Analysis of variance for random models: theory, methods, applications, and data analysis 1 Balanced data
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
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Boston, Mass. [u.a.]
Birkhäuser
2004
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXV, 484 S. |
| ISBN: | 0817632301 3764332301 |
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| 245 | 1 | 0 | |a Analysis of variance for random models |b theory, methods, applications, and data analysis |n 1 |p Balanced data |c Hardeo Sahai ; Mario Miguel Ojeda |
| 264 | 1 | |a Boston, Mass. [u.a.] |b Birkhäuser |c 2004 | |
| 300 | |a XXV, 484 S. | ||
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| adam_text |
Titel: Bd. 1. Analysis of variance for random models. Balanced data
Autor: Sahai, Hardeo
Jahr: 2004
Contents
List of Figures xiii
List of Tables xv
Preface xix
Acknowledgments xxiii
1 Introduction 1
1.1 Analysis of Variance Models. 2
1.2 Fixed Effects Models. 4
1.3 Random Effects Models . 5
1.4 Mixed Effects Models. 7
1.5 Variance Components and Their Applications. 9
1.6 Scope of the Book. 12
1.7 Organization of the Book. 13
Bibliography . 13
2 One-Way Classification 21
2.1 Mathematical Model.21
2.2 Analysis of Variance .23
2.3 Minimal Sufficient Statistics and Distribution Theory.25
2.4 Classical Estimation.30
2.4.1 Analysis of Variance Estimators.31
2.4.2 Maximum Likelihood Estimators .34
2.4.3 Restricted Maximum Likelihood Estimators.37
2.4.4 Modifications of the ML Estimators.41
2.4.5 Stein-Type Estimators.41
2.4.6 Federer's Nontruncated Exponential Corrector
Estimators .41
2.4.7 Naqvi's Goodness-of-Fit Estimators.42
2.4.8 Hodges-Lehmann-Type Estimators of cr?.43
2.4.9 MVU Estimators of o^fa^ and er^/fcr^ + a^) . 44
2.4.10 A Numerical Example.47
2.5 Bayesian Estimation .51
2.5.1 Prior and Posterior Distribution Analysis .52
2.5.2 Some Formal Bayes Estimators.60
2.5.3 A Numerical Example.64
v
Contents
2.6 Sampling Distribution and Moments of Estimators .
2.6.1 Distribution and Moments of Estimators of ae .
2.6.2 Distribution and Moments of Estimators of cr2 . 67
2.6.3 Distribution and Moments of Estimators of
+ CTa).Z?
2.7 Comparison of Estimators Using Mean Squared Error Criterion 72
2.8 Interval Estimation.75
2.8.1 Confidence Interval for a2.75
2.8.2 Confidence Intervals forCertain Parametric Functions
of a2 and o2.7^
2.8.3 Confidence Intervals for a~ .79
2.8.4 Confidence Interval for the Fixed Mean /r.86
2.8.5 A Numerical Example.86
2.9 Tests of Hypotheses.89
2.9.1 Test for er2 = r20.89
2.9.2 Test for a2 = 0.90
2.9.3 Test for 0 60.92
2.9.4 Test for (i = /xq.93
2.9.5 A Numerical Example.94
2.10 Optimum Sample Sizes for First and Second Stage Units . 94
2.11 An Alternate Formulation of the Model in (2.1.1).96
