Linear model methodology:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton [u.a.]
CRC Press
2010
|
| Schriftenreihe: | A Chapman & Hall book
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XIX, 542 S. graph. Darst. |
| ISBN: | 9781584884811 |
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| 035 | |a (OCoLC)149649998 | ||
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| 100 | 1 | |a Khuri, André I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Linear model methodology |c André I. Khuri |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2010 | |
| 300 | |a XIX, 542 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a A Chapman & Hall book | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Linear models (Statistics) |v Textbooks | |
| 650 | 0 | 7 | |a Lineares Modell |0 (DE-588)4134827-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Lineares Modell |0 (DE-588)4134827-8 |D s |
| 689 | 0 | |5 DE-604 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-018668817 | ||
Datensatz im Suchindex
| _version_ | 1804140760613781504 |
|---|---|
| adam_text | Contents
Preface
..................................... xv
Author
..................................... xix
1
Linear Models: Some Historical Perspectives
............ 1
1.1
The Invention of Least Squares
.................. 3
1.2
The Gauss-Markov Theorem
................... 4
1.3
Estimability
............................. 4
1.4
Maximum Likelihood Estimation
................ 5
1.5
Analysis of Variance
........................ 6
1.5.1
Balanced and Unbalanced Data
.............. 7
1.6
Quadratic Forms and Craig s Theorem
............. 8
1.7
The Role of Matrix Algebra
.................... 9
1.8
The Geometric Approach
..................... 10
2
Basic Elements of Linear Algebra
................... 13
2.1
Introduction
............................. 13
2.2
Vector Spaces
............................ 13
2.3
Vector Subspaces
.......................... 14
2.4
Bases and Dimensions of Vector Spaces
............. 16
2.5
Linear Transformations
...................... 17
Exercises
.................................. 20
3
Basic Concepts in Matrix Algebra
................... 23
3.1
Introduction and Notation
.................... 23
3.1.1
Notation
........................... 24
3.2
Some Particular Types of Matrices
................ 24
3.3
Basic Matrix Operations
...................... 25
3.4
Partitioned Matrices
........................ 27
3.5
Determinants
............................ 28
3.6
The Rank of a Matrix
........................ 31
3.7
The Inverse of a Matrix
...................... 33
3.7.1
Generalized Inverse of a Matrix
.............. 34
3.8
Eigenvalues and Eigenvectors
.................. 34
3.9
Idempotent and Orthogonal Matrices
.............. 36
3.9.1
Parameterization of Orthogonal Matrices
........ 36
3.10
Quadratic Forms
.......................... 39
vii
yjjj
Contents
3.11
Decomposition Theorems
..................... 40
3.12
Some Matrix Inequalities
..................... 43
3.13
Function of Matrices
........................ 46
3.14
Matrix Differentiation
....................... 48
Exercises
.................................. 52
4
The Multivariate Normal Distribution
................ 59
4.1
History of the Normal Distribution
............... 59
4.2
The Univariate Normal Distribution
............... 60
4.3
The Multivariate Normal Distribution
.............. 61
4.4
The Moment Generating Function
................ 63
4.4.1
The General Case
...................... 63
4.4.2
The Case of the Multivariate Normal
.......... 65
4.5
Conditional Distribution
...................... 67
4.6
The Singular Multivariate Normal Distribution
........ 69
4.7
Related Distributions
........................ 69
4.7.1
The Central Chi-Squared Distribution
.......... 70
4.7.2
The
Noncentral Chi-Squared
Distribution
........ 70
4.7.3
The r-Distribution
...................... 73
4.7.4
The F-Distribution
..................... 74
4.7.5
The
Wishart
Distribution
................. 75
4.8
Examples and Additional Results
................ 75
4.8.1
Some Misconceptions about the
Normal Distribution
.................... 77
4.8.2
Characterization Results
.................. 78
Exercises
.................................. 80
5
Quadratic Forms in Normal Variables
................. 89
5.1
The Moment Generating Function
................ 89
5.2
Distribution of Quadratic Forms
................. 94
5.3
Independence of Quadratic Forms
................ 103
5.4
Independence of Linear and Quadratic Forms
......... 108
5.5
Independence and Chi-Squaredness of Several
Quadratic Forms
..........................
