Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences

by Cohen; Jacob

• ISBN13: 9780805822236
• ISBN10: 0805822232
• Edition: 3rd
• Format: Hardcover
• Copyright: 2002-08-01
• Publisher: Routledge

Summary

This classic text on multiple regression is noted for its non-mathematical, applied, and data-analytic approach. Readers profit from its verbal-conceptual exposition and frequent use of examples. The applied emphasis provides clear illustrations of the principles and provides worked examples of the types of applications that are possible. Researchers learn how to specify regression models that directly address their research questions. An overview of the fundamental ideas of multiple regression and a review of bivariate correlation and regression and other elementary statistical concepts provide a strong foundation for a solid understanding of the rest of the text. The third edition features an increased emphasis on graphics and the use of confidence intervals and effect size measures and an accompanying CD with data for most of the numerical examples along with the computer code for SPSS, SAS, and SYSTAT. Applied Multiple Regression serves as both a textbook for graduate students and as a reference tool for researchers in psychology, education, health sciences, communications, business, sociology, political science, anthropology, and economics. An introductory knowledge of statistics is required. Self-standing chapters minimize the need for researchers to refer to previous chapters. The book is an ideal text for courses on multiple regression and correlational methods.

Author Biography

Jacob Cohen (deceased) New York University Patricia Cohen New York State Psychiatric Institute and Columbia University College of Physicians and Surgeons Stephen G. West Arizona State University Leona S. Aiken Arizona State University

