Variance Components (Wiley Series in Probability and Statistics)

Variance Components......Page 5
CONTENTS......Page 13
1. Introduction......Page 29
1.1. Factors, levels, cells and effects......Page 30
-i. Planned unbalancedness......Page 32
c. Unbalanced data......Page 34
Example 1 (Tomato varieties)......Page 35
Example 2 (Medications)......Page 36
Example 4 (Clinics)......Page 37
Example 5 (Dairy bulls)......Page 40
Example 6 (Ball bearings and calipers)......Page 41
Example 8 (Varieties and gardens)......Page 42
Example 9 (Mice and technicians)......Page 43
1.5. Finite populations......Page 44
b. Examples......Page 45
c. Fixed or random......Page 46
2.1. Analysis of variance......Page 47
a. Sources......Page 50
b. Pre-1900......Page 51
-i. R. A. Fisher......Page 53
-ii. L. C. Tippett......Page 55
-iii. The late 1930s......Page 57
-iv. Unbalanced data......Page 59
d. The 1940s......Page 60
2.3. Great strides: 1950-1969......Page 61
a. The Henderson methods......Page 62
-i. Negative estimates......Page 63
-ii. Unhiasedness......Page 64
-iii. Best unbiasedness......Page 65
-v. Lack of uniqueness......Page 66
a. Maximum likelihood (ML)......Page 68
c. Minimum norm estimation......Page 69
e. Bayes estimation......Page 70
f: The recent decade......Page 71
a. The model equation......Page 72
c. Second moments......Page 73
a. Example 1......Page 75
b. The general case......Page 76
-i. The traditional random model......Page 77
-ii. Other alternatives......Page 78
-i. Example 2......Page 79
3.3 Estimating the mean......Page 80
3.4 Predicting random effects......Page 82
a. Expected sums of squares......Page 85
-ii. Using the matrix formulation......Page 86
b. ANOVA estimators......Page 87
c. Negative estimates......Page 88
-i. X2-distributions of sums of squares......Page 90
-iii. Sampling variances of estimators......Page 91
-iv. An F-statistic to test H:σ 2/α = 0......Page 92
-v. Confidence intervals......Page 93
-vi. Probability of a negative estimate......Page 94
-i. A direct derivation......Page 97
-ii. Using the matrix formulation......Page 98
b. ANOVA estimators......Page 99
c. Negative estimates......Page 100
-ii. Independence of sums of squares......Page 101
-iii. Sampling variances of estimators......Page 102
-iv. The eflect of unhalancedness on sampling variances......Page 103
-vi. Confidence intervals......Page 104
3.7. Maximum likelihood estimation......Page 106
-i. Likelihood......Page 107
-ii. ML equations and their solutions......Page 108
-iii. ML estimators......Page 109
-iv. Expected values and bias......Page 112
-v. Sampling variances......Page 113
-i. Likelihood......Page 114
-ii. ML equations and their solutions......Page 115
-v. Sampling variances......Page 116
3.8. Restricted maximum likelihood estimation (REML)......Page 118
-ii. REML equations and their solutions......Page 119
-iv. Comparison with ANOVA and ML......Page 120
b. Unbalanced data......Page 121
a. A simple sample......Page 122
b. The 1-way classifcation, random model......Page 125
c. Balanced data......Page 127
a. Balanced data......Page 131
b. Unbalanced data......Page 134
3.11. Exercises......Page 136
4. Balanced Data......Page 140
c. Interactions......Page 141
e. Sums of squares......Page 142
f. Calculating sums qf squares......Page 143
4.2. Expected mean squares, E(MS)......Page 144
b. Analysis of variance table......Page 146
c. Expected mean squares,......Page 147
-i. The fixed effects model......Page 149
-iii. The mixed model......Page 150
-iv. A mixed model with Σ-restrictions......Page 151
d. ANOVA estimators of variance components......Page 155
4.4. ANOVA estimation......Page 156
a. Distribution of mean squares......Page 159
b. Distribution of estimators......Page 160
c. Tests of hypotheses......Page 161
d. Confidence intervals......Page 163
f. Sampling variances of estimators......Page 165
a. The general mixed model......Page 166
-i. Model equation......Page 168
c. The 2-way nested classifcation......Page 170
e. The general case......Page 171
-ii. Dispersion matrix......Page 172
a. Estimating the mean in random models......Page 174
-i. The 1-way random model......Page 175
-ii. The 2-way nested random model......Page 176
-iii. The 2-way crossed, with interaction, mixed model......Page 177
-iv. The 2-way crossed, no interaction, mixed model......Page 178
d. The 2-way crossed classification, random model......Page 179
-i. With interaction......Page 180
e. Existence of explicit solutions......Page 181
f. Asymptotic sampling variances ,for the 2-way crossed classification......Page 182
-ii. The 2-way crossed classification, no interaction, random model......Page 183
-iv. The 2-way crossed classification, no interaction, mixed model......Page 184
-i. The 2-way nested classification, random model......Page 185
4.8. Restricted maximum likelihood......Page 186
4.9. Estimating fixed effects in mixed models......Page 187
4.10. Summary......Page 189
4.11. Exercises......Page 191
