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Modelisation and estimation of heterogeneous variances in nonlinear mixed models

Mylene Duval
Abstract : Nonlinear mixed models stand apart in mixed models methodology. Contrary to linear and generalized linear models, often used as black boxes, the trajectory function t in nonlinear models generally comes from the integration of dierential equations. This provides a biological interpretation for the parameters, whereas models are often more parsimonious. However, the estimation of the parameters in nonlinear mixed models is complex because random eects cannot be integrated out of the likelihood in closed form. As in all mixed models, especially those used to analyze longitudinal data, nonlinear models are well adapted to take into account between and within-cluster variation. However, one of the common assumptions of such models is that of independent, identically distributed residuals with a common variance; this assumption is unrealistic in many elds of applications. The objective of this study was to propose some models for the residual variance to take into account the potential heterogeneity of variance of the residuals, while limiting the number of parameters in these models. In this sense, we used a parametric approach based on a linear mixed model on the logvariance, as well as the classical power \meanvariance" function. A classical inference method based on maximum likelihood theory was selected and we considered a stochastic EM algorithm, the SAEM-MCMC algorithm. The mixed model structure applied to the position and dispersion parameters is well adapted to the implementation of EM algorithms. Some instrumental distributions adapted to the analysis of these models, as well as some convergence criteria, were proposed in the MCMC step. The overall algorithm was numerically validated in both linear and nonlinear models, by comparing its results with those of an analytical EM algorithm (in the linear case) or other algorithms like those based on Gaussian quadrature. Finally, an application to the analysis of somatic cell scores in dairy cattle was presented. Several linear and nonlinear models were compared showing a clear gain obtained taking into account the heterogeneity of variances.
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Submitted on : Thursday, April 9, 2009 - 8:00:00 AM
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Mylene Duval. Modelisation and estimation of heterogeneous variances in nonlinear mixed models. Mathematics [math]. AgroParisTech, 2008. English. ⟨NNT : 2008AGPT0082⟩. ⟨pastel-00004846⟩

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