Preface xi
1 The R-Package, Sampling Procedures, and Random Variables 1
1.1 Introduction 1
1.2 The Statistical Software Package R 1
1.3 Sampling Procedures and Random Variables 4
References 10
2 Point Estimation 11
2.1 Introduction 11
2.2 Estimating Location Parameters 12
2.2.1 Maximum Likelihood Estimation of Location Parameters 17
2.2.2 Estimating Expectations from Censored Samples and Truncated Distributions 20
2.2.3 Estimating Location Parameters of Finite Populations 23
2.3 Estimating Scale Parameters 24
2.4 Estimating Higher Moments 27
2.5 Contingency Tables 29
2.5.1 Models of Two-Dimensional Contingency Tables 29
2.5.1.1 Model I 29
2.5.1.2 Model II 29
2.5.1.3 Model III 30
2.5.2 Association Coefficients for 2 ×2 Tables 30
References 38
3 Testing Hypotheses - One- and Two-Sample Problems 39
3.1 Introduction 39
3.2 The One-Sample Problem 41
3.2.1 Tests on an Expectation 41
3.2.1.1 Testing the Hypothesis on the Expectation of a Normal Distribution with Known Variance 41
3.2.1.2 Testing the Hypothesis on the Expectation of a Normal Distribution with Unknown Variance 47
3.2.2 Test on the Median 51
3.2.3 Test on the Variance of a Normal Distribution 54
3.2.4 Test on a Probability 56
3.2.5 Paired Comparisons 57
3.2.6 Sequential Tests 59
3.3 The Two-Sample Problem 63
3.3.1 Tests on Two Expectations 63
3.3.1.1 The Two-Sample t-Test 63
3.3.1.2 The Welch Test 66
3.3.1.3 The Wilcoxon Rank Sum Test 70
3.3.1.4 Definition of Robustness and Results of Comparing Tests by Simulation 72
3.3.1.5 Sequential Two-Sample Tests 74
3.3.2 Test on Two Medians 76
3.3.2.1 Rationale 77
3.3.3 Test on Two Probabilities 78
3.3.4 Tests on Two Variances 79
References 81
4 Confidence Estimations - One- and Two-Sample Problems 83
4.1 Introduction 83
4.2 The One-Sample Case 84
4.2.1 A Confidence Interval for the Expectation of a Normal Distribution 84
4.2.2 A Confidence Interval for the Variance of a Normal Distribution 91
4.2.3 A Confidence Interval for a Probability 93
4.3 The Two-Sample Case 96
4.3.1 A Confidence Interval for the Difference of Two Expectations - Equal Variances 96
4.3.2 A Confidence Interval for the Difference of Two Expectations - Unequal Variances 98
4.3.3 A Confidence Interval for the Difference of Two Probabilities 100
References 104
5 Analysis of Variance (ANOVA) - Fixed Effects Models 105
5.1 Introduction 105
5.1.1 Remarks about Program Packages 106
5.2 Planning the Size of an Experiment 106
5.3 One-Way Analysis of Variance 108
5.3.1 Analysing Observations 109
5.3.2 Determination of the Size of an Experiment 112
5.4 Two-Way Analysis of Variance 115
5.4.1 Cross-Classification (A× B) 115
5.4.1.1 Parameter Estimation 117
5.4.1.2 Testing Hypotheses 119
5.4.2 Nested Classification (A¿B) 131
5.5 Three-Way Classification 134
5.5.1 Complete Cross-Classification (A×B ×C) 135
5.5.2 Nested Classification (C ¿B¿A) 144
5.5.3 Mixed Classifications 147
5.5.3.1 Cross-Classification between Two Factors where One of Them Is Sub-Ordinated to a Third Factor ((B¿A)xC) 148
5.5.3.2 Cross-Classification of Two Factors, in which a Third Factor is Nested (C¿(A× B)) 153
References 157
6 Analysis of Variance -Models with Random Effects 159
6.1 Introduction 159