5. Analysis of Variance Estimation for Unbalanced Data......Page 196
a. Data......Page 197
-i. Example 1-the 2-way crossed classification random model......Page 198
-ii. Dispersion matrix......Page 199
-i. The general case......Page 200
-iii. Example 1 (continued)......Page 201
c. Unbiasedness......Page 203
-i. A general result......Page 204
-ii. Example 2 (continued)......Page 205
-iii. A direct approach......Page 206
e. Unbiased estimation of sampling variances......Page 207
a. The quadratic forms......Page 209
b. Estimation......Page 211
d. Sampling variances......Page 212
e. A general coeflcient for Method......Page 214
f. Synthesis......Page 215
g. Mixed models......Page 216
i. A numerical example......Page 217
5.4. Henderson\'s Method II......Page 218
a. Estimating the fixed effects......Page 219
b. Calculation......Page 220
-i. The matrix XLZ is null......Page 222
-iii. All rows of X—XLX are the same......Page 223
-i. Rank properties......Page 224
-ii. An estimable function......Page 225
e. Coefficients of σ 2/e......Page 226
f. No fixed-by-random interactions......Page 227
g. Merits and demerits......Page 229
-i. Reductions in sum of squares......Page 230
-ii. Expected sums of squares......Page 231
b. Mixed models......Page 232
c. A general result......Page 234
d. Sampling variances......Page 235
5.6. Method III applied to the 2-way crossed classification......Page 236
-i. One set of sums of squares......Page 237
-ii. Three sets of sums of squares......Page 238
c. With interaction, random model......Page 241
-i. One set of sums of squares......Page 242
-ii. Three sets of sums of squares......Page 243
-iv. Sampling variances......Page 245
5.7. Nested models......Page 246
a. Unweighted means method: all cells filled......Page 247
b. Weighted squares of means: all cells filled......Page 248
5.9 Comparing different forms of ANOVA estimation......Page 249
5.10 Estimating fixed effects in mixed models......Page 253
5.11 Summary......Page 254
5.12 Exercises......Page 255
6. Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML)......Page 260
6.1. The model and likelihood function......Page 261
a. A direct derivation......Page 262
b. An alternative form......Page 264
c. The Hartley-Rao form......Page 265
a. For variance components......Page 266
b. For ratios of components......Page 268
c. Maximum?......Page 269
6.4. Some remarks on computing......Page 270
-i. Notation......Page 271
-ii. lnverse of V......Page 272
-iii. The estimation equations......Page 273
-iv. Information matrix......Page 275
6.6. Restricted maximum likelihood (REML)......Page 277
a. Linear combinations of observations......Page 278
c. An alternative form......Page 279
e. The information matrix......Page 280
g. Using cell means models for the fixed effects......Page 281
6.8 ML or REML?......Page 282
6.9 Summary......Page 283
6.10 Exercises......Page 284
7.1. Introduction......Page 286
a. The best predictor......Page 289
b. Mean and variance properties......Page 290
c. Two properties of the best predictor of a scalar......Page 291
-ii. Maximizing the mean of a selected proportion......Page 292
a. BLP(u)......Page 293
b. Example......Page 294
c. Derivation......Page 295
d. Ranking......Page 296
a.Combining Jixed and random effects......Page 297
b. Example......Page 298
c. Derivation of BLUP......Page 299
d. Variances and covariances......Page 300
a. A two-stage derivation......Page 301
c. Partitioning y into two parts......Page 302
a. Derivation......Page 303
b. Solutions......Page 304
-i. The estimation equations......Page 306
-ii. The information matrix......Page 308
-i. The estimation equations......Page 310
-ii. Using the MME......Page 312
-iv. Iterating for REML......Page 313
7.7. Summary......Page 314
7.8. Exercises......Page 315
8.1. Introduction......Page 318
a. The basis of the methods......Page 320
b. The Newton-Raphson and Marquardt methods......Page 321
e. Obtaining starting values......Page 323
h. Easing the computational burden......Page 324
a. A general formulation......Page 325
b. Distributional derivations needed for the EM algorithm......Page 326
c. EM algorithm for ML estimation (Version 1)......Page 328
d. EM algorithm for ML estimation (Version 2)......Page 329
f. EM algorithm for REML estimation......Page 330
g. A Bayesian justification for REML......Page 331
8.4. General methods that converge rapidly for balanced data......Page 332
8.5. Pooling estimators from subsets of a large data set......Page 333
8.6 Example: the 1-way random model......Page 335
a. The EM algorithm (Version 1)......Page 336
b. The method of scoring algorithm......Page 338
a. Computing packages......Page 339
b. Elialuation of algorithms......Page 340
8.8. Summary......Page 341
8.9. Exercises......Page 342
a. Introduction......Page 343
b. Simple examples......Page 344
c. The mixed model hierarchy......Page 345
d. The normal hierarchy......Page 346