Exercises.98
Bibliography .105
Two-Way Crossed Classification without Interaction 115
3.1 Mathematical Model.115
3.2 Analysis of Variance .116
3.3 Minimal Sufficient Statistics and Distribution Theory.118
3.4 Classical Estimation.119
3.4.1 Analysis of Variance Estimators.119
3.4.2 Maximum Likelihood Estimators.120
3.4.3 Restricted Maximum Likelihood Estimators.122
3.4.4 Some Improvements over the ANOVA Estimators . . . 126
3.4.5 Some Improvements over the REML Estimators . 127
3.4.6 Hodges-Lehmann-Type Estimators of er2.127
3.4.7 A Numerical Example.128
3.5 Bayesian Estimation .-.133
3.5.1 Prior and Posterior Distribution Analysis .133
3.5.2 Some Formal Bayes Estimators.134
3.5.3 A Numerical Example.139
3.6 Sampling Distribution and Moments of Estimators .142
3.7 Comparison of Estimators using Mean Squared Error Criterion 143
3.8 Interval Estimation.244
3.8.1 Confidence Interval for ct2.144
3.8.2 Confidence Intervals for aj and ct2 /.145
Contents vii
3.8.3 Confidence Intervals for ct2 + ctp + 0"2.146
3.8.4 Confidence Intervals for o^/ct2 and o^/cr2.146
3.8.5 Confidence Intervals for ct2/(ct2+CT£+ct2),0£/(ot2+
°1 + al )' and Vef e + a} + °l) .147
3.8.6 Simultaneous Confidence Intervals for 0^/0} and
.149
3.8.7 Simultaneous Confidence Intervals for r2 and . . . 149
3.8.8 Confidence Intervals for o^/a^y cr^/(cr^+o^), and
+ al).150
3.8.9 Confidence Interval for the Fixed Mean fi.151
3.8.10 A Numerical Example.152
3.9 Tests of Hypotheses.156
3.9.1 Test for o*2 = r20.156
3.9.2 Test for 7^ = 0.156
3.9.3 Test for a2 = 0.157
3.9.4 A Numerical Example.158
Exercises .158
Bibliography .167
4 Two-Way Crossed Classification with Interaction 171
4.1 Mathematical Model.171
4.2 Analysis of Variance .172
4.3 Minimal Sufficient Statistics and Distribution Theory.175
4.4 Classical Estimation.176
4.4.1 Analysis of Variance Estimators.177
4.4.2 Maximum Likelihood Estimators .177
4.4.3 Restricted Maximum Likelihood Estimators.182
4.4.4 Other Estimators.185
4.4.5 A Numerical Example.185
4.5 Bayesian Estimation .188
4.5.1 Prior and Posterior Distribution Analysis .189
4.5.2 Some Formal Bayes Estimators.190
4.5.3 A Numerical Example.193
4.6 Sampling Distribution and Moments of Estimators .198
4.7 Interval Estimation.199
4.7.1 Confidence Interval for cr2.199
4.7.2 Confidence Intervals for cr2^, cr^, and cr2.200
4.7.3 Confidence Intervals for a2 4- a2^ + Op+o%.202
4.7.4 Confidence Intervals for o"2/ r2 and er^/cr2.203
4.7.5 Confidence Intervals for ^/( 72 + o-2^),
and a2/(a2 + o-2^).203
Contents
4.7.6 Confidence Intervals for o-2/(o-f2 + a£p + tp +CTa)
jj j(Ge+Gap Jr° j +aa )' Uaf$ / ^ ^~0p ^'
and all {a} + cr^ + aj + a*) . - - - • • .204
4.7.7 Confidence Intervals for a^/ap, ol/(crj+v«), and
^/(^ + f7«)./ ' 2/ V *2/ V '