Ill
5.6
Computing the Distribution of Quadratic Forms
........ 118
5.6.1
Distribution of a Ratio of Quadratic Forms
....... 119
Appendix 5.A: Positive Definiteness of the Matrix Wt-1 in
(5.2) . . 120
Appendix 5.B:
ΑΣ
Is Idempotent If and Only If
Ζ1/2ΑΣ1/2
Is
Idempotent
............................. 121
Exercises
.................................. 121
6
Full-Rank Linear Models
........................ 127
6.1
Least-Squares Estimation
..................... 128
6.1.1
Estimation of the Mean Response
............ 130
6.2
Properties of Ordinary Least-Squares Estimation
....... 132
Contents ix
6.2.1
Distributional Properties
.................. 132
6.2.1.1
Properties under the Normality
Assumption
................... 133
6.2.2
The Gauss-Markov Theorem
............... 134
6.3
Generalized Least-Squares Estimation
.............. 137
6.4
Least-Squares Estimation under Linear
Restrictions on
β
.......................... 137
6.5
Maximum Likelihood Estimation
................ 140
6.5.1
Properties of Maximum Likelihood Estimators
..... 141
6.6
Inference Concerning
β
...................... 146
6.6.1
Confidence Regions and Confidence Intervals
..... 148
6.6.1.1
Simultaneous Confidence Intervals
...... 148
6.6.2
The Likelihood Ratio Approach to Hypothesis
Testing
............................ 149
6.7
Examples and Applications
.................... 151
6.7.1
Confidence Region for the Location of
the Optimum
........................ 151
6.7.2
Confidence Interval on the True Optimum
....... 154
6.7.3
Confidence Interval for a Ratio
.............. 157
6.7.4
Demonstrating the Gauss-Markov Theorem
...... 159
6.7.5
Comparison of Two Linear Models
........... 162
Exercises
.................................. 169
7
Less-Than-Full-Rank Linear Models
................. 179
7.1
Parameter Estimation
....................... 179
7.2
Some Distributional Properties
.................. 180
7.3
Reparameterized Model
...................... 181
7.4
Estimable Linear Functions
.................... 184
7.4.1
Properties of Estimable Functions
............ 185
7.4.2
Testable Hypotheses
.................... 187
7.5
Simultaneous Confidence Intervals on Estimable Linear
Functions
.............................. 192
7.5.1
The Relationship between
Scheffé s
Simultaneous
Confidence Intervals and the F-Test Concerning
Η0:Αβ = 0
......................... 194
7.5.2
Determination of an Influential Set of Estimable
Linear Functions
...................... 196
7.5.3
Bonferroni s Intervals
................... 199
7.5.4
Šidák s
Intervals
....................... 200
7.6
Simultaneous Confidence Intervals on All Contrasts among
the Means with Heterogeneous Group Variances
....... 202
7.6.1
The Brown-Forsythe Intervals
.............. 202
7.6.2
Spjotvoll s Intervals
.................... 203
7.6.2.1
The Special Case of Contrasts
......... 205
7.6.3
Exact Conservative Intervals
............... 206
Contents
7.7
Further Results Concerning Contrasts and Estimable
Linear Functions
.......................... 209
7.7.1
A Geometrical Representation of Contrasts
....... 209
7.7.2
Simultaneous Confidence Intervals for Two Estimable
Linear Functions and Their Ratio
............. 213
7.7.2.1
Simultaneous Confidence Intervals Based on
Scheffé s
Method
................. 213
7.7.2.2
Simultaneous Confidence Intervals Based on
the Bonferroni Inequality
............ 214
7.7.2.3
Conservative Simultaneous Confidence
Intervals
...................... 214
Exercises
.................................. 216
Balanced Linear Models
......................... 225
8.1
Notation and Definitions
..................... 225
8.2
The General Balanced Linear Model
............... 229
8.3
Properties of Balanced Models
.................. 232
8.4
Balanced Mixed Models
...................... 237
8.4.1
Distribution of Sums of Squares
............. 238
8.4.2
Estimation of Fixed Effects
................ 240
8.5
Complete and Sufficient Statistics
................ 249
8.6
ANOVA Estimation of Variance Components
......... 254
8.6.1