6.2 One-Way Classification 159
6.2.1 Estimation of the Variance Components 160
6.2.1.1 ANOVA Method 160
6.2.1.2 Maximum Likelihood Method 164
6.2.1.3 REML - Estimation 166
6.2.2 Tests of Hypotheses and Confidence Intervals 169
6.2.3 Expectation and Variances of the ANOVA Estimators 174
6.3 Two-Way Classification 176
6.3.1 Two-Way Cross Classification 176
6.3.2 Two-Way Nested Classification 182
6.4 Three-Way Classification 186
6.4.1 Three-Way Cross-Classification with Equal Sub-Class Numbers 186
6.4.2 Three-Way Nested Classification 192
6.4.3 Three-Way Mixed Classifications 195
6.4.3.1 Cross-Classification Between Two Factors Where One of Them is Sub-Ordinated to a Third Factor ((B¿A)×C) 195
6.4.3.2 Cross-Classification of Two Factors in Which a Third Factor is Nested (C¿(A×B)) 197
References 199
7 Analysis of Variance -Mixed Models 201
7.1 Introduction 201
7.2 Two-Way Classification 201
7.2.1 Balanced Two-Way Cross-Classification 201
7.2.2 Two-Way Nested Classification 214
7.3 Three-Way Layout 223
7.3.1 Three-Way Analysis of Variance - Cross-Classification A × B × C 223
7.3.2 Three-Way Analysis of Variance - Nested Classification A¿B¿C 230
7.3.2.1 Three-Way Analysis of Variance - Nested Classification - Model III - Balanced Case 230
7.3.2.2 Three-Way Analysis of Variance - Nested Classification - Model IV - Balanced Case 232
7.3.2.3 Three-Way Analysis of Variance - Nested Classification - Model V - Balanced Case 234
7.3.2.4 Three-Way Analysis of Variance - Nested Classification - Model VI - Balanced Case 236
7.3.2.5 Three-Way Analysis of Variance - Nested Classification - Model VII - Balanced Case 237
7.3.2.6 Three-Way Analysis of Variance - Nested Classification - Model VIII - Balanced Case 238
7.3.3 Three-Way Analysis of Variance - Mixed Classification - (A× B)¿C 239
7.3.3.1 Three-Way Analysis of Variance - Mixed Classification - (A× B)¿C Model III 239
7.3.3.2 Three-Way Analysis of Variance - Mixed Classification - (A× B)¿C Model IV 242
7.3.3.3 Three-Way Analysis of Variance - Mixed Classification - (A× B)¿C Model V 243
7.3.3.4 Three-Way Analysis of Variance - Mixed Classification - (A× B)¿C Model VI 245
7.3.4 Three-Way Analysis of Variance - Mixed Classification - (A¿B) ×C 247
7.3.4.1 Three-Way Analysis of Variance - Mixed Classification - (A¿B) ×C Model III 247
7.3.4.2 Three-Way Analysis of Variance - Mixed Classification - (A¿B) ×C Model IV 249
7.3.4.3 Three-Way Analysis of Variance - Mixed Classification - (A¿B) ×C Model V 251
7.3.4.4 Three-Way Analysis of Variance - Mixed Classification - (A¿B) ×C Model VI 253
7.3.4.5 Three-Way Analysis of Variance - Mixed Classification - (A¿B) ×C model VII 254
7.3.4.6 Three-Way Analysis of Variance - Mixed Classification - (A¿B) ×C Model VIII 255
References 256
8 Regression Analysis 257
8.1 Introduction 257
8.2 Regression with Non-Random Regressors - Model I of Regression 262
8.2.1 Linear and Quasilinear Regression 262
8.2.1.1 Parameter Estimation 263
8.2.1.2 Confidence Intervals and Hypotheses Testing 274
8.2.2 Intrinsically Non-Linear Regression 282
8.2.2.1 The Asymptotic Distribution of the Least Squares Estimators 283
8.2.2.2 The Michaelis-Menten Regression 285
