Regression (eBook)

Linear Models in Statistics

N. H. Bingham, John M. Fry (Autoren)

eBook Download: PDF

2010 | 1. Auflage
293 Seiten
Springer London (Verlag)
978-1-84882-969-5 (ISBN)

Infos

Regression is the branch of Statistics in which a dependent variable of interest is modelled as a linear combination of one or more predictor variables, together with a random error. The subject is inherently two- or higher- dimensional, thus an understanding of Statistics in one dimension is essential.


Regression: Linear Models in Statistics fills the gap between introductory statistical theory and more specialist sources of information. In doing so, it provides the reader with a number of worked examples, and exercises with full solutions.


The book begins with simple linear regression (one predictor variable), and analysis of variance (ANOVA), and then further explores the area through inclusion of topics such as multiple linear regression (several predictor variables) and analysis of covariance (ANCOVA). The book concludes with special topics such as non-parametric regression and mixed models, time series, spatial processes and design of experiments.


Aimed at 2nd and 3rd year undergraduates studying Statistics, Regression: Linear Models in Statistics requires a basic knowledge of (one-dimensional) Statistics, as well as Probability and standard Linear Algebra. Possible companions include John Haigh's Probability Models, and T. S. Blyth & E.F. Robertsons' Basic Linear Algebra and Further Linear Algebra.

Inhalt

Preface 
8 
Contents 12
1. Linear Regression 16
1.1 Introduction 16
1.2 The Method of Least Squares 18
1.2.1 Correlation version 22
1.2.2 Large-sample limit 23
1.3 The origins of regression 24
1.4 Applications of regression 26
1.5 The Bivariate Normal Distribution 29
1.6 Maximum Likelihood and Least Squares 36
1.7 Sums of Squares 38
1.8 Two regressors 41
Exercises 43
2. The Analysis of Variance (ANOVA) 48
2.1 The Chi-Square Distribution 48
2.2 Change of variable formula and Jacobians 51
2.3 The Fisher F-distribution 52
2.4 Orthogonality 53
2.5 Normal sample mean and sample variance 54
2.6 One-Way Analysis of Variance 57
2.7 Two-Way ANOVA No Replications
2.8 Two-Way ANOVA: Replications and Interaction 67
Exercises 71
3. Multiple Regression 75
3.1 The Normal Equations 75
3.2 Solution of the Normal Equations 78
3.3 Properties of Least-Squares Estimators 84
3.4 Sum-of-Squares Decompositions 87
3.4.1 Coefficient of determination 93
3.5 Chi-Square Decomposition 94
3.5.1 Idempotence, Trace and Rank 95
3.5.2 Quadratic forms in normal variates 96
3.5.3 Sums of Projections 96
3.6 Orthogonal Projections and Pythagoras's Theorem 99
3.7 Worked examples 103
Exercises 108
4. Further Multilinear Regression 112
4.1 Polynomial Regression 112
4.1.1 The Principle of Parsimony 115
4.1.2 Orthogonal polynomials 116
4.1.3 Packages 116
4.2 Analysis of Variance 117
4.3 The Multivariate Normal Distribution 118
4.4 The Multinormal Density 124
4.4.1 Estimation for the multivariate normal 126
4.5 Conditioning and Regression 128
4.6 Mean-square prediction 134
4.7 Generalised least squares and weighted regression 136
Exercises 138
5. Adding additional covariates and the Analysisof Covariance 141
5.1 Introducing further explanatory variables 141
5.1.1 Orthogonal parameters 145
5.2 ANCOVA 147
Interactions 148
5.2.1 Nested Models 151
Update 151
Akaike Information Criterion (AIC) 152
Step 152
5.3 Examples 152
Exercises 157
6. Linear Hypotheses 161
6.1 Minimisation Under Constraints 161
6.2 Sum-of-Squares Decomposition and F-Test 164
6.3 Applications: Sequential Methods 169
6.3.1 Forward selection 169
6.3.2 Backward selection 170
6.3.3 Stepwise regression 171
Exercises 172
7. Model Checking and Transformation of Data 175
7.1 Deviations from Standard Assumptions 175
Residual Plots 175
Scatter Plots 175
Non-constant Variance 176
Unaccounted-for Structure 176
Outliers 176
Detecting outliers via residual analysis 177
Influential Data Points 178
Cook's distance 179
Non-additive or non-Gaussian errors 180
Correlated Errors 180
7.2 Transformation of Data 180
Dimensional Analysis 183
7.3 Variance-Stabilising Transformations 183
Taylor's Power Law 184
Delta Method 185
7.4 Multicollinearity 186
Regression Diagnostics 189
Exercises 189
8. Generalised Linear Models 193
8.1 Introduction 193
8.2 Definitions and examples 195
8.2.1 Statistical testing and model comparisons 197
8.2.2 Analysis of residuals 199
8.2.3 Athletics times 200
8.3 Binary models 202
8.4 Count data, contingency tables and log-linear models 205
8.5 Over-dispersion and the Negative Binomial Distribution 209
8.5.1 Practical applications: Analysis of over-dispersed models in R"472 211
Exercises 212
9. Other topics 214
9.1 Mixed models 214
9.1.1 Mixed models and Generalised Least Squares 217
9.2 Non-parametric regression 222
9.2.1 Kriging 224
9.3 Experimental Design 226
9.3.1 Optimality criteria 226
9.3.2 Incomplete designs 227
9.4 Time series 230
9.4.1 Cointegration and spurious regression 231
9.5 Survival analysis 233
9.5.1 Proportional hazards 235
9.6 p > >
Solutions 237
Dramatis Personae: Who did what when 279
Bibliography 281
Index 288

Produkt-Details

• Erscheint lt. Verlag: 29.9.2010
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• ISBN-10: 1-84882-969-8 / 1848829698
• ISBN-13: 978-1-84882-969-5 / 9781848829695