Statistical Tests for Mixed Linear Models

by Khuri, André; I.; Mathew, Thomas; Sinha, Bimal K.

• ISBN13: 9780471156536
• ISBN10: 0471156531
• Edition: 1st
• Format: Hardcover
• Copyright: 1998-01-29
• Publisher: Wiley-Interscience

Summary

An advanced discussion of linear models with mixed or random effects. In recent years a breakthrough has occurred in our ability to draw inferences from exact and optimum tests of variance component models, generating much research activity that relies on linear models with mixed and random effects. This volume covers the most important research of the past decade as well as the latest developments in hypothesis testing. It compiles all currently available results in the area of exact and optimum tests for variance component models and offers the only comprehensive treatment for these models at an advanced level. Statistical Tests for Mixed Linear Models: * Combines analysis and testing in one self-contained volume. * Describes analysis of variance (ANOVA) procedures in balanced and unbalanced data situations. * Examines methods for determining the effect of imbalance on data analysis. * Explains exact and optimum tests and methods for their derivation. * Summarizes test procedures for multivariate mixed and random models. * Enables novice readers to skip the derivations and discussions on optimum tests. * Offers plentiful examples and exercises, many of which are numerical in flavor. * Provides solutions to selected exercises. Statistical Tests for Mixed Linear Models is an accessible reference for researchers in analysis of variance, experimental design, variance component analysis, and linear mixed models. It is also an important text for graduate students interested in mixed models.

Author Biography

Andr&eacute; I. Khuri, PhD, is Professor of Statistics at the University of Florida in Gainesville and author of Advanced Calculus with Applications in Statistics and Response Surfaces: Designs and Analyses. <p> Mathew and Bimal K. Sinha are Professors of Statistics at the University of Maryland. Professor Sinha is coauthor of Robustness of Statistical Tests.