8.2.2.3 Exponential Regression 290
8.2.2.4 The Logistic Regression 298
8.2.2.5 The Bertalanffy Function 306
8.2.2.6 The Gompertz Function 312
8.2.3 Optimal Experimental Designs 315
8.2.3.1 Simple Linear and Quasilinear Regression 316
8.2.3.2 Intrinsically Non-linear Regression 317
8.2.3.3 The Michaelis-Menten Regression 319
8.2.3.4 Exponential Regression 319
8.2.3.5 The Logistic Regression 320
8.2.3.6 The Bertalanffy Function 321
8.2.3.7 The Gompertz Function 321
8.3 Models with Random Regressors 322
8.3.1 The Simple Linear Case 322
8.3.2 The Multiple Linear Case and the Quasilinear Case 330
8.3.2.1 Hypotheses Testing - General 333
8.3.2.2 Confidence Estimation 333
8.3.3 The Allometric Model 334
8.3.4 Experimental Designs 335
References 335
9 Analysis of Covariance (ANCOVA) 339
9.1 Introduction 339
9.2 Completely Randomised Design with Covariate 340
9.2.1 Balanced Completely Randomised Design 340
9.2.2 Unbalanced Completely Randomised Design 350
9.3 Randomised Complete Block Design with Covariate 358
9.4 Concluding Remarks 365
References 366
10 Multiple Decision Problems 367
10.1 Introduction 367
10.2 Selection Procedures 367
10.2.1 The Indifference Zone Formulation for Selecting Expectations 368
10.2.1.1 Indifference Zone Selection, ¿2 Known 368
10.2.1.2 Indifference Zone Selection, ¿2 Unknown 371
10.3 The Subset Selection Procedure for Expectations 371
10.4 Optimal Combination of the Indifference Zone and the Subset Selection Procedure 372
10.5 Selection of the Normal Distribution with the Smallest Variance 375
10.6 Multiple Comparisons 375
10.6.1 The Solution of MC Problem 10.1 377
10.6.1.1 The F-test for MC Problem 10.1 377
10.6.1.2 Scheffé's Method for MC Problem 10.1 378
10.6.1.3 Bonferroni's Method for MC Problem 10.1 379
10.6.1.4 Tukey's Method for MC Problem 10.1 for ni = n 382
10.6.1.5 Generalised Tukey's Method for MC Problem 10.1 for ni ¿n 383
10.6.2 The Solution of MC Problem 10.2 - the Multiple t-Test 384
10.6.3 The Solution of MC Problem 10.3 - Pairwise and Simultaneous Comparisons with a Control 385
10.6.3.1 Pairwise Comparisons - The Multiple t-Test 385
10.6.3.2 Simultaneous Comparisons -The Dunnett Method 387
References 390
11 Generalised Linear Models 393
11.1 Introduction 393
11.2 Exponential Families of Distributions 394
11.3 Generalised Linear Models - An Overview 396
11.4 Analysis - Fitting a GLM - The Linear Case 398
11.5 Binary Logistic Regression 399
11.5.1 Analysis 400
11.5.2 Overdispersion 408
11.6 Poisson Regression 411
11.6.1 Analysis 411
11.6.2 Overdispersion 417
11.7 The Gamma Regression 417
11.8 GLM for Gamma Regression 418
11.9 GLM for the Multinomial Distribution 425
References 428
12 Spatial Statistics 429
12.1 Introduction 429
12.2 Geostatistics 431
12.2.1 Semi-variogram Function 432
12.2.2 Semi-variogram Parameter Estimation 439
12.2.3 Kriging 440
12.2.4 Trans-Gaussian Kriging 446
12.3 Special Problems and Outlook 450
12.3.1 Generalised Linear Models in Geostatistics 450
12.3.2 Copula Based Geostatistical Prediction 451
References 451
Appendix A List of Problems 455
Appendix B Symbolism 483
Appendix C Abbreviations 485
Appendix D Probability and Density Functions 487
Index 489