4.7.8 Simultaneous Confidence Intervals for cra/aey fae,
and a^pl a}.^ \ ^
4.7.9 Simultaneous Confidence Intervals for ag, and a~ 209
4.7.10 Confidence Intervals for the Fixed Mean fi.209
4.7.11 A Numerical Example.210
4.8 Tests of Hypotheses.214
4.8.1 Test for a} ~ j}q.21^
4.8.2 Test for = 0 .215
4.8.3 Test for aj = 0.216
4.8.4 Test for = 0.218
4.8.5 A Numerical Example.218
Exercises .219
Bibliography .231
Three-Way and Higher-Order Crossed Classifications 235
5.1 Three-Way Crossed Classification with Interaction .235
5.1.1 Mathematical Model and Analysis of Variance . 235
5.1.2 Estimation of Variance Components.238
5.1.3 Interval Estimation.238
5.1.4 Tests of Hypotheses .242
5.2 Three-Way Crossed Classification with One Observation
Per Cell.246
5.3 Three-Way Crossed Classification without Interaction.248
5.4 Four-Way Crossed Classification.250
5.5 General r-Way Crossed Classification.252
5.6 The r-Way Crossed Classification without Interaction.254
5.6.1 Mathematical Model and Analysis of Variance . 255
5.6.2 Estimation of Variance Components.256
5.6.3 Interval Estimation.257
5.6.4 Bayesian Analysis.258
5.7 A Numerical Example.261
Exercises .267
Bibliography .272
Two-Way Nested Classification 277
6.1 Mathematical Model.277
6.2 Analysis of Variance .278
6.3 Minimal Sufficient Statistics and Distribution Theory.279
Contents ix
6.4 Classical Estimation.280
6.4.1 Analysis of Variance Estimators.280
6.4.2 Maximum Likelihood Estimators .280
6.4.3 Restricted Maximum Likelihood Estimators.283
6.4.4 Some Modifications of the ANOVA Estimators . 285
6.4.5 Some Modifications of the Maximum Likelihood
Estimators .286
6.4.6 Stein-Type Estimators.286
6.4.7 Hodges-Lehmann-Type Estimators of 7e2.287
6.4.8 Minimum Variance Unbiased Estimators of o^jo^
and ffa/cTf .287
6.4.9 A Numerical Example.288
6.5 Bayesian Estimation .294
6.5.1 Prior and Posterior Distribution Analysis .294
6.5.2 Some Formal Bayes Estimators.296
6.5.3 A Numerical Example.301
6.6 Sampling Distribution and Moments of Estimators .304
6.7 Comparison of Estimators Using Mean Squared Error Criterion 305
6.8 Interval Estimation.306
6.8.1 Confidence Interval for a}.306
6.8.2 Confidence Intervals for ajj and cr^ .307
6.8.3 Confidence Intervals for .308
6.8.4 Confidence Intervalsforcr^/cre2, cr^ficand
?/( ? + aj).308
6.8.5 Confidence Intervals for /a ?, and
°e/(0? + aa).309
6.8.6 Simultaneous Confidence Intervals for and
.310
6.8.7 Simultaneous Confidence Intervals for and a* . 311
6.8.8 Confidence intervals for 7^/(0^+
)' and °e!(ae +°l+aa) .312
6.8.9 Confidence Interval for the Fixed Mean fj,.315
6.8.10 A Numerical Example.315
6.9 Tests of Hypotheses.319
6.9.1 A Numerical Example.320
6.10 Estimation of Optimum Sample Sizes.321
Exercises .321
Bibliography .329
7 Three-Way and Higher-Order Nested Classifications 333
7.1 Mathematical Model and Analysis of Variance.333
7.2 Minimal Sufficient Statistics and Distribution Theory.335
7.3 Estimation of Variance Components.335
Contents
335
7.3.1 Analysis of Variance Estimators.
7.3.2 Maximum Likelihood Estimators .
7.3.3 Restricted Maximum Likelihood Estimators.
7.3.4 Sampling Distribution of the ANOVA Estimators . 343
7.3.5 Probability of Negative Estimates.344
7.3.6 A Numerical Example.347
7.4 Interval Estimation.
7.4.1 Confidence Intervals for a2, and oa.351
7.4.2 Confidence Intervals for o] + Jy + 4- .353
7.4.3 Confidence Intervals for 72/cr2, +o2), and
*e/( !?+ay).353
7.4.4 Confidence Intervals for + er2 + 4- ),
o2pl{o*+ol+ol + o%),o*J{o} + o^+ol + al),
and r2/( r2 + o-2 + -f r2).354
7.4.5 Simultaneous Confidence Intervals for r2/cr2, crj/a},
andcr2/(r2 .355
7.4.6 Confidence Interval for the Fixed Mean //.357
7.4.7 A Numerical Example.357
7.5 Tests of Hypotheses.360
7.5.1 A Numerical Example.360
7.6 Four-Way Nested Classification .361
7.7 Higher-Order Nested Classifications.363
7.7.1 Mathematical Model and Analysis of Variance . 363
