Regression Analysis : A Constructive Critique

Regression Analysis: A Constructive Critique identifies a wide variety of problems with regression analysis as it is commonly used and then provides a number of ways in which practice could be improved. Regression is most useful for data reduction, leading to relatively simple but rich and precise descriptions of patterns in a data set. The emphasis on description provides readers with an insightful rethinking from the ground up of what regression analysis can do, so that readers can better match regression analysis with useful empirical questions and improved policy-related research."An interesting and lively text, rich in practical wisdom, written for people who do empirical work in the social sciences and their graduate students."--David A. Freedman, Professor of Statistics, University of California, Berkeley

Richard Berk is Professor in the Departments of Statistics and Sociology at the University of California, Los Angeles.

Series Editor's Introduction

xi

Preface

xvii

1. Prologue: Regression Analysis as Problematic

1

(4)

2. A Grounded Introduction to Regression Analysis

5

(16)

2.1 Some Examples of Regression Analysis

5

(5)

2.1.1 Abortion and Subsequent Crime

5

(1)

2.1.2 Mandatory Basic Education for Welfare Recipients

6

(1)

2.1.3 Gender and Academic Salaries

7

(1)

2.1.4 Climate Change and Water Resources in India

8

(1)

2.1.5 Deforestation and Soil Erosion in the Yangtze River Valley

8

(1)

2.1.6 Epidemics of Hepatitis C

9

(1)

2.1.7 Onward and Upward

9

(1)

2.2 What Is Regression Analysis?

10

(8)

2.2.1 A Simple Illustration

10

(3)

2.2.2 Controlling for a Third Variable

13

(3)

2.2.3 Imposing a Smoother

16

(2)

2.3 Getting From Data to Stories

18

(3)

3. Simple Linear Regression

21

(18)

3.1 Introduction

21

(1)

3.2 Describing a Conditional Relationship With a Straight Line

22

(2)

3.3 Defining the "Best" Line

24

(3)

3.4 Some Useful Formulas

27

(1)

3.5 Standardized Slopes

28

(2)

3.6 Using Transformations for a Nonlinear Fit

30

(5)

3.7 What About the Variance Function?

35

(2)

3.8 Summary and Conclusions

37

(2)

4. Statistical Inference for Simple Linear Regression

39

(42)

4.1 The Role of Sampling

39

(19)

4.1.1 Random Sampling

39

(3)

4.1.2 Strategy I: Treating the Data as Population

42

(2)

4.1.3 Strategy II: Treating the Data as If They Were Generated by Random Sampling From a Population

44

(7)

4.1.4 Strategy III: Inventing an Imaginary Population

51

(2)

4.1.5 Strategy IV: Model-Based Sampling-Inventing a Friendly Natural Process Responsible for the Data

53

(3)

4.1.6 A Note on Randomization Inference

56

(2)

4.1.7 Summing Up

58

(1)

4.2 Simple Linear Regression Under Random Sampling

58

(11)

4.2.1 Estimating the Population Regression Line

58

(3)

4.2.2 Estimating the Standard Errors

61

(1)

4.2.3 Estimation Under Model-Based Sampling

62

(1)

4.2.4 Some Things That Can Go Wrong

62

(3)

4.2.5 Tests and Confidence Intervals

65

(4)

4.3 Statistical Power

69

(1)

4.4 Stochastic Predictors

69

(4)

4.5 Measurement Error

73

(1)

4.6 Can Resampling Techniques Help?

74

(5)

4.6.1 Percentile Confidence Intervals

76

(1)

4.6.2 Hypothesis Testing

77

(1)

4.6.3 Bootstrapping Regression

77

(1)

4.6.4 Possible Benefits From Resampling

78

(1)

4.7 Summary and Conclusions

79

(2)

5. Causal Inference for the Simple Linear Model

81

(22)

5.1 Introduction

81

(1)

5.2 Some Definitions: What's a Causal Effect?

82

(15)

5.2.1 The Neyman-Rubin Model

84

(4)

5.2.2 Thinking About Causal Effects as Response Schedules

88

(2)

5.2.3 What's an Intervention?

90

(7)

5.3 Studying Cause and Effect With Data

97

(4)

5.3.1 Using Nonstatistical Solutions for Making Causal Inferences

97

(1)

5.3.2 Using Statistical Solutions for Making Causal Inferences

98

(1)

5.3.3 Using the Simple Linear Model for Making Causal Inferences

99

(2)

5.4 Summary and Conclusions

101

(2)

6. The Formalities of Multiple Regression

103

(8)

6.1 Introduction

103

(1)

6.2 Terms and Predictors

103

(2)

