Contents preface PART ONE SIMPLE LINEAR REGRESSION 1 Chapter 1 Linear Regression with One Predictor Variable 2 1.1 Relations between Variables 2 Functional Relation between Two Variables 2 Statistical Relation between Two Variables 3 1.2 Regression Models and Their Uses 5 Historical Origins 5 Basic Concepts 5 Construction of Regression Models 7 Uses of Regression Analysis 8 Regression and Causality 8 Use of Computers 9 1.3 Simple Linear Regression Model with Distribution of Error Terms Unspecified 9 Formal Statement of Model 9 Important Features of Model 9 Meaning of Regression Parameters 11 Alternative Versions of Regression Model 12 1.4 Data for Regression Analysis 12 Observational Data 12 Experimental Data 13 Completely Randomized Design 13 1.5 Overview of Steps in Regression Analysis 13 1.6 Estimation of Regression Function 15 Method of Least Squares 15 Point Estimation of Mean Response 21 Residuals 22 Properties of Fitted Regression Line 23 1.7 Estimation of Error Terms Variance ?2 24 Point Estimator of ?2 24 1.8 Normal Error Regression Model 26 Model 26 Estimation of Parameters by Method of Maximum Likelihood 27 Cited References 33 Problems 33 Exercises 37 Projects 38 Chapter 2 Inferences in Regression and Correlation Analysis 40 2.1 Inferences Concerning ?1 40 Sampling Distribution of b1 41 Sampling Distribution of (b1 -?1)/s{b1} 44 Confidence Interval for ?1 45 Tests Concerning ?1 47 2.2 Inferences Concerning ?0 48 Sampling Distribution of b0 48 Sampling Distribution of (b0 -?0)/s{b0} 49 Confidence Interval for ?0 49 2.3 Some Considerations on Making Inferences Concerning ?0 and ?1 50 Effects of Departures from Normality 50 Interpretation of Confidence Coefficient and Risks of Errors 50 Spacing of the X Levels 50 Power of Tests 50 2.4 Interval Estimation of E{Yh} 52 Sampling Distribution of ?Y h 52 Sampling Distribution of ( ?Y h - E{Yh})/s{ ?Y h} 54 Confidence Interval for E{Yh} 54 2.5 Prediction of New Observation 55 Prediction Interval for Yh(new) when Parameters Known 56 Prediction Interval for Yh(new) when Parameters Unknown 57 Prediction of Mean of m New Observations for Given Xh 60 2.6 Confidence Band for Regression Line 61 2.7 Analysis of Variance Approach to Regression Analysis 63 Partitioning of Total Sum of Squares 63 Breakdown of Degrees of Freedom 66 x Contents xi Mean Squares 66 Analysis of Variance Table 67 Expected Mean Squares 68 F Test of ?1 = 0 versus ?1 _= 0 69 2.8 General Linear Test Approach 72 Full Model 72 Reduced Model 72 Test Statistic 73 Summary 73 2.9 Descriptive Measures of Linear Association between X and Y 74 Coefficient of Determination 74 Limitations of R2 75 Coefficient of Correlation 76 2.10 Considerations in Applying Regression Analysis 77 2.11 Normal Correlation Models 78 Distinction between Regression and Correlation Model 78 Bivariate Normal Distribution 78 Conditional Inferences 80 Inferences on Correlation Coefficients 83 Spearman Rank Correlation Coefficient 87 Cited References 89 Problems 89 Exercises 97 Projects 98 Chapter 3 Diagnostics and Remedial Measures 100 3.1 Diagnostics for Predictor Variable 100 3.2 Residuals 102 Properties of Residuals 102 Semistudentized Residuals 103 Departures from Model to Be Studied by Residuals 103 3.3 Diagnostics for Residuals 103 Nonlinearity of Regression Function 104 Nonconstancy of Error Variance 107 Presence of Outliers 108 Nonindependence of Error Terms 108 Nonnormality of Error Terms 110 Omission of Important Predictor Variables 112 Some Final Comments 114 3.4 Overview of Tests Involving Residuals 114 Tests for Randomness 114 Tests for Constancy of Variance 115 Tests for Outliers 115 Tests for Normality 115 3.5 Correlation Test for Normality 115 3.6 Tests for Constancy of Error Variance 116 Brown-Forsythe Test 116 Breusch-Pagan Test 118 3.7 F Test for Lack of Fit 119 Assumptions 119 Notation 121 Full Model 121 Reduced Model 123 Test Statistic 123 ANOVA Table 124 3.8 Overview of Remedial Measures 127 Nonlinearity of Regression Function 128 Nonconstancy of Error Variance 128 Nonindependence of Error Terms 128 Nonnormality of Error Terms 128 Omission of Important Predictor Variables 129 Outlying Observations 129 3.9 Transformations 129 Transformations for Nonlinear Relation Only 129 Transformations for Nonnormality and Unequal Error Variances 132 Box-Cox Transformations 134 3.10 Exploration of Shape of Regression Function 137 Lowess Method 138 Use of Smoothed Curves to Confirm Fitted Regression Function 139 3.11 Case Example—Plutonium Measurement 141 Cited References 146 Problems 146 Exercises 151 Projects 152 Case Studies 153 xii Contents Chapter 4 Simultaneous Inferences and Other Topics in Regression Analysis 154 4.1 Joint Estimation of ?0 and ?1 154 Need for Joint Estimation 154 Bonferroni Joint Confidence Intervals 155 4.2 Simultaneous Estimation of Mean Responses 157 Working-Hotelling Procedure 158 Bonferroni Procedure 159 4.3 Simultaneous Prediction Intervals for New Observations 160 4.4 Regression through Origin 161 Model 161 Inferences 161 Important Cautions for Using Regression through Origin 164 4.5 Effects of Measurement Errors 165 Measurement Errors in Y 165 Measurement Errors in X 165 Berkson Model 167 4.6 Inverse Predictions 168 4.7 Choice of X Levels 170 Cited References 172 Problems 172 Exercises 175 Projects 175 Chapter 5 Matrix Approach to Simple Linear Regression Analysis 176 5.1 Matrices 176 Definition of Matrix 176 Square Matrix 178 Vector 178 Transpose 178 Equality of Matrices 179 5.2 Matrix Addition and Subtraction 180 5.3 Matrix Multiplication 182 Multiplication of a Matrix by a Scalar 182 Multiplication of a Matrix by a Matrix 182 5.4 Special Types of Matrices 185 Symmetric Matrix 185 Diagonal Matrix 185 Vector and Matrix with All Elements Unity 187 Zero Vector 187 5.5 Linear Dependence and Rank of Matrix 188 Linear Dependence 188 Rank of Matrix 188 5.6 Inverse of a Matrix 189 Finding the Inverse 190 Uses of Inverse Matrix 192 5.7 Some Basic Results for Matrices 193 5.8 Random Vectors and Matrices 193 ......
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