7.7.2 Estimation of Variance Components.364
7.7.3 Confidence Intervals for Variance Components . 365
7.7.4 Tests of Hypotheses .367
7.7.5 Bayesian Analysis .367
Exercises .370
Bibliography .380
General Balanced Random Effects Model 383
8.1 Mathematical Model.383
8.2 Analysis of Variance .384
8.3 Estimation of Variance Components.384
8.4 Interval Estimation.385
8.4.1 Confidence Intervals for an Expected Mean Square
or a Ratio of Two Expected Mean Squares.385
8.4.2 Confidence Intervals for a linear Combination of
Expected Mean Squares with Positive Coefficients . . 386
8.4.3 Confidence Intervals for a Linear Combination of
Expected Mean Squares with Both Positive and
Negative Coefficients. 339
Contents xi
8.4.4 Confidence Intervals for a Ratio of Two Linear
Combinations of Expected Mean Squares.391
8.4.5 Confidence Regions for Variance Ratios.393
8.5 Tests of Hypotheses.395
8.5.1 Test of the Difference Between a*2 and a J2.397
8.5.2 Test of a Linear Combination of r*2s.398
8.5.3 A Numerical Example.400
8.6 Sampling Variances and Covariances of Estimators.402
8.7 Bayesian Analysis.404
Exercises .405
Bibliography .406
Appendices 409
A Two Useful Lemmas in Distribution Theory.409
B Some Useful Lemmas for a Certain Matrix.411
Bibliography.411
C Incomplete Beta Function.412
Bibliography.413
D Incomplete Inverted Dirichlet Function.413
Bibliography.414
E Inverted Chi-Square Distribution.415
F Satterthwaite Procedure.415
Bibliography.418
G Maximum Likelihood Estimation.418
H Some Useful Lemmas on Invariance Property of the ML
Estimators.420
I Complete Sufficient Statistics and the Rao-Blackwell and
Lehmann-Scheffe Theorems.421
J Point Estimators and MSE Criterion.421
Bibliography .422
K Likelihood Ratio Test.422
Bibliography .423
L Definition of Interaction .423
M Some Basic Results on Matrix Algebra.424
N Newton-Raphson, Fisher Scoring, and EM Algorithms . 433
Bibliography .435
O Software for Variance Component Analysis.436
Bibliography.440
General Bibliography 441
Author Index 465
Subject Index 473 |
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| spelling | Sahai, Hardeo 1942- Verfasser (DE-588)122112318 aut Analysis of variance for random models theory, methods, applications, and data analysis 1 Balanced data Hardeo Sahai ; Mario Miguel Ojeda Boston, Mass. [u.a.] Birkhäuser 2004 XXV, 484 S. txt rdacontent n rdamedia nc rdacarrier Ojeda, Mario Miguel Verfasser aut (DE-604)BV019810645 1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013136088&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sahai, Hardeo 1942- Ojeda, Mario Miguel Analysis of variance for random models theory, methods, applications, and data analysis |
| title | Analysis of variance for random models theory, methods, applications, and data analysis |
| title_auth | Analysis of variance for random models theory, methods, applications, and data analysis |
| title_exact_search | Analysis of variance for random models theory, methods, applications, and data analysis |
| title_full | Analysis of variance for random models theory, methods, applications, and data analysis 1 Balanced data Hardeo Sahai ; Mario Miguel Ojeda |
| title_fullStr | Analysis of variance for random models theory, methods, applications, and data analysis 1 Balanced data Hardeo Sahai ; Mario Miguel Ojeda |
| title_full_unstemmed | Analysis of variance for random models theory, methods, applications, and data analysis 1 Balanced data Hardeo Sahai ; Mario Miguel Ojeda |
| title_short | Analysis of variance for random models |
| title_sort | analysis of variance for random models theory methods applications and data analysis balanced data |
| title_sub | theory, methods, applications, and data analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013136088&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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