Table of Contents

Preface	xiii
1. Nature of Exact and Optimum Tests in Mixed Linear Models	1	(16)
1.1. Introduction	1	(1)
1.2. Exact F-Tests	2	(3)
1.3. Optimality of Tests	5	(12)
1.3.1. Uniformly Most Powerful Similar and Uniformly Most Powerful Unbiased Tests	8	(3)
1.3.2. Uniformly Most Powerful Invariant and Locally Most Powerful or Locally Best Invariant Tests	11	(6)
Appendix 1.1. Distribution of a Maximal Invariant T (x): Wijsman's Representation Theorem	17	(1)
Bibliography	18	(1)
2. Balanced Random and Mixed Models	19	(34)
2.1. Introduction	19	(1)
2.2. Balanced Models -- Notations and Definitions	20	(7)
2.3. Balanced Model Properties	27	(3)
2.4. Balanced Mixed Models: Distribution Theory	30	(3)
2.5. Derivation of Optimum Tests	33	(7)
2.5.1. A Numerical Example	39	(1)
2.6. Approximate and Exact Tests	40	(6)
2.6.1. Satterthwaite's Approximation	40	(3)
2.6.2. Exact Unbiased Tests of Bartlett-Scheffe Type	43	(3)
Exercises	46	(5)
Bibliography	51	(2)
3. Measures of Data Imbalance	53	(36)
3.1. Introduction	53	(1)
3.2. The Effects of Imbalance	53	(9)
3.2.1. The Variance of XXX(2)(T)	56	(2)
3.2.2. The Probability of a Negative XXX(2)(T)	58	(3)
3.2.3. Power of the Test Concerning XXX(2)(T)	61	(1)
3.3. Measures of Imbalance for the One-Way Model	62	(4)
3.3.1. The Effect of Imbalance on Var(XXX(2)(T))	63	(1)
3.3.2. The Effect of Imbalance on the Test Concerning XXX(2)(T)	64	(2)
3.4. A General Procedure For Measuring Imbalance	66	(5)
3.4.1. The One-Way Classification Model	66	(1)
3.4.2. The Two-Way Classification Model	67	(2)
3.4.3. The Three-Way Classification Model	69	(2)
3.5. Special Types of Imbalance	71	(5)
3.5.1. The Two-Fold Nested Classification Model	71	(3)
3.5.2. A Model With a Mixture of Cross-Classified and Nested Effects	74	(2)
3.6. A General Method for Determining the Effect of Imbalance	76	(11)
3.6.1. Generation of Designs Having a Specified Degree of Imbalance for the One-Way Model	76	(5)
3.6.2. An Example	81	(6)
3.7. Summary	87	(2)
Appendix 3.1. Hirotsu's Approximation	89	(1)
Exercises	90	(3)
Bibliography	93	(2)
4. Unbalanced One-Way and Two-Way Random Models	95	(24)
4.1. Introduction	95	(1)
4.2. Unbalanced One-Way Random Models	96	(4)
4.3. Two-Way Random Models	100	(13)
4.3.1. Models Without Interaction: Exact Tests	101	(2)
4.3.2. Models Without Interaction: Optimum Tests	103	(1)
4.3.3. Models With Interaction: Exact Tests	104	(7)
4.3.4. A Numerical Example	111	(2)
4.4. Random Two-Fold Nested Models	113	(4)
4.4.1. Testing HXXX(T): XXX(2)(XXX(T)) = 0	114	(1)
4.4.2. Testing H(T): XXX(2)(T) = 0	114	(3)
Exercises	117	(1)
Bibliography	118	(1)
5. Random Models with Unequal Cell Frequencies in the Last Stage	119	(22)
5.1. Introduction	119	(1)
5.2. Unbalanced Random Models With Imbalance In The Last Stage Only-Notation	119	(3)
5.3. Unbalanced Random Models With Imbalance In The Last Stage Only--Analysis	122	(8)
5.3.1. Derivation of Exact Tests	125	(5)
5.4. More on Exact Tests	130	(5)
5.4.1. Power of the Exact Tests	130	(2)
5.4.2. Sufficient Statistics Associated With the Exact Tests	132	(3)
5.5. A Numerical Example	135	(3)
Exercises	138	(2)
Bibliography	140	(1)
6. Tests in Unbalanced Mixed Models	141	(26)
6.1. Introduction	141	(1)
6.2. Mixed Models With Two Variance Components	141	(8)
6.2.1. Test for H(T): T(1) = ... = T(v)	142	(2)
6.2.2. Optimum Test for H(T): XXX(2)(T) = 0	144	(5)
6.3. Mixed Two-Way Crossed-Classification Models With Interactions	149	(10)
6.3.1. Derivation of Exact Tests for Variance Components	150	(3)
6.3.2. Derivation of an Exact Test for Fixed Effects	153	(6)
6.4. General Unbalanced Mixed Models: Exact Tests	159	(5)
6.5. General Unbalanced Mixed Models: Approximate Tests	164	(3)
Appendix 6.1. Proof of Lemma 6.3.2	167	(1)
Exercises	168	(1)
Bibliography	169	(3)
7. Recovery of Inter-block Information	172	(34)
7.1. Introduction	172	(1)
7.2. Notations and Test Statistics	173	(3)
7.3. BIBD With Fixed Treatment Effects	176	(13)
7.3.1. Combined Tests When b greater than v	177	(10)
7.3.2. Combined Tests When b = v	187	(1)
7.3.3. A Numerical Example	188	(1)
7.4. BIBD With Random Effects	189	(3)
7.5. General Incomplete Block Designs	192	(11)
7.5.1. The Combined Test	193	(7)
7.5.2. Some Computational Formulas	200	(1)
7.5.3. A Numerical Example	201	(2)
Exercises	203	(1)
Bibliography	204	(2)
8. Split-plot Designs Under Mixed and Random Models	206	(26)
8.1. Introduction	206	(7)
8.2. Derivation of Exact and Optimum Tests: Balanced Case	213	(5)
8.3. Derivation of Exact and Optimum Tests: Unbalanced Case	218	(12)
8.4. A Numerical Example	230	(2)
Appendix 8.1. Some Results on the Validity and Optimality of F-tests	232	(3)
Exercises	235	(1)
Bibliography	235	(1)
9. Tests Using Generalized P-Values	236	(20)
9.1. Introduction	236	(2)
9.2. The Generalized P-Value	238	(2)
9.3. Tests Using Generalized P-Values in the Balanced Case	240	(5)
9.4. Extensions to the Unbalanced Case	245	(2)
9.5. Two Examples	247	(3)
9.6. Comparison With Satterthwaite's Approximation	250	(3)
Exercises	253	(1)
Bibliography	254	(2)
10. Multivariate Mixed and Random Models	256	(41)
10.1. Introduction	256	(1)
10.2. The General Balanced Model	256	(3)
10.3. Properties of Balanced Multivariate Mixed Models	259	(15)
10.3.1. Distribution of Y'P(i)Y	260	(4)
10.3.2. Hypothesis Testing	264	(3)
10.3.3. Satterthwaite's Approximation	267	(4)
10.3.4. Closeness of Satterthwaite's Approximation	271	(3)
10.4. A Multivariate Approach to Univariate Balanced Mixed Models	274	(9)
10.4.1. Testing the Fixed Effects in a Balanced Mixed Two-Way Model	275	(2)
10.4.2. Testing the Random Effects Under Model B's Assumptions	277	(6)
10.5. The Derivation of Exact Tests	283	(2)
10.6. Optimum Tests	285	(5)
Exercises	290	(4)
Bibliography	294	(3)
Appendix: Solutions to Selected Exercises	297	(38)
General Bibliography	335	(10)
Author Index	345	(4)
Subject Index	349