An up-to-date, rigorous, and lucid treatment of the theory, methods, and applications of regression analysis, and thus ideally suited for those interested in the theory as well as those whose interests lie primarily with applications. It is further enhanced through real-life examples drawn from many disciplines, showing the difficulties typically encountered in the practice of regression analysis. Consequently, this book provides a sound foundation in the theory of this important subject.

Contenu

1 Introduction.- 1.1 Relationships.- 1.2 Determining Relationships: A Specific Problem.- 1.3 The Model.- 1.4 Least Squares.- 1.5 Another Example and a Special Case.- 1.6 When Is Least Squares a Good Method?.- 1.7 A pleasure of Fit for Simple Regression.- 1.8 Mean and Variance of b0 and b1.- 1.9 Confidence Intervals and Tests.- 1.10 Predictions.- Appendix to Chapter 1.- Problems.- 2 Multiple Regression.- 2.1 Introduction.- 2.2 Regression Model in Matrix Notation.- 2.3 Least Squares Estimates.- 2.4 Examples 31 2..- Gauss-Markov Conditions.- 2.6 Mean and Variance of Estimates Under G-M Conditions.- 2.7 Estimation of ?.- 2.8 Measures of Fit 39?2.- 2.9 The Gauss-Markov Theorem.- 2.10 The Centered Model.- 2.11 Centering and Scaling.- 2.12 *Constrained Least Squares.- Appendix to Chapter 2.- Problems.- 3 Tests and Confidence Regions.- 3.1 Introduction.- 12 Linear Hypothesis.- 3.3 *Likelihood Ratio Test.- 3.4 *Distribution of Test Statistic.- 3.5 Two Special Cases.- 3.6 Examples.- 3.7 Comparison of Repression Equations.- 3.8 Confidence Intervals and Regions.- 3.8.1 C.I. for the Expectation of a Predicted Value.- 3.8.2 C.I for a Future Observation.- 3.8.3 *Confidence Region for Regression Parameters.- 3.8.4 *C.I's for Linear Combinations of Coefficients.- Problem.- 4 Indicator Variables.- 4.1 Introduction.- 4.2 A Simple Application.- 4.3 Polychotomous Variables.- 4.4 Continuous and Indicator Variables.- 4.5 Broken Line Regression.- 4.6 Indicators as Dependent Variables.- Problems.- 5 The Normality Assumption.- 5.1 Introduction.- 5.2 Checking for Normality.- 5.2.1 ProbahilItV Plots.- 5.2.2 Tests for Normalitv.- 5.3 Invoking Large Sample Theory.- 5.4 *Bootstrapping.- 5.5 *Asymptotic Theory.- Problems.- 6 Unequal Variances.- 6.1 Introduction.- 6.2 Detecting Heteroscedasticity.- 6.2.1 Formal Tests.- 6.3 Variance Stabilizing Transformations.- 6.4 Weighing.- Problems.- 7 *Correlated Errors.- 7.1 Introduction.- 7.2 Generalized Least Squares: Case When ? Is Known.- 7.3 Estimated Generalized Least Squares.- 7.3.1 Error Variances Unequal and Unknown.- 7.4 Nested Errors.- 7.5 The Growth Curve Model.- 7.6 Serial Correlation.- 7.6.1 The Durbin-Watson Test.- 7.7 Spatial Correlation.- 7.7. 1 Testing for Spatial Correlation.- 7.7.2 Estimation of Parameters.- Problems.- 8 Outliers and Influential Observations.- 8.1 Introduction.- 8.2 The Leverage.- 8.2.1 *Leverage as Description of Remoteness.- 8.3 The Residuals.- 8.4 Detecting Outliers and Points That Do Not Belong to the Model 157.- 8.5 Influential Observations.- 8.5.1 Other Measures of Influence.- 8.6 Examples.- Appendix to Chapter 8.