Joint analysis of genotypic and environmental effects.

A definition of jointly contributing genotypic and environmental effects is introduced, from which a new concept of genotype × environment interactions is derived. Interaction is defined to be the failure of genotypic or environmental response functions to be separable. For separable response functions, the contributions of the genotypic and environmental effects must be related in terms of an operator which can describe their joint actions. A scale-free method of determining the simplest operator is developed in terms of comparative norms of reaction and a characteristic of the operator is given for several operators. With a defined operator, the genetic and environmental contributions can be derived, and biologically interpreted. These methods are applied to published data on Pinus caribaea.