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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 requries 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.
Industry Reviews
From the reviews:
"The present book is intended for a second undergraduate or beginning graduate course in statistics providing further study of this single topic. ... Complete, mathematically rigorous proofs are routinely provided for theorems. The fully-worked examples and solutions to the exercises are detailed. ... Linear Models in Statistics is highly suitable for a theoretical statistics course for advanced undergraduate math majors, beginning math graduate students or others interested in using the book for independent study." (Susan D'Agostino, The Mathematical Association of America, December, 2010)
"Intended primarily for advanced undergraduate and beginning graduate students with knowledge of the basic concepts of statistics, probability, and linear algebra, this student-friendly book provides a lucid presentation of numerous regression analysis topics. ... A salient feature is the numerous, carefully selected worked examples and complete solutions to all the problems in various chapters. Includes a useful index and bibliography. Summing Up: Recommended. Upper-division undergraduates, graduate students, and professionals." (D. V. Chopra, Choice, Vol. 48 (8), April, 2011)
"This book describes the linear regression statistical models as a core of statistics, from simple linear regression (with one predictor variable) and analysis of variance (ANOVA) to more extended topics as multiple linear regression (with two or more predictor variables) and analysis of covariance (ANCOVA). ... The contents of the book are addressed in most part to the undergraduates students (but with some chapters appropriate for master level) having a basic knowledge of linear algebra, probability and statistics." (Nicoleta Breaz, Zentralblatt MATH, Vol. 1245, 2012)
| Linear Regression | p. 1 |
| Introduction | p. 1 |
| The Method of Least Squares | p. 3 |
| Correlation version | p. 7 |
| Large-sample limit | p. 8 |
| The origins of regression | p. 9 |
| Applications of regression | p. 11 |
| The Bivariate Normal Distribution | p. 14 |
| Maximum Likelihood and Least Squares | p. 21 |
| Sums of Squares | p. 23 |
| Two regressors | p. 26 |
| Exercises | p. 28 |
| The Analysis of Variance (ANOVA) | p. 33 |
| The Chi-Square Distribution | p. 33 |
| Change of variable formula and Jacobians | p. 36 |
| The Fisher F-distribution | p. 37 |
| Orthogonality | p. 38 |
| Normal sample mean and sample variance | p. 39 |
| One-Way Analysis of Variance | p. 42 |
| Two-Way ANOVA; No Replications | p. 49 |
| Two-Way ANOVA: Replications and Interaction | p. 52 |
| Exercises | p. 56 |
| Multiple Regression | p. 61 |
| The Normal Equations | p. 61 |
| Solution of the Normal Equations | p. 64 |
| Properties of Least-Squares Estimators | p. 70 |
| Sum-of-Squares Decompositions | p. 73 |
| Coefficient of determination | p. 79 |
| Chi-Square Decomposition | p. 80 |
| Idempotence, Trace and Rank | p. 81 |
| Quadratic forms in normal variates | p. 82 |
| Sums of Projections | p. 82 |
| Orthogonal Projections and Pythagoras's Theorem | p. 85 |
| Worked examples | p. 89 |
| Exercises | p. 94 |
| Further Multilinear Regression | p. 99 |
| Polynomial Regression | p. 99 |
| The Principle of Parsimony | p. 102 |
| Orthogonal polynomials | p. 103 |
| Packages | p. 103 |
| Analysis of Variance | p. 104 |
| The Multivariate Normal Distribution | p. 105 |
| The Multinormal Density | p. 111 |
| Estimation for the multivariate normal | p. 113 |
| Conditioning and Regression | p. 115 |
| Mean-square prediction | p. 121 |
| Generalised least squares and weighted regression | p. 123 |
| Exercises | p. 125 |
| Adding additional covariates and the Analysis of Covariance | p. 129 |
| Introducing further explanatory variables | p. 129 |
| Orthogonal parameters | p. 133 |
| ANCOVA | p. 135 |
| Nested Models | p. 139 |
| Examples | p. 140 |
| Exercises | p. 145 |
| Linear Hypotheses | p. 149 |
| Minimisation Under Constraints | p. 149 |
| Sum-of-Squares Decomposition and F-Test | p. 152 |
| Applications: Sequential Methods | p. 157 |
| Forward selection | p. 157 |
| Backward selection | p. 158 |
| Stepwise regression | p. 159 |
| Exercises | p. 160 |
| Model Checking and Transformation of Data | p. 163 |
| Deviations from Standard Assumptions | p. 163 |
| Transformation of Data | p. 168 |
| Variance-Stabilising Transformations | p. 171 |
| Multicollinearity | p. 174 |
| Exercises | p. 177 |
| Generalised Linear Models | p. 181 |
| Introduction | p. 181 |
| Definitions and examples | p. 183 |
| Statistical testing and model comparisons | p. 185 |
| Analysis of residuals | p. 187 |
| Athletics times | p. 188 |
| Binary models | p. 190 |
| Count data, contingency tables and log-linear models | p. 193 |
| Over-dispersion and the Negative Binomial Distribution | p. 197 |
| Practical applications: Analysis of over-dispersed models in R® | p. 199 |
| Exercises | p. 200 |
| Other topics | p. 203 |
| Mixed models | p. 203 |
| Mixed models and Generalised Least Squares | p. 206 |
| Non-parametric regression | p. 211 |
| Kriging | p. 213 |
| Experimental Design | p. 215 |
| Optimality criteria | p. 215 |
| Incomplete designs | p. 216 |
| Time series | p. 219 |
| Cointegration and spurious regression | p. 220 |
| Survival analysis | p. 222 |
| Proportional hazards | p. 224 |
| p>>n | p. 225 |
| Solutions | p. 227 |
| Dramatis Personae: Who did what when | p. 269 |
| Bibliography | p. 271 |
| Index | p. 279 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9781848829688
ISBN-10: 184882968X
Series: Springer Undergraduate Mathematics Series
Published: 29th September 2010
Format: Paperback
Language: English
Number of Pages: 300
Audience: College, Tertiary and University
Publisher: Springer Nature B.V.
Country of Publication: GB
Dimensions (cm): 23.5 x 15.24 x 1.91
Weight (kg): 0.42
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