Between and beyond additivity and non-additivity : the statistical modelling of genotype by environment interaction in plant breeding

In plant breeding it is a common observation to see genotypes react differently to environmental changes. This phenomenon is called genotype by environment interaction. Many statistical approaches for analysing genotype by environment interaction rely heavily on the analysis of variance model. Genotype by environment interaction is then taken to be equivalent to non-additivity. This thesis criticizes the analysis of variance approach. Modelling genotype by environment interaction by non-additivity is little parsimonious and interaction patterns remain hard to interpret. Interpretation is hindered by the multitude of parameters that require interpretation and the fact that these parameters do not refer to external genotypic and environmental information. A viable alternative is presented in the form of multiplicative models for interaction. The latter can be distinguished in two classes; factorial regression models and multilinear models. Factorial regression models describe genotype by environment interaction in direct relation to explicit external genotypic and environmental covariables. They are ordinary linear models that allow the testing of biologically interesting hypotheses about the mechanisms responsible for genotype by environment interaction. Multilinear models are based on low rank approximations to the tables of non-additivity parameters. Parameter estimates can be obtained from multiplicative decompositions of the non-additivity tables. Multilinear models guarantee a parsimonious description of the interaction. When genotypic and environmental interaction parameters are plotted simultaneously in so-called biplots, the emerging patterns often allow biologically interesting conclusions. The successful application of multiplicative models for interaction is illustrated for a number of variables in a number of crops like white cabbage, sugar beet perennial ryegrass, lettuce, wheat, potato and maize. The data came from plant breeding, resistance breeding, variety trials, and seed technology research. Theoretical contributions include the introduction of reduced rank factorial regression models in plant breeding, the development of generalized bilinear models, and the implementation of quadrilinear models for three-way non-additivity. In addition, the use of diagnostic biplots as a model screening device for two-way tables is described and evaluated. Besides applied and theoretical papers, the thesis contains extensive reviews of the possibilities of linear and bilinear models for modelling genotype by evironment interaction. Two opinion papers provide conceptual clarifications. The thesis not only addresses plant breeders interested in modelling genotype by environment interaction, but also statisticians and researchers interested in parsimonious modelling of interactions.