Table of Contents

Preface	xxv
Introduction	1	(18)
Multiple Regression/Correlation as a General Data-Analytic System	1	(3)
Overview	1	(1)
Testing Hypotheses Using Multiple Regression/Correlation: Some Examples	2	(1)
Multiple Regression/Correlation in Prediction Models	3	(1)
A Comparison of Multiple Regression/Correlation and Analysis of Variance Approaches	4	(2)
Historical Background	4	(1)
Hypothesis Testing and Effect Sizes	5	(1)
Multiple Regression/Correlation and the Complexity of Behavioral Science	6	(4)
Multiplicity of Influences	6	(1)
Correlation Among Research Factors and Partialing	6	(1)
Form of Information	7	(1)
Shape of Relationship	8	(1)
General and Conditional Relationships	9	(1)
Orientation of the Book	10	(4)
Nonmathematical	11	(1)
Applied	11	(1)
Data-Analytic	12	(1)
Inference Orientation and Specification Error	13	(1)
Computation, the Computer, and Numerical Results	14	(2)
Computation	14	(1)
Numerical Results: Reporting and Rounding	14	(1)
Significance Tests, Confidence Intervals, and Appendix Tables	15	(1)
The Spectrum of Behavioral Science	16	(1)
Plan for the Book	16	(2)
Content	16	(1)
Structure: Numbering of Sections, Tables, and Equations	17	(1)
Summary	18	(1)
Bivariate Correlation and Regression	19	(45)
Tabular and Graphic Representations of Relationships	19	(4)
The Index of Linear Correlation Between Two Variables: The Pearson Product Moment Correlation Coefficient	23	(5)
Standard Scores: Making Units Comparable	23	(3)
The Product Moment Correlation as a Function of Differences Between z Scores	26	(2)
Alternative Formulas for the Product Moment Correlation Coefficient	28	(4)
r as the Average Product of z Scores	28	(1)
Raw Score Formulas for r	29	(1)
Point Biserial r	29	(1)
Phi (φ) Coefficient	30	(1)
Rank Correlation	31	(1)
Regression Coefficients: Estimating Y From X	32	(4)
Regression Toward the Mean	36	(1)
The Standard Error of Estimate and Measures of the Strength of Association	37	(4)
Summary of Definitions and Interpretations	41	(1)
Statistical Inference With Regression and Correlation Coefficients	41	(9)
Assumptions Underlying Statistical Inference With Byx, B0, Yi, and rXY	41	(1)
Estimation With Confidence Intervals	42	(5)
Null Hypothesis Significance Tests (NHSTs)	47	(3)
Confidence Limits and Null Hypothesis Significance Testing	50	(1)
Precision and Power	50	(3)
Precision of Estimation	50	(1)
Power of Null Hypothesis Significance Tests	51	(2)
Factors Affecting the Size of r	53	(9)
The Distributions of X and Y	53	(2)
The Reliability of the Variables	55	(2)
Restriction of Range	57	(2)
Part-Whole Correlations	59	(1)
Ratio or Index Variables	60	(2)
Curvilinear Relationships	62	(1)
Summary	62	(2)
Multiple Regression/Correlation With Two or More Independent Variables	64	(37)
Introduction: Regression and Causal Models	64	(2)
What Is a Cause?	64	(1)
Diagrammatic Representation of Causal Models	65	(1)
Regression With Two Independent Variables	66	(3)
Measures of Association With Two Independent Variables	69	(6)
Multiple R and R2	69	(3)
Semipartial Correlation Coefficients and Increments to R2	72	(2)
Partial Correlation Coefficients	74	(1)
Patterns of Association Between Y and Two Independent Variables	75	(4)
Direct and Indirect Effects	75	(1)
Partial Redundancy	76	(1)
Suppression in Regression Models	77	(1)
Spurious Effects and Entirely Indirect Effects	78	(1)
Multiple Regression/Correlation With k Independent Variables	79	(7)
Introduction: Components of the Prediction Equation	79	(1)
Partial Regression Coefficients	80	(2)
R, R2, and Shrunken R2	82	(2)
sr and sr2	84	(1)
pr and pr2	85	(1)
Example of Interpretation of Partial Coefficients	85	(1)
Statistical Inference With k Independent Variables	86	(4)
Standard Errors and Confidence Intervals for B and β	86	(2)
Confidence Intervals for R2	88	(1)
Confidence Intervals for Differences Between Independent R2s	88	(1)
Statistical Tests on Multiple and Partial Coefficients	88	(2)
Statistical Precision and Power Analysis	90	(5)
Introduction: Research Goals and the Null Hypothesis	90	(1)
The Precision and Power of R2	91	(2)
Precision and Power Analysis for Partial Coefficients	93	(2)
Using Multiple Regression Equations in Prediction	95	(4)
Prediction of Y for a New Observation	95	(1)
Correlation of Individual Variables With Predicted Values	96	(1)
Cross-Validation and Unit Weighting	97	(1)
Multicollinearity	98	(1)
Summary	99	(2)
Data Visualization, Exploration, and Assumption Checking: Diagnosing and Solving Regression Problems I	101	(50)
Introduction	101	(1)
Some Useful Graphical Displays of the Original Data	102	(15)
Univariate Displays	103	(7)
Bivariate Displays	110	(5)
Correlation and Scatterplot Matrices	115	(2)
Assumptions and Ordinary Least Squares Regression	117	(8)
Assumptions Underlying Multiple Linear Regression	117	(7)
Ordinary Least Squares Estimation	124	(1)
Detecting Violations of Assumptions	125	(16)
Form of the Relationship	125	(2)
Omitted Independent Variables	127	(2)
Measurement Error	129	(1)
Homoscedasticity of Residuals	130	(4)
Nonindependence of Residuals	134	(3)
Normality of Residuals	137	(4)
Remedies: Alternative Approaches When Problems Are Detected	141	(9)
Form of the Relationship	141	(2)
Inclusion of All Relevant Independent Variables	143	(1)
Measurement Error in the Independent Variables	144	(1)
Nonconstant Variance	145	(2)
Nonindependence of Residuals	147	(3)
Summary	150	(1)
Data-Analytic Strategies Using Multiple Regression/Correlation	151	(42)
Research Questions Answered by Correlations and Their Squares	151	(3)
Net Contribution to Prediction	152	(1)
Indices of Differential Validity	152	(1)
Comparisons of Predictive Utility	152	(1)
Attribution of a Fraction of the XY Relationship to a Third Variable	153	(1)
