详解线性回归分析理论与计算实践

线性回归

理论分析

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《线性回归分析理论与计算》是一本由Xin Yan和Xiao Gang Su合著的专业统计学著作，它全面探讨了线性回归这一关键的统计分析方法。线性回归是数据分析中的一种基础技术，主要关注如何通过一个或多个自变量（独立变量）来预测或解释一个因变量（依赖变量）的变化。在本书中，作者深入解析了线性回归的理论原理，包括但不限于： 1. 基本概念：介绍了线性回归模型的基本形式，即y = ax + b + ε，其中y是响应变量，x是解释变量，a是斜率代表变量间的关系强度，b是截距，ε是误差项，反映了模型无法完全解释的实际变异性。 2. 参数估计：涵盖了最小二乘法（Least Squares Method），这是估计线性回归系数最常用的方法，它通过最小化残差平方和来找到最佳拟合直线。 3. 假设检验：讨论了统计显著性和假设检验在评估回归模型的有效性和预测能力中的应用，如F检验、t检验等。 4. 多重共线性：当自变量之间存在高度相关时，可能会影响回归分析的结果，作者详细阐述了如何识别和处理这个问题。 5. 回归诊断：涵盖了残差分析、杠杆值检验、异方差性检查等工具，帮助用户确保模型的稳健性。 6. 实际应用：书中还包含了许多实际案例，展示了如何将线性回归理论应用于经济学、社会科学、工程学等各种领域的数据建模。 7. 计算技术：针对不同的软件包（如R、Python等），书中提供了对线性回归计算的实用指导，使读者能够有效地实施分析。 8. 理论发展：本书紧跟最新研究动态，不仅涵盖经典线性回归，还包括了现代技术如岭回归、lasso回归等高级线性模型。作为一本深入讲解理论与实践的教材，这本书旨在帮助读者掌握线性回归的精髓，并能熟练地在实际问题中应用这一强大的统计工具。无论是初学者还是经验丰富的数据分析师，都能从中受益匪浅。同时，该书的图书馆分类信息和版权信息也表明了其专业性和权威性。

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April 29, 2009 11:50 World Scientiﬁc Book - 9in x 6in Regression˙master

List of Figures

2.1 (a) Regression Line and Scatter Plot. (b) Residual Plot, (c)

95% Conﬁdence Band for Regression Mean. (d) 95% Conﬁdence

Band for Regression Prediction. . . . . . . . . . . . . . . . . . . 33

2.2 Q-Q Plot for Regression Model density=β

+ β

stiﬀness + ε . 34

3.1 Response Curves of Y Versus X

at Diﬀerent Values of X

Models (3.33) and (3.32). . . . . . . . . . . . . . . . . . . . . . 74

3.2 Regression on Dummy Variables . . . . . . . . . . . . . . . . . 82

3.3 Various Shapes of Residual Plots . . . . . . . . . . . . . . . . . 98

3.4 Residual Plots of the Regression Mo del of the US Population . 100

4.1 Partial Residual Plots for Regression Models . . . . . . . . . . 145

5.1 (a) Plot of the Average and Standard Deviation of SSPE Pre-

diction from 100 Simulation Runs; (b) A Hypothetical Plot of

Prediction Error Versus Model Complexity in General Regres-

sion Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.2 Variable Importance for the Quasar Data: Random Forests

vs. RELIEF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

5.3 A Simulation Example on Variable Importance: Random

Forests vs. RELIEF. . . . . . . . . . . . . . . . . . . . . . . . . 185

6.1 Box-Cox Power Transformation of Y in the Navy Manpower

Data: Plot of the Log-likelihood Score vs. λ. . . . . . . . . . . 212

6.2 Detecting the Nonlinear Eﬀect of x

and x

via Additive Models

in the Navy Manpower Data. . . . . . . . . . . . . . . . . . . . 213

6.3 AVAS Results for the Navy Manpower Data: (a) Plot of g(Y );

(b) Plot of f

); (c) Plot of f

). . . . . . . . . . . . . . . 214

April 29, 2009 11:50 World Scientiﬁc Book - 9in x 6in Regression˙master

xvi Linear Regression Analysis: Theory and Computing

6.4 AVAS Results for the Navy Manpower Data: (a) 3-D Scat-

ter Plot of the Original Data; (b) 3-D Scatter Plot of g(Y )

vs. f

) and f

) after Transformation. . . . . . . . . . . 215

7.1 (a) The Scatter Plot of DBP Versus Age; (b) Residual ver-

sus Age; (c) Absolute Values of Residuals |ˆε

| Versus Age; (d)

