may thus be recommended especially when the sample sizes of the populations are either low on average or highly unbalanced among populations.

Litter characteristics at birth were re- corded in 4 genetic types of sows with differing maternal abilities. Eighty-two litters from F1 Duroc × Large White sows, 651 litters from Large White sows, 63 lit- ters from Meishan sows, and 173 litters from Laconie sows were considered. Statistical models included ran- dom effects of sow, litter, or both; fixed effects of sow genetic type, parity, birth assistance, and piglet sex, as well as gestation length, farrowing duration, piglet birth weight, and litter size as linear covariates. The quadratic components of the last 2 factors were also considered. For statistical analyses, GLM were first considered, assuming a binomial distribution of still- birth. Hierarchical models were also fitted to the data to take into account correlations among piglets from the same litter. Model selection was performed based on deviance and deviance information criterion. Finally, standard and robust generalized estimating equations(GEE) procedures were applied to quantify the impor- tance of each effect on a piglet's probability of stillbirth. The 5 most important factors involved were, in decreas- ing order (contribution of each effect to variance reduc- tion): difference between piglet birth weight and the litter mean (2.36%), individual birth weight (2.25%), piglet sex (1.01%), farrowing duration (0.99%), and sow genetic type (0.94%). Probability of stillbirth was greater for lighter piglets, for male piglets, and for pig- lets from small or very large litters. Probability of still- birth increased with sow parity number and with far- rowing duration. Piglets born from Meishan sows had a lower risk of stillbirth (P < 0.0001) and were little affected by the sources of variation mentioned above compared with the 3 other sow genetic types. Standard and robust GEE approaches gave similar results de- spite some disequilibrium in the data set structure highlighted with the robust GEE approach.

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.

Between-breed genetic diversity is classically considered as a major criterion to be taken into account when setting priorities for conservation of domestic animal breeds. However, it has been argued that methods based on the between-breed component of genetic diversity may not be optimal because they ignore the within-breed component of variation. The paper considers the most common methods used to evaluate those two components when genetic diversity is evaluated on the basis of genetic markers, and proposes to define an aggregate diversity combining linearly the two components. This implies defining for each breed (or population) its contributions to the between-breed and to the within-breed diversity. When defining an aggregate diversity, one can weight these contributions by FST and 1−FST, respectively, since the fixation index FST of Wright represents the proportion of the total genetic variation which is due to differences in allelic frequencies between populations. Such an approach is valid when the objective is genetic improvement by selection within a so-called “meta-population”. However, in a more general context of animal breeding, when heterosis and complementarities between breeds have to be considered, as well as adaptation to specific environments, more weight should be given to the between-breed variation. The proper weight to apply may require solutions adapted to each particular situation. In a long-term conservation perspective, priorities should also take into account the degree of endangerment of each breed. By combining diversity contributions and probability of extinction, a cryopreservation potential (or priority) may be estimated for each breed. The problem is illustrated on a sample of 11 European pig breeds typed for 18 microsatellite loci.

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.