- Problems.- 9 Transformations.- 9.1 Introduction.- 9.1.1 An Important Word of Warning.- 9.2 Some Common Transformations.- 9.2.1 Polynomial Regression.- 9.2.2 Spline.- 9.2.3 Multiplicative Models.- 9.2.4 The Logit Model for Proportions.- 9.3 Deciding on the Need for Transformations.- 9.3.1 Examining Residual Plots.- 9.3.2 Use of Additional Terms.- 9.3.3 Use of Repeat Measurements.- 9.3.4 Daniel and Wood Near-Neighbor Approach.- 9.3.5 Another Method Based on Near Neighbors.- 9.4 Choosing Transformations.- 9.4.1 Graphical Method: One Independent. Variable.- 9.4.2 Graphical Method: Many Independent Variables.- 9.4.3 Analytic Methods: Transforming the Response.- 9.4.4 Analytic Methods: Transforming the Predictors.- 9.3.5 Simultaneous Power Transformations for Predictors and Response.- Appendix to Chapter 9.- Problems.- 10 Multicollinearity.- 10.1 Introduction.- 10.2 Multicollinearity and Its Effects.- 10.3 Detecting Multicollinearity.- 10.3.1 Tolerances and Variance Inflation Factors.- 10.3.2 Eigenvalues and Condition Numbers.- 10.3.3 Variance Components.- 10.4 Examples.- Problems.- 11 Variable Selection.- 11.1 Introduction.- 11.2 Some Effects of Dropping Variables.- 11.2.1 Effects on Estimates of ßj.- 11.2.2 *Effect on Estimation of Error Variance.- 11.2.3 *Effect on Covariance Matrix of Estimates.- 11.2.4 *Effect on Predicted Values: Mallows' Cp.- 11.3 Variable Selection Procedures.- 11.3.1 Search Over All Possible Subsets.- 11.3.2 Stepwise Procedures.- 11.3.3 Stagewise and Modified Stagewise Procedures.- 11.4 Examples.- Problems.- 12 *Biased Estimation.- 12.1 Introduction 2..- 12.2 Principal Component. Regression.- 12.2.1 Bias and Variance of Estimates.- 12.3 Ridge Regression.- 12.3.1 Physical Interpretations of Ridge Regression.- 12.3.2 Bias and Variance of Estimates.- 12.4 Shrinkage Estimator.- Problems.- A Matrices.- A.1 Addition and Multiplication.- A.2 The Transpose of a Matrix.- A.3 Null and Identity Matrices.- A.4 Vectors.- A.5 Rank of a Matrix.- A.6 Trace of a Matrix.- A.7 Partitioned Matrices.- A.8 Determinants.- A.9 Inverses.- A.10 Characteristic Roots and Vectors.- A.11 Idempotent Matrices.- A.12 The Generalized Inverse.- A.13 Quadratic Forms.- A.14 Vector Spaces.- Problems.- B Random Variables and Random Vectors.- B.1 Random Variables.- B.1.1 Independent. Random Variables.- B.1.2 Correlated Random Variables.- B.1.3 Sample Statistics.- B.1.4 Linear Combinations of Random Variables.- B.2 Random Vectors.- B.3 The Multivariate Normal Distribution.- B.4 The Chi-Square Distributions.- B.5 The F and t Distributions.- B.6 Jacobian of Transformations.- B.7 Multiple Correlation.- Problems.- C Nonlinear Least Squares.- C.1 Gauss-Newton Type Algorithms.- C.1.1 The Gauss-Newton Procedure.- C.1.2 Step Halving.- C.1.3 Starting Values and Derivatives.- C.1.4 Marquardt Procedure.- C.2 Some Other Algorithms.- C.2.1 Steepest Descent Method.- C.2.2 Quasi-Newton Algorithms.- C.2.3 The Simplex Method.- C.2.4 Weighting.- C.3 Pitfalls.- C.4 Bias, Confidence Regions and Measures of Fit.- C.5 Examples.- Problems.- Tables.- References.- Author Index.