e. Point estimator of variance or variance of point estimator?......Page 347
b. Likelihood methods......Page 349
-ii. Estimation......Page 353
-iii. Connections with likelihood......Page 354
a. Hierarchical estimation......Page 355
-i. Estimation of β......Page 356
-ii. Estimation of u......Page 358
-i. Exploiting the multivariate normal structure......Page 359
c. The I-way classifcation, random model......Page 361
i. Estimation of μ......Page 362
ii. Estimation of α......Page 363
-i. The 1-way classifcation......Page 365
-ii. Cautions......Page 366
-iii. Variance approximations......Page 369
9.4. Other types of hierarchies......Page 371
a. A beta-binomial model......Page 372
b. A generalized linear model......Page 375
a. Computational problems......Page 377
b. Hierarchical EM......Page 378
9.6. Philosophical considerations in hierarchical modeling......Page 380
9.7. Summary......Page 383
9.8. Exercises......Page 387
10.1. Introduction......Page 395
a. Introduction......Page 397
b. Model specification......Page 398
d. Discussion......Page 399
10.4. Logit-normal models......Page 400
b. An example......Page 401
10.6. Discussion......Page 402
10.7. Summary......Page 403
10.8. Exercises......Page 404
11.1. Estimating components of covariance......Page 406
a. Easy ANOVA estimation for certain models......Page 407
b. Examples of covariance components models......Page 408
-ii. Covariances between effects of different random factors......Page 409
c. Combining variables into a single vector......Page 410
-i. Estimation equations......Page 411
-ii. Large-sample dispersion matrix......Page 413
-i. Estimation equations......Page 414
11.2. Modeling variance components as covariances......Page 415
a. All-cells-filled data......Page 416
c. Diagnostic opportunities......Page 417
d. Some-cells-empty data......Page 418
11.3. Criteria-based procedures......Page 419
-ii. Translation invariance......Page 420
-i. Class C0: unrestricted......Page 421
c. Minimum variance estimation (MINVAR)......Page 422
d. Minimum norm estimation (MINQUE)......Page 425
e. REML, MlNQUE and I-MINQUE......Page 426
h. MINQUE for the 1-way classification......Page 427
11.4 Summary......Page 428
11.5 Exercises......Page 430
12.1. The model......Page 433
12.2. Ordinary least squares (OLS) yields MINQUEO......Page 434
-i. A normalizing transformation......Page 435
-ii. Example......Page 436
-iii. The general form of E(ww\' [omitted] ww\')......Page 438
-i. General case......Page 439
d. Variance of a translation-invariant quadratic form......Page 440
a. GLS yields REML equations under normality......Page 441
c. REML is BLUE......Page 443
12.5 Modified GLS yields ML......Page 444
-i. History......Page 445
-ii. The model......Page 447
-i. The model......Page 449
-ii. ANOVA estimation......Page 450
12.7. Non-negative estimation......Page 452
12.8 Summary......Page 453
12.9. Exercises......Page 454
a. Model......Page 455
e. Large-sample variances of maximum likelihood estimators (under normality)......Page 456
c. Variances of analysis of variance estimators (under normality)......Page 457
d. Large-snmple variances of maximum likelihood estimators (under normality)......Page 458
b. Analysis of variance estimators......Page 459
c. Variances of analysis oj variance estimators (under normality)......Page 460
b. Henderson Method I estimators......Page 462
c. Variances of Henderson Method I estimators (under normality)......Page 463
d. Henderson Method III estimators......Page 465
b. Henderson Method Ill......Page 466
c. Variances of Henderson Method I estimators (under normality)......Page 467
e. Variances of Henderson Method Ill estimators (under normality)......Page 468
b. Henderson Method III......Page 469
M.1. Summing vectors, and J-matrices......Page 470
M.2. Direct sums and products......Page 471
M.3. A matrix notation in terms of elements......Page 473
a. Definitions......Page 475
b. Generalized inverses of X\'X......Page 476
c. Partitioning X\'X......Page 478
f. A theorem involving K\' of maximum row rank for K\'X being null......Page 479
M.5. The Schur complement......Page 481
a. Scalars......Page 482
d. Quadratic jorms......Page 483
f. Determinants......Page 484
g. Traces......Page 485
M.8. The operators vec and vech......Page 486
M.9. Vec permutation matrices......Page 487
M.10 The equality VV-X = X......Page 488
S.1. Conditional first and second moments......Page 489
S.2 Least squares estimation......Page 490
S.3 Normal and x2-distributions......Page 491
b. Mean squares......Page 492
S.4 F-distributions......Page 493
S.5 Quadratic forms......Page 494
a. Density junctions......Page 495
b. Bayes Theorem......Page 496
d. Example......Page 497
e. Empirical Bayes estimation......Page 499
b. Maximum likelihood estimation......Page 500
c. Asymptotic dispersion matrix......Page 501
d. Transforming parameters......Page 502
References......Page 503
List of  Tables and Figures......Page 518
Author Index......Page 521
Subject Index......Page 525