Analysis of variance for random models, vol.2: Unbalanced data

Contents......Page 5
List of Figures......Page 13
List of Tables......Page 15
Preface......Page 19
Acknowledgments......Page 23
9 Matrix Preliminaries and General Linear Model......Page 27
9.1 GENERALIZED INVERSE OF A MATRIX......Page 29
9.3 QUADRATIC FORMS......Page 30
9.4 GENERAL LINEAR MODEL......Page 32
EXERCISES......Page 35
Bibliography......Page 36
10 Some General Methods for Making Inferences about Variance Components......Page 39
10.1 HENDERSON’S METHOD I......Page 40
10.2 HENDERSON’S METHOD II......Page 42
10.3 HENDERSON’S METHOD III......Page 44
10.4 ANALYSIS OF MEANS METHOD......Page 48
10.5 SYMMETRIC SUMS METHOD......Page 49
10.6 ESTIMATION OF POPULATION MEAN IN A RANDOM EFFECTS MODEL......Page 51
10.7 MAXIMUM LIKELIHOOD ESTIMATION......Page 53
10.8 RESTRICTED MAXIMUM LIKELIHOOD ESTIMATION......Page 59
10.10 MINIMUM-NORM AND MINIMUM-VARIANCE QUADRATIC UNBIASED ESTIMATION......Page 66
10.11 MINIMUM MEAN SQUARED ERROR QUADRATIC ESTIMATION......Page 76
10.12 NONNEGATIVE QUADRATIC UNBIASED ESTIMATION......Page 79
10.13 OTHER MODELS, PRINCIPLES AND PROCEDURES......Page 81
10.14 RELATIVE MERITS AND DEMERITS OF GENERAL METHODS OF ESTIMATION......Page 88
10.15 COMPARISONS OF DESIGNS AND ESTIMATORS......Page 90
10.16 METHODS OF HYPOTHESIS TESTING......Page 92
10.17 METHODS FOR CONSTRUCTING CONFIDENCE INTERVALS......Page 95
EXERCISES......Page 96
Bibliography......Page 102
11.1 MATHEMATICAL MODEL......Page 119
11.2 ANALYSIS OF VARIANCE......Page 120
11.3 MINIMAL SUFFICIENT STATISTICS AND DISTRIBUTION THEORY......Page 122
11.4 CLASSICAL ESTIMATION......Page 123
11.5 BAYESIAN ESTIMATION......Page 143
11.6 DISTRIBUTION AND SAMPLING VARIANCES OF ESTIMATORS......Page 148
11.7 COMPARISONS OF DESIGNS AND ESTIMATORS......Page 157
11.8 CONFIDENCE INTERVALS......Page 162
11.9 TESTS OF HYPOTHESES......Page 172
EXERCISES......Page 175
Bibliography......Page 181
12.2 ANALYSIS OF VARIANCE......Page 191
12.3 EXPECTED MEAN SQUARES......Page 193
12.4 ESTIMATION OF VARIANCE COMPONENTS......Page 195
12.5 VARIANCES OF ESTIMATORS......Page 212
12.6 CONFIDENCE INTERVALS......Page 216
12.7 TESTS OF HYPOTHESES......Page 219
EXERCISES......Page 221
Bibliography......Page 224
13.2 ANALYSIS OF VARIANCES......Page 227
13.3 EXPECTED MEAN SQUARES......Page 230
13.4 ESTIMATION OF VARIANCE COMPONENTS......Page 232
13.5 VARIANCES OF ESTIMATORS......Page 252
13.6 COMPARISONS OF DESIGNS AND ESTIMATORS......Page 256
13.7 CONFIDENCE INTERVALS......Page 264
13.8 TESTS OF HYPOTHESES......Page 266
EXERCISES......Page 269
Bibliography......Page 276
14.1 MATHEMATICAL MODEL......Page 281
14.2 ANALYSIS OF VARIANCE......Page 282
14.3 EXPECTED MEAN SQUARES......Page 284
14.4 ESTIMATION OF VARIANCE COMPONENTS......Page 287
14.5 VARIANCES OF ESTIMATORS......Page 290
14.6 GENERAL r-WAY CROSSED CLASSIFICATION......Page 292
14.7 A NUMERICAL EXAMPLE......Page 294
EXERCISES......Page 298
APPENDIX: COEFFICIENTS Aij OF PRODUCTS OF VARIANCE COMPONENTS IN COVARIANCE MATRIX OF T......Page 302
Bibliography......Page 311
15.1 MATHEMATICAL MODEL......Page 313
15.2 ANALYSIS OF VARIANCE......Page 314
15.3 EXPECTED MEAN SQUARES......Page 315
15.4 DISTRIBUTION THEORY......Page 318
15.5 UNWEIGHTED MEANS ANALYSIS......Page 319
15.6 ESTIMATION OF VARIANCE COMPONENTS......Page 321
15.7 VARIANCES OF ESTIMATORS......Page 329
15.8 COMPARISONS OF DESIGNS AND ESTIMATORS......Page 332
15.9 CONFIDENCE INTERVALS......Page 336
15.10 TESTS OF HYPOTHESES......Page 343
EXERCISES......Page 347
Bibliography......Page 351
16.1 MATHEMATICAL MODEL......Page 355
16.2 ANALYSIS OF VARIANCE......Page 356
16.3 EXPECTED MEAN SQUARES......Page 357
16.4 UNWEIGHTED MEANS ANALYSIS......Page 360
16.5 ESTIMATION OF VARIANCE COMPONENTS......Page 362
16.6 VARIANCES OF ESTIMATORS......Page 373
16.7 COMPARISONS OF DESIGNS AND ESTIMATORS......Page 379
16.8 CONFIDENCE INTERVALS AND TESTS OF HYPOTHESES......Page 385
EXERCISES......Page 388
Bibliography......Page 393
17.1 MATHEMATICAL MODEL......Page 397
17.2 ANALYSIS OF VARIANCE......Page 398
17.3 EXPECTED MEAN SQUARES......Page 401
17.4 ESTIMATION OF VARIANCE COMPONENTS......Page 402
17.6 CONFIDENCE INTERVALS AND TESTS OF HYPOTHESES......Page 408
17.7 A NUMERICAL EXAMPLE......Page 411
EXERCISES......Page 413
Bibliography......Page 414
A TWO USEFUL LEMMAS IN DISTRIBUTION THEORY......Page 417
B SOME USEFUL LEMMAS FOR A CERTAIN MATRIX......Page 419
C INCOMPLETE BETA FUNCTION......Page 420
D INCOMPLETE INVERTED DIRICHLET FUNCTION......Page 421
F THE SATTERTHWAITE PROCEDURE......Page 423
G MAXIMUM LIKELIHOOD ESTIMATION......Page 426
H SOME USEFUL LEMMAS ON THE INVARIANCE PROPERTY OF THE ML ESTIMATORS......Page 428
J POINT ESTIMATORS AND THE MSE CRITERION......Page 429
K LIKELIHOOD RATIO TEST......Page 431
SOME DEFINITIONS......Page 432
SUMS AND PRODUCTS OF MATRICES......Page 434
THE RANK OF A MATRIX......Page 436
THE INVERSE OF A MATRIX......Page 437
PARTITIONED MATRICES......Page 438
DIFFERENTIATION OF MATRICES AND VECTORS......Page 439
NEWTON–RAPHSON......Page 441
THE EM ALGORITHM......Page 442
O SOFTWARE FOR VARIANCE COMPONENT ANALYSIS......Page 444
General Bibliography......Page 451
Author Index......Page 485
Subject Index......Page 495