Analysis of three-way data for components of simultaneous interaction between all factors.

摘要

The phenomenon where different genotypes vary in their response to different environments is called genotype by environment interaction. A classical statistical approach to analyzing genotype by environment interaction, represents the genotype by environment interaction simply by a matrix of non-additivity parameters. This approach has been superceded by the recently developed mixed additive and multiplicative model, which represents the interaction by a low rank approximation of the matrix of non-additive parameters. Because of its two merits, parsimoniousness and high interpretability, this multiplicative model is rapidly becoming popular. Applications of the multiplicative model have been mainly confined to the analysis of two-way data, although in agriculture there are abundant data sets of higher dimensionality. This is primarily because the mathematical tool used in applying the multiplicative model for two-way data sets, singular value decomposition, has no counterpart for high dimensional data sets. One attempt at applying a three-way multiplicative model to three-way data sets uses the Tucker method. It has not been very successful because of the complexity of the analysis output.;This thesis explores the viability of using two alternative methods, Pi method and Extended Power method, for multiplicative models for three-way data analysis, by comparing their performances with that of Tucker. It shows that using the Tucker method in analyzing genotype by environment interaction unnecessarily complicates the model without providing additional information. Although, theoretically, the Pi method provides the possibility of more precisely identifying the underlying patterns when data is of very high signal to noise ratio and the interaction function is simple, this may not be of importance for agriculture data. Theoretically, the Extended Power method cannot exactly decompose the data matrix into axes of its true rank. Again this drawback is seldom significant due to the noisy nature of agriculture data. Orthogonality among the axes of decomposition, is an important property in graphical presentation of the analysis output, but is not imposed by the Pi method. This suggests the Extended Power method as the most usable method among the three. Both synthetic and real data are used for comparison.