Statistical tests for mixed linear models.

*(English)*Zbl 0893.62009
Wiley Series in Probability and Mathematical Statistics, Applied Section. New York, NY: Wiley. xv, 352 p. (1998).

This book deals entirely with hypothesis testing under general linear mixed models. Both balanced and unbalanced data situations are considered. According to the authors, this book is intended to provide the state of the art in this field. It contains many of the recent results pertaining to exact and optimum tests of hypotheses in mixed linear models using normality assumptions. The book is written at a level suitable for a graduate topics course in mixed linear models. Appropriate prerequisites are (1) knowledge of statistical theory equivalent to what is usually covered in a graduate course in mathematical statistics and (2) a graduate course in linear models. The book contains ten chapters, an appendix, a subject index, an author index, and a bibliography.

Chapter 1 develops Wald’s variance components test in a general framework. There is also a brief discussion of optimum tests. By optimum tests the authors usually mean either locally or uniformly best tests among the class of relevant unbiased/invariant tests.

Chapter 2 considers balanced data situations and provides a comprehensive account of the main results pertaining to testing. Optimality of the standard \(F\)-tests is demonstrated. It is shown that for balanced data, the \(F\)-tests constructed from the ANOVA table are optimum. For some important hypotheses in balanced mixed models for which optimum tests are unavailable, the authors discuss exact unbiased tests of Bartlett-Scheffé type and the Satterthwaite approximation.

Chapter 3 describes a measure for quantifying the degree of imbalance of an unbalanced data set. The effect of imbalance on data analysis is explained using examples. The authors also discuss a general method for determining the effect of imbalance on inference. The method is based on an approach for generating designs having a specified degree of imbalance.

Chapter 4 discusses exact and optimum tests for the unbalanced one-way and unbalanced random two-way crossed and two-fold nested models. A general methodology for the derivation of exact tests is presented. The approach is further discussed in subsequent chapters.

Chapter 5 presents an exact test for a general unbalanced random model when the imbalance occurs in the last stage of the design only and there are no missing cells.

Chapter 6 discusses exact, approximate, and optimum tests for variance components as well as fixed effects in unbalanced mixed models. Mixed models with two variance components are discussed in detail. Exact tests for some mixed models with three variance components are also described. The chapter concludes with some exact tests of Bartlett-Scheffé type for a general mixed model with arbitrary number of variance components.

Chapter 7 deals with hypothesis tests that are designed to recover interblock information in general incomplete block designs. Several such tests are derived and compared. Tests are discussed for balanced incomplete block designs where the treatment factor may be fixed or random.

Chapter 8 is concerned with the analysis of balanced and unbalanced split-plot designs under mixed and random models. The imbalance may occur due to incomplete blocks or incomplete whole plots or due to missing data. Exact and optimum tests are derived for the balanced and unbalanced split-plot design.

Chapter 9 introduces the concept of generalized \(P\)-values and applies this approach for testing hypotheses in some balanced and unbalanced mixed models where exact tests are unavailable – for instance, testing whether or not a variance component equals a specified value, or if one variance component is equal to another variance component, etc. The approach of generalized \(P\)-values is compared with methods that use the Satterthwaite approximation.

Chapter 10 considers multivariate mixed and random models and provides a summary of the available testing procedures. It also discusses approaches for testing the adequacy of a multivariate Satterthwaite approximation.

The book contains several exercises at the end of each chapter beginning with Chapter 2. Solutions to selected exercises in Chapters 2 through 10 are given in the Appendix.

This book is an important resource for researchers in the field of mixed linear models. It complements other books in this field as it focusses on hypothesis tests whereas most other books deal either with point estimation or with confidence interval estimation.

Chapter 1 develops Wald’s variance components test in a general framework. There is also a brief discussion of optimum tests. By optimum tests the authors usually mean either locally or uniformly best tests among the class of relevant unbiased/invariant tests.

Chapter 2 considers balanced data situations and provides a comprehensive account of the main results pertaining to testing. Optimality of the standard \(F\)-tests is demonstrated. It is shown that for balanced data, the \(F\)-tests constructed from the ANOVA table are optimum. For some important hypotheses in balanced mixed models for which optimum tests are unavailable, the authors discuss exact unbiased tests of Bartlett-Scheffé type and the Satterthwaite approximation.

Chapter 3 describes a measure for quantifying the degree of imbalance of an unbalanced data set. The effect of imbalance on data analysis is explained using examples. The authors also discuss a general method for determining the effect of imbalance on inference. The method is based on an approach for generating designs having a specified degree of imbalance.

Chapter 4 discusses exact and optimum tests for the unbalanced one-way and unbalanced random two-way crossed and two-fold nested models. A general methodology for the derivation of exact tests is presented. The approach is further discussed in subsequent chapters.

Chapter 5 presents an exact test for a general unbalanced random model when the imbalance occurs in the last stage of the design only and there are no missing cells.

Chapter 6 discusses exact, approximate, and optimum tests for variance components as well as fixed effects in unbalanced mixed models. Mixed models with two variance components are discussed in detail. Exact tests for some mixed models with three variance components are also described. The chapter concludes with some exact tests of Bartlett-Scheffé type for a general mixed model with arbitrary number of variance components.

Chapter 7 deals with hypothesis tests that are designed to recover interblock information in general incomplete block designs. Several such tests are derived and compared. Tests are discussed for balanced incomplete block designs where the treatment factor may be fixed or random.

Chapter 8 is concerned with the analysis of balanced and unbalanced split-plot designs under mixed and random models. The imbalance may occur due to incomplete blocks or incomplete whole plots or due to missing data. Exact and optimum tests are derived for the balanced and unbalanced split-plot design.

Chapter 9 introduces the concept of generalized \(P\)-values and applies this approach for testing hypotheses in some balanced and unbalanced mixed models where exact tests are unavailable – for instance, testing whether or not a variance component equals a specified value, or if one variance component is equal to another variance component, etc. The approach of generalized \(P\)-values is compared with methods that use the Satterthwaite approximation.

Chapter 10 considers multivariate mixed and random models and provides a summary of the available testing procedures. It also discusses approaches for testing the adequacy of a multivariate Satterthwaite approximation.

The book contains several exercises at the end of each chapter beginning with Chapter 2. Solutions to selected exercises in Chapters 2 through 10 are given in the Appendix.

This book is an important resource for researchers in the field of mixed linear models. It complements other books in this field as it focusses on hypothesis tests whereas most other books deal either with point estimation or with confidence interval estimation.

Reviewer: H.Iyer (Fort Collins)

##### MSC:

62F03 | Parametric hypothesis testing |

62J10 | Analysis of variance and covariance (ANOVA) |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62E17 | Approximations to statistical distributions (nonasymptotic) |

62F05 | Asymptotic properties of parametric tests |