Multiple Regression : Testing and

This successful book, now available in paperback, provides academics and researchers with a clear set of prescriptions for estimating, testing and probing interactions in regression models. Including the latest research in the area, such as Fuller's work on the corrected/constrained estimator, the book is appropriate for anyone who uses multiple regression to estimate models, or for those enrolled in courses on multivariate statistics.

Preface

ix

Introduction

1

(8)

Interactions Between Continuous Predictors in Multiple Regression

9

(19)

What Interactions Signify in Regression

9

(1)

Data Set for Numerical Examples

10

(2)

Probing Significant Interactions in Regression Equations

12

(15)

Plotting the Interaction

12

(2)

Post Hoc Probing

14

(8)

Ordinal Versus Disordinal Interactions

22

(2)

Optional Section: The Derivation of Standard Errors of Simple Slopes

24

(3)

Summary

27

(1)

The Effects of Predictor Scaling on Coefficients of Regression Equations

28

(21)

The Problem of Scale Invariance

28

(8)

Linear Regression with no Higher Order Terms

29

(1)

Regression Equations with Higher Order Terms

30

(1)

Simple Slopes of Simple Regression Equations

31

(1)

Ordinal Versus Disordinal Interactions

31

(1)

Numerical Example---Centered Versus Uncentered Data

32

(3)

Should the Criterion Y Be Centered?

35

(1)

Multicollinearity: Essential Versus Nonessential Ill-Conditioning

35

(1)

Interpreting the Regression Coefficients

36

(4)

The Interaction Term XZ

36

(1)

The First Order Terms X and Z

37

(2)

A Geometric Interpretation

39

(1)

Standardized Solutions with Multiplicative Terms

40

(7)

Appropriate Standardized Solution with Interaction Terms

43

(1)

Simple Slope Analysis from the Standardized Solution

44

(1)

Relationship Between Raw and ``Standardized'' Solution

45

(2)

Summary

47

(2)

Testing and Probing Three-Way Interactions

49

(13)

Specifying, Testing, and Interpreting Three-Way Interactions

49

(1)

Probing Three-Way Interactions

50

(9)

Simple Regression Equation

50

(1)

Numerical Example

50

(2)

Graphing the Three-Way Interaction

52

(2)

Testing Simple Slopes for Significance

54

(1)

Standard Errors by Computer

54

(4)

Crossing Point of Simple Regression Equations with Three-Predictor Interaction

58

(1)

Simple Slopes and Their Variances in a Series of Regression Equations

59

(2)

Summary

61

(1)

Structuring Regression Equations to Reflect Higher Order Relationships

62

(38)

Structuring and Interpreting Regression Equations Involving Higher Order Relationships

63

(9)

Curvilinear X Relationship

63

(2)

A Progression of More Complex Equations with Curvilinear Relationships

65

(5)

Representation of Curvilinearity in ANOVA Versus MR

70

(2)

Post Hoc Probing of More Complex Regression Equations

72

(17)

Curvilinear X Equation

72

(6)

The Progression of More Complex Curvilinear Equations Revisited

78

(11)

Coefficients of Simple Slopes by Computer

89

(3)

Three Final Issues

92

(5)

Curvilinearity Versus Interaction

92

(1)

What Terms Should Be Included in the Regression Equation?

93

(2)

Other Methods of Representing Curvilinearity

95

(2)

Summary

97

(3)

Model and Effect Testing with Higher Order Terms

100

(16)

Some Issues in Testing Lower Order Effects in Models Containing Higher Order Terms

100

(5)

Interpretation of Lower Order Terms When b3 Is Significant

102

(1)

Should Lower Order Coefficients Be Tested in Reduced Models When b3 Is Nonsignificant?

103

(2)

Exploring Regression Equations Containing Higher Order Terms with Global Tests

105

(6)

Some Global Tests of Models with Higher Order Terms

107

(3)

Structuring Regression Equations with Higher Order Terms

110

(1)

Sequential Model Revision of Regression Equations Containing Higher Order Terms: Exploratory Tests

111

(3)

Application of Sequential Testing Following a Global Test

112

(1)

General Application of Sequential Testing

113

(1)

Present Approach Versus That Recommended by Cohen (1978)

113

(1)

Variable Selection Algorithms

114

(1)

Summary

114

(2)

Interactions Between Categorical and Continuous Variables

116

(23)

Coding Categorical Variables

116

(14)

Dummy Variable Coding

116

(11)

Unweighted Effects Coding

127

(2)

Choice of Coding System

129

(1)

Centering Revisited

130

(1)

Post Hoc Probing of Significant Interactions

130

(7)

Testing Simple Slopes Within Groups

131

(1)

Computer Procedure

131

(1)

Differences Between Regression Lines at a Specific Point

132

(2)

Identifying Regions of Significance

134

(3)

Summary

137

(2)

Reliability and Statistical Power

139

(33)

Reliability

140

(16)

Biased Regression Coefficients with Measurement Error

140

(5)

Corrected Estimates of Regression Coefficients in Equations Containing Higher Order Terms

145

(11)

Statistical Power

156

(13)

Statistical Power Analysis

156

(4)

The Effects of Measurement Error on Statistical Power

160

(5)

Some Corroborative Evidence: Simulation Studies

165

(2)

The Median Split Approach: The Cost of Dichotomization

167

(1)

Principal Component Regression Is Not a Cure for Power Woes

168

(1)

Coming Full Circle

169

(1)

Summary

170

(2)

Conclusion: Some Contrasts Between ANOVA and MR in Practice

172

(5)

Appendix A: Mathematical Underpinnings

177

(6)

Appendix B: Algorithm for Identifying Scale-Independent Terms

183

(5)

Appendix C: SAS Program for Test of Critical Region(s)

188

(2)

References

190

(8)

Glossary of Symbols

198

(6)

Author Index

204

(3)

Subject Index

207

(5)

About the Authors

212

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