Statistical inference on variance components

In several sciences but especially in animal and plant breeding, the general mixed model with fixed and random effects plays a great role. Statistical inference on variance components means tests of hypotheses about variance components, constructing confidence intervals for them, estimating them, and using the variance components to get best estimates for fixed effects as well as to predict random effects.Many problems in the statistical inference of variance components already arise in even the most simple mixed model, describing nested designs; they are already present in the balanced nested designs, but become more pronounced in the unbalanced case. To find a guideline for solving the problems for the general mixed model, the study of the nested designs is worthwile.In chapter 1 the outline of the author's research on statistical inference of variance components is given.In chapter 2 the historical development of variance components estimation is described. More than 125 years ago the notion of variance components appears explicitly in astronomy. The expansion began with the development of quantitative genetics in 1918 after the first World War, bull the tremendous increase in research in variance components dates from after the second World War.Just as in the fixed effects model, the notions of vectors, vector spaces and projections of vectors on vector spaces, can be fruitfully used for the mixed effects model. This approach is given in chapter 3.For balanced nested designs exact tests about ratios of variance components and the calculation of their power, are well- known. For the unbalanced three-stage nested designs an exact test exists about the variance component belonging to the second stage, but for the first stage variance component no exact test was available in practice before 1974. in chapter 4 the onset of the author's research on the testing problem is described. It was possible to calculate the critical level or P-value for the exact test.The estimation of variance components was directed towards finding, for several designs, the best quadratic unbiased estimators for the variance components. In chapter 5 the onset of author's research on the estimation problem is described. On introduction of the concept of a permissible estimator it becomes clear which approaches to estimate variance components are unsatisfactory. To use permissible estimators for variance components means to use estimators which give non-negative estimates. A solution for a non-negative estimator has been found using the least squares approach as a unified procedure in variance components estimation.To demonstrate the danger of the use of the best quadratic unbiased estimator in the simplest random model, the balanced one-way classification, the probabilities for negative estimates were calculated in chapter 6.In chapter 7 a new exact test about ratios of variance components in the unbalanced three-stage nested design is given.The description of the least squares method as a unified procedure for the estimation of variance components and for the derivation of permissible (non-negative) estimators of variance components is given in chapter 8. The use of the main estimation procedures such as iterated Least Squares (or I-MINQUE), Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) estimators, is also discussed there.Finally in chapter 9 the use of variance components in predicting the random effects is discussed. Best Linear Unbiased Prediction (BLUP), which is extensively used in animal breeding, has not so far been applied on the same scale in plant breeding. How to use it is the subject of the last section 9.2.