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Mixed-Effects Models
Mixed-Effects Models

José Pinheiro

Mixed-effects models provide a powerful and flexible tool for the analysis of balanced and unbalanced grouped data. These data arise in several areas of investigation and are characterized by the presence of correlation between observations within the same group. Some examples are repeated measures data, longitudinal studies, and nested designs. Classical modeling techniques which assume independence of the observations are not appropriate for grouped data. These models are becoming increasingly popular because of the development of reliable and efficient software for fitting them, which, in recent years, have become available in commercial packages.

Ongoing research in mixed-effects models at the Statistics and Data Mining Research Department focuses on computational and methodological aspects, in the main areas listed below.

NLME: Software for Analyzing Mixed-Effects Models

Developed in collaboration with Douglas M. Bates , from the University of Wisconsin - Madison, the NLME software comprises a set of S (S-PLUS) functions, methods, and classes for the analysis of both linear and nonlinear mixed-effects models. It extends the linear and nonlinear modeling facilities available in release 3 of S and S-PLUS and is available for Unix and Windows platforms. The following research activities are related to the NLME software.

* Include functions for fitting and analyzing generalized linear and non-linear mixed-effects models, using pseudo quasi-likelihood (PQL) and marginal quasi-likelihood (MQL) estimation.
* Extend the mixed-effects model specification to allow correlation structures and variance functions at any level of random effects. This will greatly enhance the current modeling capabilities, especially for spatial data and experimental designs data.
* Develop efficient numerical algorithms for calculating prediction errors for future observations and random effects, incorporating them into the NLME library.

Robust Estimation

In collaboration with Chuanhai Liu and Yingnian Wu, from UCLA, we have proposed a robust version of the linear mixed-effects model for continuous data in which multivariate t-distributions are assumed for both the random effects and the within-group errors. Simulation results suggest that the proposed model outperforms the usual linear mixed-effects model with Gaussian distributions for both the random effects and the within-group errors, even with moderate amounts of contamination are present in the data. Our results are summarized in the paper Efficient Algorithms for Robust Estimation in Linear Mixed-Effects Models Using the Multivariate t-Distribution [ PostScript ][ PDF ].

Current research includes extending the multivariate t model to generalized linear mixed-effects models and nonlinear mixed-effects models, and investigating versions of restricted maximum likelihood estimation that can be used with the multivariate t model.

Bootstrap Methods for Mixed-Effects Models

In collaboration with José Sanchez, from the University of Barcelona, Spain, we are investigating different bootstrap algorithms to be used with mixed-effects models. These include parametric, semi-parametric, and nonparametric bootstrap methods. We are studying the computational and statistical efficiency of these methods, as well as their robustness to departures in the model assumptions.

Book: Mixed-Effects Models in S and S-PLUS

Written in collaboration with Douglas M. Bates, this book provides an overview of the theory and application of linear and nonlinear mixed-effects models in the analysis of grouped data, such as longitudinal data, repeated measures, and multilevel data. A unified model-building strategy for both linear and nonlinear models is presented and applied to the analysis of over 20 real datasets from a wide variety of areas, including pharmacokinetics, agriculture, and manufacturing. A strong emphasis is placed on the use of graphical displays at the various phases of the model-building process, starting with exploratory plots of the data and concluding with diagnostic plots to assess the adequacy of a fitted model. Over 170 figures are included in the book. Table of contents:[ PostScript ][ PDF ].

The balanced mix of real data examples, modeling software, and theory makes this book a useful reference for practitioners who use, or intend to use, mixed-effects models in their data analyses. It can also be used as a text for a one-semester graduate-level applied course in mixed-effects models. Researchers in statistical computing will also find this book appealing for its presentation of novel and efficient computational methods for fitting linear and nonlinear mixed-effects models.

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