Showing papers by "Douglas M. Bates published in 1998"

Linear and nonlinear mixed-effects models

Abstract: Douglas M. Bates Department of Statistics University of Wisconsin Madison Jose C. Pinheiro Bell Laboratories Lucent Technologies 1 Recent developments in computational methods for maximum likelihood (ML) or restricted maximum likelihood (REML) estimation of parameters in general linear mixed-effects models have made the analysis of data in typical agricultural settings much easier. With software such as SAS PROC MIXED we are able to handle da~ from random-effects one-way classifications, from blocked designs including incomplete blocked designs, from hierarchical designs such as splitplot designs, and other types of data that may be described as repeated measures or longitudinal data or growth-curve data. It is especially helpful that the new computational methods do not depend on balance in the data so we are able to deal more easily with observational studies or with randomly missing data in a designed experiment. We describe some of the new computational approaches and how they are implemented in the nlme3.0 library for the S-PLUS language. One of the most powerful features of this language is the graphics capabilities, especially the trellis graphics facilities developed by Bill Cleveland and his coworkers at Bell Labs. Although most participants in this conference may be more familiar with SAS, and most of the models described here can be fit with PROC MIXED or the NLiNMIX macro or new P ROC N LM IXED in SAS version 7, some exposure to the combination of graphical display and model-fitting approaches from S-PLUS may be informative. 1 Annual Conference on Applied Statistics in Agriculture Kansas State University New Prairie Press http://newprairiepress.org/agstatconference/1998/proceedings/2 2 Kansas State University We show how data exploration with trellis graphics, followed by fitting and comparing mixedeffects models, followed by graphical assessment of the fitted model can be used in a variety of situations. On some occasions, such as modeling growth curves, a linear trend or polynomial trend or other types of linear statistical models for the within-subject time dependence are just not going to do an adequate job of representing the data. In those cases, a nonlinear model is more appropriate. We show how the concept of a random coefficient model can be extended to nonlinear models so as to fit nonlinear mixed-effects models.

Computational Methods for Multilevel Modelling

Abstract: A multilevel mixed-effects model has random effects at each of several nested levels of grouping of the observed responses. We may use these, for example, when modelling observations taken over time on students who are grouped into classes that are grouped into schools that are grouped into districts. If each of the distributions of the random effects is Gaussian and if the disturbance term at the lowest level of grouping is also Gaussian it is straightforward to define a likelihood for the fixed effects and the parameters defining the random effects distribution. We show that by expressing the random effects distribution in terms of relative precision factors and using matrix decompositions, this likelihood can be profiled and can be compactly expressed. The same decompositions provide rapid evaluation of the profiled log-restricted-likelihood for REML estimation. The conditional distribution of the random effects given the data can be derived from the decomposed matrices. From this a compact and rapidly evaluated expression for the EM iterations can be derived. Reasonable starting estimates for the relative precision factors can be derived from the design alone. These starting estimates, refined by a moderate number of EM iterations, provide excellent starting values for a Newton-Raphson or quasi-Newton optimization of the log-likelihood or the log-restricted-likelihood. The methods we describe extend easily to models with non-spherical distributions for the within-group errors and to nonlinear multilevel models.