Linear Models: A Mean Model Approach

Linear models, normally presented in a highly theoretical and mathematical style, are brought down to earth in this comprehensive textbook. Linear Models examines the subject from a mean model perspective, defining simple and easy-to-learn rules for building mean models, regression models, mean vectors, covariance matrices and sums of squares matrices for balanced and unbalanced data sets. The author includes both applied and theoretical discussions of the multivariate normal distribution, quadratic forms, maximum likelihood estimation, less than full rank models, and general mixed models. The mean model is used to bring all of these topics together in a coherent presentation of linear model theory.

Provides a versatile format for investigating linear model theory, using the mean modelUses examples that are familiar to the student: Design of experiments, analysis of variance, regression, and normal distribution theoryIncludes a review of relevant linear algebra conceptsContains fully worked examples which follow the theorem/proof presentation

Linear Algebra and Related Introductory Topics: Elementary Matrix Concepts. Kronecker Products. Random Vectors. Multivariate Normal Distribution: Multivariate Normal Distribution Function. Conditional Distributionsof Multivariate Normal Vectors. Distributions of Certain Quadratic Forms. Distributions of Quadratic Forms: Quadratic Forms of Normal Random Vectors. Independence. t and F Distributions. Bhats Lemma. Complete, Balanced Factorial Experiments: Models That Admit Restrictions (Finite Models). Models That Do Not Admit Restrictions (Infinite Models). Sum of Squares and Covariance Matrix Algorithms. Expected Mean Squares. Algorithm Applications. Least Squares Regression: Ordinary Least SquaresEstimation. Best Linear Unbiased Estimators. ANOVA Table for the Ordinary Least Squares Regression Function. Weighted Least Squares Regression. Lack of Fit Test. Partitioning the Sum of Squares Regression. The Model Y = X( + E in Complete, BalancedFactorials. Maximum Likelihood Estimation and Related Topics: Maximum Likelihood Estimators (MLEs) of ( and ( + 2. Invariance Property, Sufficiency and Completeness. ANOVA Methods for Finding Maximum Likelihood Estimators. The Likelihood Ratio Test for H( = h. Confidence Bands on Linear Combinations of (. Unbalanced Designs and Missing Data: Replication Matrices. Pattern Matrices and Missing Data. Using Replication and Pattern Matrices Together. Balanced Incomplete Block Designs: General Balanced Incomplete Block Design. Analysis of the General Case. Matrix Derivations of Kempthornes Inter- and Intra-Block Treatment Difference Estimators. Less Than Full Rank Models: Model Assumptions and Examples. The Mean Model Solution. Mean Model Analysis When cov(E) = ( + 2I - n. Estimable Functions. Mean Model Analysis When cov(E) = ( + 2V. The General Mixed Model: The Mixed Model Structure and Assumptions. Random Portion Analysis: Type I Sumof Squares Method. Random Portion Analysis: Restricted Maximum Likelihood Method. Random Portion Analysis: A Numerical Example. Fixed Portion Analysis. Fixed Portion Analysis: A Numerical Example. Appendixes. References. Subject Index.