6.3 Some Notation for Multiple Regression

105

(1)

6.4 Estimation

105

(2)

6.5 How Multiple Regression "Holds Constant"

107

(3)

6.6 Summary and Conclusions

110

(1)

7. Using and Interpreting Multiple Regression

111

(14)

7.1 Introduction

111

(1)

7.2 Another Formal Perspective on Holding Constant

111

(2)

7.3 When Does Holding Constant Make Sense?

113

(4)

7.4 Standardized Regression Coefficients: Once More With Feeling

117

(2)

7.5 Variances of the Coefficient Estimates

119

(3)

7.6 Summary and Conclusions

122

(3)

8. Some Popular Extensions of Multiple Regression

125

(26)

8.1 Introduction

125

(1)

8.2 Model Selection and Stepwise Regression

126

(9)

8.2.1 Model Selection by Removing Terms

127

(1)

8.2.2 Tests to Compare Models

128

(2)

8.2.3 Selecting Terms Without Testing

130

(2)

8.2.4 Stepwise Selection Methods

132

(1)

8.2.5 Some Implications

133

(2)

8.3 Using Categorical Terms: Analysis of Variance and Analysis of Covariance

135

(6)

8.3.1 An Extended Example

135

(6)

8.4 Back to the Variance Function: Weighted Least Squares

141

(6)

8.4.1 Visualizing Lack of Fit

141

(1)

8.4.2 Weighted Least Squares as a Possible Fix

142

(3)

8.4.3 Evaluating the Mean Function

145

(2)

8.5 Locally Weighted Regression Smoother

147

(1)

8.6 Summary and Conclusions

148

(3)

9. Some Regression Diagnostics

151

(20)

9.1 Introduction

151

(1)

9.2 Transformations of the Response Variable

152

(7)

9.2.1 Box-Cox Procedures

152

(1)

9.2.2 Inverse Fitted Value Response Plots

153

(6)

9.3 Leverage and Influence

159

(3)

9.3.1 Influential Cases and Cook's Distance

159

(3)

9.4 Cross-Validation

162

(1)

9.5 Misspecification Tests

163

(5)

9.5.1 Instrumental Variables

164

(3)

9.5.2 Tests for Exogeneity

167

(1)

9.6 Conclusions

168

(3)

10. Further Extensions of Regression Analysis

171

(32)

10.1 Regression Models for Longitudinal Data

171

(6)

10.1.1 Multiple Linear Regression for Time Series Data

172

(5)

10.2 Regression Analysis With Multiple Time Series Data

177

(3)

10.2.1 Fixed Effects Models

178

(1)

10.2.2 Random Effects Models

178

(2)

10.2.3 Estimation

180

(1)

10.3 Multilevel Models

180

(3)

10.4 The Generalized Linear Model

183

(5)

10.4.1 GLM Structure

183

(1)

10.4.2 Normal Models

184

(1)

10.4.3 Poisson Models

184

(2)

10.4.4 Poisson Models for Contingency Tables

186

(1)

10.4.5 Binomial Regression

186

(2)

10.5 Multiple Equation Models

188

(8)

10.5.1 Causal Inference Once Again

191

(5)

10.5.2 A Final Observation

196

(1)

10.6 Meta-Analysis

196

(4)

10.7 Conclusions

200

(3)

11. What to Do

203

(36)

11.1 How Did We Get Into This Mess?

203

(3)

11.2 Three Cheers for Description

206

(12)

11.2.1 What's Description?

207

(1)

11.2.2 Advocacy Settings

207

(2)

11.2.3 Descriptive Regressions as Part of a Broad Research Program

209

(1)

11.2.4 Spotting Provocative Associations

210

(2)

11.2.5 Some Other Benefits of Description

212

(6)

11.3 Two Cheers for Statistical Inference

218

(5)

11.3.1 Working With Near-Random Samples

220

(2)

11.3.2 Working With Data From Nature

222

(1)

11.3.3 Working With a Nearly Correct Model

222

(1)

11.4 One Cheer for Causal Inference

223

(11)

11.4.1 Special-Purpose Estimators

226

(4)

11.4.2 Propensity Scores

230

(1)

11.4.3 Sensitivity Analysis of the Selection Process

231

(1)

11.4.4 Bounding Treatment Effects

232

(2)

11.4.5 Some Forecasts

234

(1)

11.5 Some Final Observations

234

(5)

11.5.1 A Police Story

234

(3)

11.5.2 Regression Analysis as Too Little, Too Late

237

(2)

References

239

(12)

Index

251

(8)

About the Author

259

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Regression Analysis : A Constructive Critique

Summary

Author Biography

Table of Contents

Supplemental Materials