Relationships of BLUP under transformed general linear mixed models

A general linear mixed model includes both fixed and random effects. In some cases, we need to take various transformations over these models. One of the most simple and direct transformations is to pre-multiply a certain matrix to both sides of given the model, which is called a linear transformation of the model. In this study, two transformed general linear mixed models are considered without making any restrictions regarding random effects and matrix ranks in models.These two transformed models are not have to be equal unless proper dimensional matrices, which are used to obtain these models, are in full column rank. Therefore, predictors/estimators of unkonwn parameters under these two transformed models do not have to be similar. Although these estimators have different algebraic expressions and different properties, they might show similar features in some circumstances. In this study, the relations among best linear unbiased predictors (BLUPs) of joint unknown parameters have been analyzed by presenting an overall approach to the problem of comparing BLUPs under two transformed general linear mixed models. Matrix equations encountered while the results are obtained particularly include Moore-Penrose generalized inverse of matrices and come up with a very complex structure. Consequently, equations and expressions obtanied from column spaces and ranks of matrixes have been simplified to reach solutions. The results of some special conditions regarding predictors/estimators of joint unknown parameters under transformed linear mixed models have been given. In the first chapter, general linear mixed model and its transformed model have been been given. Some basic concepts and theorems are mentioned in the second chapter. In the third chapter, predictability/estimatability in general linear mixed models and transformed general linear mixed models, matrix equations concerning BLUP and best linear unbiased estimator (BLUE), analytic expressions and properties of predictors/estimators have been reviewed. In the fourth chapter, equality of BLUPs of vectors of joint unknown parameter under transformed general linear mixed models have been examined. The results are obtained by using some matrix ranks and column space properties. The fifth chapter consists of conclusion and recommendations.