Regression

The Springer Undergraduate Mathematics Series (SUMS) is designed for undergraduates in the mathematical sciences. From core foundational material to final year topics, SUMS books take a fresh and modern approach and are ideal for self-study or for a one-or two-semester course. Each book includes numerous examples, problems and fully-worked solutions. N. H. Bingham. John M. Fry Regression

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.

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.

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)

1 Linear Regression		1	(32)
1.1 Introduction		1	(2)
1.2 The Method of Least Squares		3	(6)
1.2.1 Correlation version		7	(1)
1.2.2 Large-sample limit		8	(1)
1.3 The origins of regression		9	(2)
1.4 Applications of regression		11	(3)
1.5 The Bivariate Normal Distribution		14	(7)
1.6 Maximum Likelihood and Least Squares		21	(2)
1.7 Sums of Squares		23	(3)
1.8 Two regressors		26	(7)
Exercises		28	(5)
2 The Analysis of Variance (ANOVA)		33	(28)
2.1 The Chi-Square Distribution		33	(3)
2.2 Change of variable formula and Jacobians		36	(1)
2.3 The Fisher F-distribution		37	(1)
2.4 Orthogonality		38	(1)
2.5 Normal sample mean and sample variance		39	(3)
2.6 One-Way Analysis of Variance		42	(7)
2.7 Two-Way ANOVA; No Replications		49	(3)
2.8 Two-Way ANOVA: Replications and Interaction		52	(9)
Exercises		56	(5)
3 Multiple Regression		61	(38)
3.1 The Normal Equations		61	(3)
3.2 Solution of the Normal Equations		64	(6)
3.3 Properties of Least-Squares Estimators		70	(3)
3.4 Sum-of-Squares Decompositions		73	(7)
3.4.1 Coefficient of determination		79	(1)
3.5 Chi-Square Decomposition		80	(5)
3.5.1 Idempotence, Trace and Rank		81	(1)
3.5.2 Quadratic forms in normal variates		82	(1)
3.5.3 Sums of Projections		82	(3)
3.6 Orthogonal Projections and Pythagoras's Theorem		85	(4)
3.7 Worked examples		89	(10)
Exercises		94	(5)
4 Further Multilinear Regression		99	(30)
4.1 Polynomial Regression		99	(5)
4.1.1 The Principle of Parsimony		102	(1)
4.1.2 Orthogonal polynomials		103	(1)
4.1.3 Packages		103	(1)
4.2 Analysis of Variance		104	(1)
4.3 The Multivariate Normal Distribution		105	(6)
4.4 The Multinormal Density		111	(4)
4.4.1 Estimation for the multivariate normal		113	(2)
4.5 Conditioning and Regression		115	(6)
4.6 Mean-square prediction		121	(2)
4.7 Generalised least squares and weighted regression		123	(6)
Exercises		125	(4)
5 Adding additional covariates and the Analysis of Covariance		129	(20)
5.1 Introducing further explanatory variables		129	(6)
5.1.1 Orthogonal parameters		133	(2)
5.2 ANCOVA		135	(5)
5.2.1 Nested Models		139	(1)
5.3 Examples		140	(9)
Exercises		145	(4)
6 Linear Hypotheses		149	(14)
6.1 Minimisation Under Constraints		149	(3)
6.2 Sum-of-Squares Decomposition and F-Test		152	(5)
6.3 Applications: Sequential Methods		157	(6)
6.3.1 Forward selection		157	(1)
6.3.2 Backward selection		158	(1)
6.3.3 Stepwise regression		159	(1)
Exercises		160	(3)
7 Model Checking and Transformation of Data		163	(18)
7.1 Deviations from Standard Assumption		163	(5)
7.2 Transformation of Data		168	(3)
7.3 Variance-Stabilising Transformations		171	(3)
7.4 Multicollinearity		174	(7)
Exercises		177	(4)
8 Generalised Linear Models		181	(22)
8.1 Introduction		181	(2)
8.2 Definitions and examples		183	(7)
8.2.1 Statistical testing and model comparions		185	(2)
8.2.2 Analysis of residuals		187	(1)
8.2.3 Athletics times		188	(2)
8.3 Binary models		190	(3)
8.4 Count data, contingency tables and log-linear models		193	(4)
8.5 Over-dispersion and the Negative Binomial Distribution		197	(6)
8.5.1 Practical applications: Analysis of over-dispersed models in R®		199	(1)
Exercises		200	(3)
9 Other topics		203	(24)
9.1 Mixed models		203	(8)
9.1.1 Mixed models and Generalised Least Squares		206	(5)
9.2 Non-parametric regression		211	(4)
9.2.1 Kriging		213	(2)
9.3 Experimental Design		215	(4)
9.3.1 Optimality criteria		215	(1)
9.3.2 Incomplete designs		216	(3)
9.4 Time series		219	(3)
9.4.1 Cointegration and spurious regression		220	(2)
9.5 Survival analysis		222	(3)
9.5.1 Proportional hazards		224	(1)
9.6 p > > n		225	(2)
Solutions		227	(42)
Dramatis Personae: Who did What when		269	(2)
Bibliography		271	(8)
Index		279