Formulating Mixed Models for Experiments, Including Longitudinal Experiments

• Published in: Journal of agricultural, biological, and environmental statistics Vol. 14; no. 3; pp. 253 - 280
• Main Authors: Brien, C. J., Demétrio, C. G. B.
• Format: Journal Article
• Language: English
• Published: New York American Statistical Association and the International Biometric Society 01.09.2009; Springer-Verlag; American Statistical Association; International Biometric Society
• Subjects: Agriculture; Agronomy. Soil science and plant productions; Biological and medical sciences; Biometrics, statistics, experimental designs, modeling, agricultural computer applications; Biostatistics; Covariance; Experiment design; Fundamental and applied biological sciences. Psychology; Generalities. Biometrics, experimentation. Remote sensing; Health Sciences; Mathematics and Statistics; Medicine; Monitoring/Environmental Analysis; Multilevel models; Observational terms; Planting; Random allocation; Statistical discrepancies; Statistics; Statistics for Life Sciences; Subject terms; Three dimensional modeling; Two dimensional modeling; Multiphase experiments; Repeated measures; Longitudinal experiments; Multitiered experiments; Analysis of variance; Mixed models; Mixed model; Variance analysis
• ISSN: 1085-7117; 1537-2693
• DOI: 10.1198/jabes.2009.08001

Mixed models have become important in analyzing the results of experiments, particularly those that require more complicated models (e.g., those that involve longitudinal data). This article describes a method for deriving the terms in a mixed model. Our approach extends an earlier method by Brien and Bailey to explicitly identify terms for which autocorrelation and smooth trend arising from longitudinal observations need to be incorporated in the model. At the same time we retain the principle that the model used should include, at least, all the terms that are justified by the randomization. This is done by dividing the factors into sets, called tiers, based on the randomization and determining the crossing and nesting relationships between factors. The method is applied to formulate mixed models for a wide range of examples. We also describe the mixed model analysis of data from a three-phase experiment to investigate the effect of time of refinement on Eucalyptus pulp from four different sources. Cubic smoothing splines are used to describe differences in the trend over time and unstructured covariance matrices between times are found to be necessary.