Linear and Nonlinear Mixed-Effects Models

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 data 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 PROC NLM IXED in SAS version 7, some exposure to the combination of graphical display and model-fitting approaches from S-PLUS may be informative . 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 .