Squared Residuals ˆε

Versus Age. . . . . . . . . . . . . . . . . . 237

7.2 Estimation in Constrained Least Squares in the Two-

dimensional Case: Contours of the Residual Sum of Squares and

the Constraint Functions in (a) Ridge Regression; (b) LASSO. 240

7.3 Paired Scatter Plots for Longley’s (1967) Economic Data. . . . 251

7.4 Ridge Regression for Longley’s (1967) Economic Data: (a) Plot

of Parameter Estimates Versus s; (b) Plot of GCV Values Versus s.254

7.5 LASSO Pro cedure for Longley’s (1967) Economic Data: (a) Plot

of Standardized Parameter Estimates Versus Fraction of the L

Shrinkage; (b) Plot of Cross-validated SSE Versus Fraction. . . 257

7.6 Plot of the Logistic Function With C = 1: (a) When B Varies

With A = 1; (b) When A Varies With B = 1. . . . . . . . . . . 260

7.7 Parametric Nonlinear Regression with the US Population

Growth Data Reproduced from Fox (2002): (a) Scatter Plot

of the Data Superimposed by the LS Fitted Straight Line; (b)

Plot of the Residuals From the Linear Model Versus Year; (c)

Scatter Plot of the Data Superimposed by the Fitted Nonlinear

Curve; (d) Plot of the Residuals From the Nonlinear Fit Versus

Year. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

8.1 (a) Scatterplot of the CHD Data, Superimposed With the

Straight Line from Least Squares Fit; (b) Plot of the Percentage

of Subjects With CHD Within Each Age Group, Superimposed

by LOWESS Smoothed Curve. . . . . . . . . . . . . . . . . . . 270

8.2 Plot of the Log-likelihood Function: Comparison of Wald, LRT,

and Score Tests on H

: β = b. . . . . . . . . . . . . . . . . . . 280

9.1 Posterior Marginal Densities of (β, σ

): Bayesian Linear Regres-

sion for Shingle Data. . . . . . . . . . . . . . . . . . . . . . . . 308

9.2 Posterior Densities for Predicting y

: the Shingle Data. . . . . 309

9.3 The 13 Selected Models by BMA: the Quasar Data. . . . . . . 313

9.4 Posterior Distributions of the Coeﬃcients Generated by BMA:

the Quasar Data. . . . . . . . . . . . . . . . . . . . . . . . . . . 314

April 29, 2009 11:50 World Scientiﬁc Book - 9in x 6in Regression˙master

xviii Linear Regression Analysis: Theory and Computing

3.12 Parameter Estimates for Regression Model for Coal-Cleansing

Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.13 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.14 Manpower Data . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.15 Simultaneous Outlier Detection . . . . . . . . . . . . . . . . . . 113

3.16 Detection of Multiple Mean Shift Outliers . . . . . . . . . . . . 114

3.17 Stand Characteristics of Pine Tree Data . . . . . . . . . . . . . 116

3.18 Parameter Estimates and Conﬁdence Intervals Using x

, x

and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.19 Parameter Estimates and Conﬁdence Intervals after Deleting x

117

3.20 Conﬁdence Intervals on Regression Mean and Prediction With-

out Deletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.21 Conﬁdence Intervals on Regression Mean and Prediction After

Deleting x

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.22 Regression Model for Centralized Data . . . . . . . . . . . . . . 119

3.23 Test for Equal Slope Among 3 Groups . . . . . . . . . . . . . . 121

3.24 Regression by Group . . . . . . . . . . . . . . . . . . . . . . . . 121

3.25 Data Set for Calculation of Conﬁdence Interval on Regression

Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.26 Propellant Grain Data . . . . . . . . . . . . . . . . . . . . . . . 124

