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E-raamat: Linear Models and Design

  • Formaat: EPUB+DRM
  • Ilmumisaeg: 14-Nov-2022
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783031081767
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Linear Models and DesignSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 14-Nov-2022
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783031081767
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This book is designed as a textbook for graduate students and as a resource for researchers seeking a thorough mathematical treatment of its subject. It develops the main results of regression and the analysis of variance, as well as the central results on confounded and fractional factorial experiments. Matrix theory is deemphasized; its role is taken instead by the theory of linear transformations between vector spaces.

The text gives a carefully paced and unified presentation of two topics, linear models and experimental design. Students are assumed to have a solid background in linear algebra, basic knowledge of regression and analysis of variance, and some exposure to experimental design, and should be comfortable with reading and constructing mathematical proofs.

The book leads students into the mathematical theory, including many examples both for motivation and for illustration. Over 130 exercises of varying difficulty are included. An extensive mathematical appendix and a detailed index make the text especially accessible.

Linear Models and Design can serve as a textbook for a year-long course in the topics covered, or for a one-semester course in either linear model theory or experimental design. It prepares students for more advanced topics in the field, and assists in developing a thoughtful approach to the existing literature. It includes a guide to terminology as well as discussion of the history and development of ideas, and offers a fresh perspective on the fundamental concepts and results of the subject.

Arvustused

The present book offers a unified introduction to general linear models and design of experiments with special attention to the connections between the two topics. The material is organized originally and pleasantly and the style is always agreeable. Some special features make this book unique among the dozens of other books on the same topic. this book is intended as a textbook for students with a solid mathematical background. (Fabio Rapallo, zbMATH 1532.62003, 2024)

