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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.

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.

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.

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

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.