3.27 Data Set for Testing Linear Hypothesis . . . . . . . . . . . . . 124

3.28 Algae Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.1 Coal-cleansing Data . . . . . . . . . . . . . . . . . . . . . . . . 141

4.2 Parameter Estimates for y = b

+ b

. . . . . . 141

4.3 Residuals, Leverage, and t

. . . . . . . . . . . . . . . . . . . . 142

4.4 Navy Manpower Data . . . . . . . . . . . . . . . . . . . . . . . 144

4.5 Residuals for Model With Regressors x

and x

. . . . . . . . . 146

4.6 Residuals for Model With Regressors x

and x

. . . . . . . . 146

4.7 Sales Data for Asphalt Shingles . . . . . . . . . . . . . . . . . . 151

4.8 Values of PRESS Statistic for Various Regression Models . . . 152

4.9 PRESS Residuals and Leverages for Models Including x

, x

(press12, h12) and Including x

, x

(press123, h123) . . . . 153

4.10 Data Set for Multiple Mean Shift Outliers . . . . . . . . . . . 154

4.11 Fish Biomass Data . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.1 The Quasar Data from Ex. 4.9 in Mendenhall and Sinich (2003) 174

5.2 All Possible Regression Selection for the Quasar Data. . . . . . 175

5.3 Backward Elimination Procedure for the Quasar Data. . . . . . 190

April 29, 2009 11:50 World Scientiﬁc Book - 9in x 6in Regression˙master

List of Tables xix

5.4 Forward Addition Procedure for the Quasar Data. . . . . . . . 191

5.5 The Clerical Data from Ex. 6.4 in Mendenhall and Sinich (2003) 192

5.6 Backward Elimination Results for the Clerical Data. . . . . . . 193

5.7 Results of a Simulated Example for Inspecting Overall Signiﬁ-

cance in Stepwise Selection . . . . . . . . . . . . . . . . . . . . 193

5.8 Female Teachers Eﬀectiveness Data . . . . . . . . . . . . . . . . 194

5.9 All-subset-regression for Female Teacher Eﬀectiveness Data . . 194

6.1 Results of Eﬃciency Tests . . . . . . . . . . . . . . . . . . . . . 216

7.1 Layouts for Data Collected in One-way ANOVA Experiments. 220

7.2 Analysis of the Diastolic Blood Pressure Data: (a) Ordinary

Least Squares (OLS) Estimates; (b) Weighted Least Squares

(WLS) Fitting with Weights Derived from Regressing Absolute

Values |ˆε

| of the Residuals On Age. . . . . . . . . . . . . . . . 238

7.3 The Macro economic Data Set in Longley (1967). . . . . . . . . 249

7.4 Variable Description for Longley’s (1967) Macroeconomic Data. 250

7.5 Least Squares Results for Longley’s (1967) Economic Data. . . 252

7.6 Ridge Regression and LASSO for Longley’s (1967) Data. . . . . 255

7.7 Nonlinear Regression Results of the US Population Data from

PROC NLIN and PROC AUTOREG. . . . . . . . . . . . . . . 265

8.1 Frequency Table of AGE Group by CHD. . . . . . . . . . . . . 271

8.2 Variable Description for the Kyphosis Data: Logistic Regression

Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

8.3 Analysis Results for the Kyphosis Data from PROC LOGISTIC. 286

8.4 Variable Description for the Log-Linear Regression Example. . 288

8.5 Analysis Results for the School Absence Data from PROC GEN-

MOD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

8.6 Variable Description for the Prostate Cancer Data. . . . . . . . 293

8.7 Table of Parameter Estimates and the Estimated Covariance

Matrix for Model I: the Prostate Cancer Data. . . . . . . . . . 294

8.8 Table of Parameter Estimates and the Estimated Covariance

Matrix for Model II: the Prostate Cancer Data. 295

8.9 The Maximized Loglikelihood Scores for Several Fitted Models

with the Prostate Cancer Data. . . . . . . . . . . . . . . . . . 295

9.1 Bayesian Linear Regression Results With the Shingle Data. . . 307

9.2 BMA Results for the Quasar Data. . . . . . . . . . . . . . . . . 315

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