The Probability of a Negative ANOVA
Estimator
.......................... 254
8.7
Confidence Intervals on Continuous Functions of the
Variance Components
....................... 257
8.7.1
Confidence Intervals on Linear Functions of the
Variance Components
................... 259
8.8
Confidence Intervals on Ratios of Variance Components
. . . 263
Exercises
.................................. 266
The Adequacy of Satterthwaite s Approximation
.......... 271
9.1
Satterthwaite s Approximation
.................. 271
9.1.1
A Special Case: The Behrens-Fisher Problem
...... 274
9.2
Adequacy of Satterthwaite s Approximation
.......... 278
9.2.1
Testing Departure from Condition
(9.35)........ 282
9.3
Measuring the Closeness of Satterthwaite s
Approximation
........................... 287
9.3.1
Determination of Xsup
................... 290
9.4
Examples
.............................. 290
9.4.1
The Behrens-Fisher Problem
............... 291
9.4.2
A Confidence Interval on the Total Variation
...... 293
9.4.3
A Linear Combination of Mean Squares
......... 296
Appendix 9.A: Determination of the Matrix
G
in Section
9.2.1 ... 297
Exercises
.................................. 297
Contents xi
10
Unbalanced Fixed-Effects
Models................... 301
10.1
The
R-Notation........................... 301
10.2
Two-Way Models without Interaction
.............. 304
10.2.1
Estimable Linear Functions for Model
(10.10)...... 305
10.2.2
Testable Hypotheses for Model
(10.10).......... 306
10.2.2.1
Type I Testable Hypotheses
.......... 309
10.2.2.2
Type
Π
Testable Hypotheses
.......... 310
10.3
Two-Way Models with Interaction
................ 314
10.3.1
Tests of Hypotheses
.................... 315
10.3.1.1
Testing the Interaction Effect
.......... 318
10.3.2
Type
ΙΠ
Analysis in
SAS
.................. 322
10.3.3
Other Testable Hypotheses
................ 324
10.4
Higher-Order Models
....................... 327
10.5
A Numerical Example
....................... 331
10.6
The Method of Unweighted Means
............... 336
10.6.1
Distributions of SSau, SSbu, and SSabu
.......... 338
10.6.2
Approximate Distributions of SSau, SSbu,
and SSabu
.......................... 340
Exercises
.................................. 342
11
Unbalanced Random and Mixed Models
............... 349
11.1
Estimation of Variance Components
............... 350
11.1.1
ANOVA Estimation—Henderson s Methods
...... 350
11.1.1.1
Henderson s Method
ΠΙ
............. 351
11.1.2
Maximum Likelihood Estimation
............. 357
11.1.3
Restricted Maximum Likelihood Estimation
...... 362
11.1.3.1
Properties of REML Estimators
........ 366
11.2
Estimation of Estimable Linear Functions
............ 369
11.3
Inference Concerning the Random One-Way Model
...... 373
11.3.1
Adequacy of the Approximation
............. 376
11.3.2
Confidence Intervals on
σ^
and
σ^/σ^
......... 379
11.4
Inference Concerning the Random Two-Way Model
..... 380
11.4.1
Approximate Tests Based on the Method of
Unweighted Means
..................... 380
11.4.1.1
Adequacy of the Approximation
....... 384
11.4.2
Exact Tests
.......................... 385
11.4.2.1
ExactTestConcerningHoio^p
=0...... 386
11.4.2.2
Exact Tests Concerning
σ^
and c^
...... 388
11.5
Exact Tests for Random Higher-Order Models
......... 397
11.6
Inference Concerning the Mixed Two-Way Model
....... 398
11.6.1
Exact Tests Concerning cr| and
er^ß........... 398
11.6.2
An Exact Test for the Fixed Effects
............ 401
11.7
Inference Concerning the Random Two-Fold
Nested Model
............................ 406
11.7.1
An Exact Test Concerning
σ*
............... 407
,¿1 Contents
11.8
Inference Concerning the Mixed Two-Fold
Nested Model
............................ 411
11.8.1
An Exact Test Concerning o^(a)
............. 411
11.8.2
An Exact Test for the Fixed Effects
............ 412
11.9
Inference Concerning the General Mixed Linear Model
.... 415
11.9.1
Estimation and Testing of Fixed Effects
......... 416
11.9.2
Tests Concerning the Random Effects
.......... 417
Appendix
11.
