Estimation of heterogeneous variances in nonlinear mixed models via the SAEM-MCMC algorithm with applications to growth curves in poultry

Duval, Mylène; Robert-Granié, Christèle; Foulley, Jean-Louis

Journal de la société française de statistique, Tome 150 (2009) no. 2, pp. 65-83.

L’algorithme SAEM-MCMC est un outil puissant en vue de l’estimation des paramètres, par maximum de vraisemblance, dans les modèles non linéaires mixtes. Dans cet article, nous proposons une adaptation de cet algorithme pour estimer les paramètres dans le cadre des modèles linéaires mixtes à variances hétérogènes. Nous considérons, ici, deux modélisations possibles des variances  : un modèle structural (incluant des effets fixes et aléatoires) basé sur les log-variances et un modèle s’appuyant sur une liaison moyenne-variance. En comparaison à d’autres procédures mises en œuvre dans les logiciels R, SAS et Monolix, notre algorithme s’avère beaucoup plus flexible pour modéliser des fonctions de variance. L’algorithme que nous proposons a été numériquement validé dans le cas d’un modèle linéaire mixte à variances hétérogènes en comparant les résultats obtenus avec ceux relatifs à un algorithme EM analytique sur le jeu de données standard de Pothoff et Roy. Enfin, nous présentons une application sur des données réelles concernant une expérience de sélection sur la croissance de poulets. Sur ces données, nous avons comparé les résultats de notre algorithme à ceux obtenus avec les procédures suivantes : SAS-NLMIXED, nlme, Monolix et WinBUGS.

The SAEM-MCMC algorithm is a powerful tool for computing maximum likelihood estimators in the wide class of nonlinear mixed effects models. We propose in this article an adaptation of this algorithm to the estimation of heterogeneous variances in such models. Two residual variance models are considered: a linear mixed model on the log-variance, with fixed and random effects, and a mean-variance relationship. As compared to other procedures implemented in R, SAS and Monolix, our algorithm provides more flexibility in modelling variance functions and reliability of the estimates. This algorithm was numerically validated in the case of a heteroskedastic linear mixed model by comparing its results with those of a standard EM algorithm applied to Pothoff and Roy’s data. Finally, an application to real data involving a selection experiment on growth in chickens is presented in which that algorithm was compared to results of SAS-Nlmixed, nlme, Monolix and WinBUGS softwares.

Classification : 62H12, 62P10
Keywords: heteroskedasticity, maximum likelihood estimation, nonlinear mixed models, SAEM-MCMC algorithm
Mots clés : Hétéroscédasticité, Maximum de vraisemblance, modèle non linéaire mixte, algorithme SAEM-MCMC

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