1 Linear Models
1(26)
1.1 Random Vectors
1(6)
1.2 Some Statistical Concepts
7(3)
1.2.1 Identifiability
7(1)
1.2.2 Estimation
8(1)
1.2.3 Testing: Confidence Sets
8(2)
1.3 Linear Models
10(3)
1.4 Regression Models
13(2)
1.5 Factorial (ANOVA) Models
15(2)
1.6 Linear Parametric Functions
17(3)
1.6.1 Identifiability
18(1)
1.6.2 Contrasts
19(1)
1.7 Linear Constraints and Hypotheses
20(5)
1.8 Exercises
25(2)
2 Effects in a Factorial Experiment
27(34)
2.1 One-Factor Designs
27(1)
2.2 Two-Factor Designs
28(8)
2.2.1 Three Elementary Hypotheses
28(3)
2.2.2 The Main Effects Hypotheses
31(4)
2.2.3 Effects in a Two-Factor Design
35(1)
2.3 Multifactor Designs
36(3)
2.4 Quantitative Factors
39(2)
2.5 The Factor-Effects Parametrization
41(12)
2.5.1 The One-Factor Model
42(2)
2.5.2 Models with Two Factors
44(3)
2.5.3 Models with Three or More Factors
47(2)
2.5.4 The Different Systems of Weights
49(4)
2.6 The Cell-Means Philosophy
53(1)
2.7 Random Effects: Components of Variance
54(4)
2.8 Exercises
58(3)
3 Estimation
61(18)
3.1 The Method of Least Squares
62(4)
3.2 Properties of Least Squares Estimators
66(4)
3.2.1 Estimability
68(1)
3.2.2 The Moore-Penrose Inverse of X
69(1)
3.3 Estimating σ2: Sum of Squares, Mean Square
70(5)
3.4 T-Tests, t-Intervals
75(1)
3.5 Exercises
76(3)
4 Testing
79(72)
4.1 Testing a Linear Hypothesis
80(8)
4.1.1 Testing in Constrained Models
85(1)
4.1.2 Replication. Lack of Fit and Pure Error
85(3)
4.2 The Wald Statistic
88(3)
4.3 The Lattice of Hypotheses
91(3)
4.3.1 The Two-Predictor Regression Model
92(1)
4.3.2 The Three-Predictor Regression Model
92(1)
4.3.3 The Two-Factor Design
93(1)
4.3.4 The Hypothesis of No Model Effect
93(1)
4.3.5 The Lattice in the Observation Space
94(1)
4.4 Adjusted SS
94(4)
4.5 Nested Hypotheses. Sequential SS
98(14)
4.5.1 Adjusted or Sequential?
106(2)
4.5.2 Associated Hypotheses
108(4)
4.6 Orthogonal Hypotheses
112(12)
4.6.1 Application: Factorial Designs
119(4)
4.6.2 Application: Regression Models
123(1)
4.7 Affine Hypotheses. Confidence Sets
124(3)
4.8 Simultaneous Inference
127(10)
4.8.1 The Bonferroni Method
128(2)
4.8.2 The Scheffe Method
130(7)
4.9 Inference for Variance Components Models
137(9)
4.9.1 The One-Factor Design
137(4)
4.9.2 The Two-Factor Design
141(3)
4.9.3 Some Challenges
144(2)
4.10 Exercises
146(5)
5 Multifactor Designs
151(68)
5.1 Vectors and Functions: Notation
152(1)
5.2 The General Theory of Block Effects
153(8)
5.3 The Multifactor Design: Reprise
161(5)
5.4 The Kurkjian-Zelen Construction
166(10)
5.4.1 Multilinear Algebra
168(4)
5.4.2 Application to Factorial Designs
172(4)
5.5 Confounding with Blocks
176(4)
5.5.1 Generalities
176(4)
5.6 Confounding in Classical Symmetric Factorial Designs
180(23)
5.6.1 Special Blockings and Block Effects
180(5)
5.6.2 Components of Interaction
185(5)
5.6.3 Finding Generalized Interactions
190(3)
5.6.4 Multiplicative Notation: The Effects Group
193(5)
5.6.5 Complex Contrasts
198(5)
5.7 Confounding in Arbitrary Factorial Designs
203(10)
5.7.1 A Survey of Methods of Confounding
204(7)
5.7.2 The Problem of Confounding
211(2)
5.8 Exercises
213(6)
6 Fractional Factorial Designs
219(72)
6.1 Aliasing in Regular Fractions
223(13)
6.1.1 Multiplicative Notation Again: The Defining Subgroup
232(4)
6.2 The Resolution of a Regular Fraction
236(5)
6.3 Construction of Regular Fractions
241(3)
6.4 An Unexpected Pattern
244(5)
6.5 Aliasing and Resolution in Arbitrary Fractions
249(10)
6.5.1 Restriction Maps
249(1)
6.5.2 Strength and Aliasing
250(6)
6.5.3 Resolution
256(3)
6.6 Aliasing and Confounding
259(1)
6.7 Aliasing, Estimability and Bias
260(7)
6.8 Relative Aberration
267(12)
6.8.1 In Nonregular Designs
271(8)
6.9 Projections
279(8)
6.9.1 In Non-regular Designs
286(1)
6.10 Exercises
287(4)
A Mathematical Background
291(40)
A.1 Functions
291(1)
A.2 Relations
292(5)
A.2.1 Partially Ordered Sets and Lattices
292(3)
A.2.2 Equivalence Relations and Partitions
295(2)
A.3 Algebra
297(3)
A.4 Linear Algebra
300(31)
A.4.1 Subspaces and Bases
301(2)
A.4.2 Linear Transformations
303(5)
A.4.3 Inner Product Spaces
308(8)
A.4.4 The Transpose. Symmetric Maps
316(3)
A.4.5 Positive Maps
319(1)
A.4.6 The Spectral Theorem
319(3)
A.4.7 Orthogonal Projections
322(2)
A.4.8 Linear Algebra in W. Matrices
324(7)
Bibliography 331(6)
Index 337
Jay H. Beder is Professor Emeritus in the Department of Mathematical Sciences at the University of Wisconsin Milwaukee.  He has published papers in statistical design theory as well as on inference for Gaussian processes, with applications to evolutionary biology.  He has taught courses in linear models, design, and combinatorics for over 30 years, and is the recipient of teaching awards from his college and his university.
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