Estimation of a linear mixed model with a reduced rank variance matrix

Kelly, Alison Maree (2013). Estimation of a linear mixed model with a reduced rank variance matrix PhD Thesis, School of Mathematics and Physics, The University of Queensland.

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Author Kelly, Alison Maree
Thesis Title Estimation of a linear mixed model with a reduced rank variance matrix
School, Centre or Institute School of Mathematics and Physics
Institution The University of Queensland
Publication date 2013
Thesis type PhD Thesis
Supervisor Richard Wilson
Brian Cullis
Total pages 174
Language eng
Subjects 010405 Statistical Theory
070305 Crop and Pasture Improvement (Selection and Breeding)
010401 Applied Statistics
Formatted abstract
Selection of superior genetic material in a plant breeding program is based on experimental data which is measured in a series of small-plot field trials grown across multiple locations and years, known as multi-environment trials (METs). The underlying genetic model for these data is best formulated as a linear mixed model which includes a multiplicative term for genotype by environment (g×e) interaction. Residual maximum likelihood (REML) estimation is routinely used to estimate variance parameters in these mixed models. REML estimation relies on a positive definite variance matrix, G, for the set of random effects, as the solution equations require the inverse of G.

The genetic variance matrix for the random g×e effects in a MET, Gg, typically exhibits heterogeneity of both variance and covariance between environments and an unstructured form of the matrix is required to capture this complexity. However, there is often a high level of genetic correlation between one or more pairs of environments with the implication that Gg becomes of reduced rank. The factor analytic (FA) model provides a parsimonious approximation to the unstructured form of Gg, and a reduced rank formulation of the FA model exists. In this thesis, a simulation study is undertaken to demonstrate that the FA model provides a robust and efficient approximation to the fully unstructured form of Gg.

The genetic model for random g×e effects has been improved by including pedigree information through a genetic relationship matrix, with specific application in plant breeding where both additive and non-additive genetic variation can be measured. This model requires two extensions to the FA methodology and associated REML estimation for multiplicative terms. Estimation of these two extensions is developed in this thesis. The first considers a linear mixed model involving two multiplicative terms, each of which has an FA variance structure, but assumes independence between genotypes. The reduced rank formulation and estimation equations are developed for this model with two terms, of which either one or both may have a reduced rank variance matrix. The second extension involves replacing the assumption of independence between genotypes with a more complex (known) relationship matrix based on the additive genetic variance. This matrix can be large and both this matrix and its inverse are required when solving the mixed model equations for the reduced rank FA approach. These issues are considered in this thesis and an alternative formulation is presented in an attempt to improve computational speed and reduce memory requirements.

The hierarchical nature of the reduced rank model forms the basis of extensions to estimation methods for FA models presented in this thesis. Furthermore, it provides an approach for estimation in a more general setting involving a linear mixed model with a singular variance matrix, G. This situation is again motivated by an example of genetic relatedness, where Gg can be singular by design, due to nature of the genetic relationship for dominance in hybrid plant breeding. A novel approach to model formulation is developed in this thesis using an extended form of the random effects which ensures that any singular variance matrix is transformed to be non-singular. Under this formulation, standard REML estimation procedures can be applied. 
Keyword Linear mixed model
Residual maximum liklihood estimation
Factor analytic models
Reduced rank variance matrix
Singular variance
Multi-environment trials
Genotype By Environment Interaction

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Created: Mon, 28 Apr 2014, 21:34:50 EST by Alison Kelly on behalf of Scholarly Communication and Digitisation Service