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Mixed models have been developed to deal with repeated data made on the same statistical unit. The standard example is longitudinal data where repeated measurement in time are made on several subjects/individuals/units. Another classical example where mixed models are used is when measurements are made on clusters of related statistical units.
These models are useful in a wide variety of disciplines in the physical, biological and social sciences, such as clinical trials, epidemiology, neuronal science, environmental science, experimental phonetic, etc.
Mixed models allow to deal with the various sources of heterogeneity induced by these repeated measurements, by introducing random effects in the (regression) model. Random effects model the inter-subject variability while the measurement noise models the intra-subject variability.
SVH team work on mixed models from both theoretical and applied points of view.
Theoretical developments:
- A first theme, developed by C. Bazzoli, is the computation of the
Fisher information for non-linear mixed models. This aim it to evaluate and optimize the
population designs of clinical studies.
An R module for this, `PFIM', was developed and a scientific/technical
user-support group for it was created in collaboration with INSERM UMR
738 Paris.
- A second theme, developed by A. Leclercq-Samson, C. Dion, is the specific case of nonlinear mixed
effects models defined by stochastic differential equations. Parametric estimation methods have been developed, including
an exact maximum likelihood approach
(with M. Delattre, AgroParisTech and V. Genon-Catalot, Univ Paris
Descartes) and a particle filter algorithm
(with S. Donnet, INRA). Nonparametric estimation of the distribution of the random effects has also been proposed with an adaptive nonparametric deconvolution technique based on the recent Goldenshluger-Lepski selection method.
- A third theme focuses on estimation in non-linear mixed models. When the regression function is defined as the solution of an ODE or PDE, being difficult to calculate, we propose to resort to krieging to approximate the model. In collaboration with P. Barbillon (AgroParisTech) and Numed team in Lyon, we have shown that this approach gives good estimation results with efficient computational times.
Another direction developed by E. Ollier and A. Leclercq-Samson, in collaboration with V. Viallon (IFSSTAR, Lyon), S. Lambert-Lacroix (TIMC-IMAG) is how to deal with high dimension in mixed models.
Applied developments:
- The project Phon&Stat is a collaboration between the SVH team and the Gipsa-Lab (speech and cognition department). Data analysed at Gipsa-Lab are repeated and paired and should be analysed by mixed model to take into account inter and intra subject variabilities. Their use has allowed to find significant effect of some covariates, which was not the case with standard analysis.
Using mixed models requires to select the influent covariates among a large number of covariates but with a small number of subjects. Model selection in high dimension has been widely studied in linear model, but only a few papers are available in the context of mixed models. A workshop has been organized in Grenoble on this topic, with 6 speakers. A PhD has started in november 2014 on this subject. 3 papers have already been submitted.
- Mixed models have also been widely used in biostatistics, see page Biostat for more details.
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