Multiple Regression

by Aiken, Leona S.; West, Stephen G.

• ISBN13: 9780803936058
• ISBN10: 0803936052
• Format: Hardcover
• Copyright: 1991-05-01
• Publisher: Sage Pubns

Summary

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.

Table of Contents

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