A: The Construction of the Matrix
Q
in Section
11.4.1........................... 421
Appendix ll.B: The Construction of the Matrix
Q
in Section
11.7.1........................... 422
Exercises
.................................. 422
12
Additional Topics in Linear Models
.................. 427
12.1
Heteroscedastic Linear Models
.................. 427
12.2
The Random One-Way Model with Heterogeneous Error
Variances
.............................. 428
12.2.1
An Approximate Test Concerning Ho
:
cr^
= 0..... 430
12.2.2
Point and Interval Estimation of
σ^
........... 433
12.2.3
Detecting Heterogeneity in Error Variances
....... 435
12.3
A Mixed Two-Fold Nested Model with Heteroscedastic
Random Effects
........................... 437
12.3.1
Tests Concerning the Fixed Effects
............ 438
12.3.2
Tests Concerning the Random Effects
.......... 441
12.4
Response Surface Models
..................... 443
12.5
Response Surface Models with Random Block Effects
..... 446
12.5.1
Analysis Concerning the Fixed Effects
.......... 448
12.5.2
Analysis Concerning the Random Effects
........ 449
12.6
Linear Multiresponse Models
................... 453
12.6.1
Parameter Estimation
................... 454
12.6.2
Hypothesis Testing
..................... 456
12.6.2.1
Hypothesis of Concurrence
........... 457
12.6.2.2
Hypothesis of Parallelism
............ 458
12.6.3
Testing for Lack of Fit
................... 459
12.6.3.1
Responses Contributing to
LOF
........ 462
Exercises
.................................. 467
13
Generalized Linear Models
....................... 473
13.1
Introduction
............................. 473
13.2
The Exponential Family
...................... 474
13.2.1
Likelihood Function
.................... 478
13.3
Estimation of Parameters
..................... 479
13.3.1
Estimation of the Mean Response
............ 483
13.3.2
Asymptotic Distribution of
β
............... 484
13.3.3
Computation of
β
in
SAS
................. 485
Contents xiii
13.4
Goodness of Fit
........................... 487
13.4.1
The Deviance
........................ 487
13.4.2
Pearson s Chi-Square Statistic
............... 490
13.4.3
Residuals
.......................... 491
13.5
Hypothesis Testing
......................... 497
13.5.1 Wald
Inference
....................... 497
13.5.2
Likelihood Ratio Inference
................. 498
13.6
Confidence Intervals
........................ 499
13.6.1
Wald s Confidence Intervals
............... 499
13.6.2
Likelihood Ratio-Based Confidence Intervals
...... 500
13.7
Gamma-Distributed Response
.................. 504
13.7.1
Deviance for the Gamma Distribution
.......... 506
13.7.2
Variance-Covariance Matrix of
β
............. 506
Exercises
.................................. 509
Bibliography
................................. 515
Index
...................................... 535
|
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| isbn | 9781584884811 |
| language | English |
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| spelling | Khuri, André I. Verfasser aut Linear model methodology André I. Khuri Boca Raton [u.a.] CRC Press 2010 XIX, 542 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Includes bibliographical references and index Linear models (Statistics) Textbooks Lineares Modell (DE-588)4134827-8 gnd rswk-swf Lineares Modell (DE-588)4134827-8 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018668817&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Khuri, André I. Linear model methodology Linear models (Statistics) Textbooks Lineares Modell (DE-588)4134827-8 gnd |
| subject_GND | (DE-588)4134827-8 |
| title | Linear model methodology |
| title_auth | Linear model methodology |
| title_exact_search | Linear model methodology |
| title_full | Linear model methodology André I. Khuri |
| title_fullStr | Linear model methodology André I. Khuri |
| title_full_unstemmed | Linear model methodology André I. Khuri |
| title_short | Linear model methodology |
| title_sort | linear model methodology |
| topic | Linear models (Statistics) Textbooks Lineares Modell (DE-588)4134827-8 gnd |
| topic_facet | Linear models (Statistics) Textbooks Lineares Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018668817&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT khuriandrei linearmodelmethodology |