Which of Two Variables Accounts for More of the XY Relationship?	153	(1)
Are the Various Squared Correlations in One Population Different From Those in Another Given the Same Variables?	154	(1)
Research Questions Answered by B or β	154	(4)
Regression Coefficients as Reflections of Causal Effects	154	(1)
Alternative Approaches to Making BYX Substantively Meaningful	154	(3)
Are the Effects of a Set of Independent Variables on Two Different Outcomes in a Sample Different?	157	(1)
What Are the Reciprocal Effects of Two Variables on One Another?	157	(1)
Hierarchical Analysis Variables in Multiple Regression/Correlation	158	(4)
Causal Priority and the Removal of Confounding Variables	158	(2)
Research Relevance	160	(1)
Examination of Alternative Hierarchical Sequences of Independent Variables Sets	160	(1)
Stepwise Regression	161	(1)
The Analysis of Sets of Independent Variables	162	(9)
Types of Sets	162	(2)
The Simultaneous and Hierarchical Analyses of Sets	164	(2)
Variance Proportions for Sets and the Ballantine Again	166	(3)
B and β Coefficients for Variables Within Sets	169	(2)
Significance Testing for Sets	171	(5)
Application in Hierarchical Analysis	172	(1)
Application in Simultaneous Analysis	173	(1)
Using Computer Output to Determine Statistical Significance	174	(1)
An Alternative F Test: Using Model 2 Error Estimate From the Final Model	174	(2)
Power Analysis for Sets	176	(6)
Determining n* for the F Test of sR2B with Model 1 or Model 2 Error	177	(2)
Estimating the Population sR2 Values	179	(1)
Setting Power for n*	180	(1)
Reconciling Different n*s	180	(1)
Power as a Function of n	181	(1)
Tactics of Power Analysis	182	(1)
Statistical Inference Strategy in Multiple Regression/Correlation	182	(8)
Controlling and Balancing Type I and Type II Errors in Inference	182	(3)
Less Is More	185	(1)
Least Is Last	186	(1)
Adaptation of Fisher's Protected t Test	187	(3)
Statistical Inference and the Stage of Scientific Investigations	190	(1)
Summary	190	(3)
Quantitative Scales, Curvilinear Relationships, and Transformations	193	(62)
Introduction	193	(3)
What Do We Mean by Linear Regression?	193	(1)
Linearity in the Variables and Linear Multiple Regression	194	(1)
Four Approaches to Examining Nonlinear Relationships in Multiple Regression	195	(1)
Power Polynomials	196	(18)
Method	196	(2)
An Example: Quadratic Fit	198	(3)
Centering Predictors in Polynomial Equations	201	(3)
Relationship of Test of Significance of Highest Order Coefficient and Gain in Prediction	204	(1)
Interpreting Polynomial Regression Results	205	(2)
Another Example: A Cubic Fit	207	(2)
Strategy and Limitations	209	(4)
More Complex Equations	213	(1)
Orthogonal Polynomials	214	(7)
The Cubic Example Revisited	216	(3)
Unequal n and Unequal Intervals	219	(1)
Applications and Discussion	220	(1)
Nonlinear Transformations	221	(30)
Purposes of Transformation and the Nature of Transformations	221	(2)
The Conceptual Basis of Transformations and Model Checking Before and After Transformation---Is It Always Ideal to Transform?	223	(1)
Logarithms and Exponents; Additive and Proportional Relationships	223	(2)
Linearizing Relationships	225	(2)
Linearizing Relationships Based on Strong Theoretical Models	227	(5)
Linearizing Relationships Based on Weak Theoretical Models	232	(1)
Empirically Driven Transformations in the Absence of Strong or Weak Models	233	(1)
Empirically Driven Transformation for Linearization: The Ladder of Re-expression and the Bulging Rule	233	(3)
Empirically Driven Transformation for Linearization in the Absence of Models: Box-Cox Family of Power Transformations on Y	236	(3)
Empirically Driven Transformation for Linearization in the Absence of Models: Box-Tidwell Family of Power Transformations on X	239	(1)
Linearization of Relationships With Correlations: Fisher z Transform of r	240	(1)
Transformations That Linearize Relationships for Counts and Proportions	240	(4)
Variance Stabilizing Transformations and Alternatives for Treatment of Heteroscedasticity	244	(2)
Transformations to Normalize Variables	246	(1)
Diagnostics Following Transformation	247	(1)
Measuring and Comparing Model Fit	248	(1)
Second-Order Polynomial Numerical Example Revisited	248	(1)
When to Transform and the Choice of Transformation	249	(2)
Nonlinear Regression	251	(1)
Nonparametric Regression	252	(1)
Summary	253	(2)
Interactions Among Continuous Variables	255	(47)
Introduction	255	(6)
Interactions Versus Additive Effects	256	(3)
Conditional First-Order Effects in Equations Containing Interactions	259	(2)
Centering Predictors and the Interpretation of Regression Coefficients in Equations Containing Interactions	261	(6)
Regression with Centered Predictors	261	(1)
Relationship Between Regression Coefficients in the Uncentered and Centered Equations	262	(1)
Centered Equations With No Interaction	262	(2)
Essential Versus Nonessential Multicollinearity	264	(1)
Centered Equations With Interactions	264	(2)
The Highest Order Interaction in the Centered Versus Uncentered Equation	266	(1)
Do Not Center Y	266	(1)
A Recommendation for Centering	266	(1)
Simple Regression Equations and Simple Slopes	267	(5)
Plotting Interactions	269	(1)
Moderator Variables	269	(1)
Simple Regression Equations	269	(1)
Overall Regression Coefficient and Simple Slope at the Mean	270	(1)
Simple Slopes From Uncentered Versus Centered Equations Are Identical	271	(1)
Linear by Linear Interactions	271	(1)
Interpreting Interactions in Multiple Regression and Analysis of Variance	272	(1)
Post Hoc Probing of Interactions	272	(10)
Standard Error of Simple Slopes	272	(1)
Equation Dependence of Simple Slopes and Their Standard Errors	273	(1)
Tests of Significance of Simple Slopes	273	(1)
Confidence Intervals Around Simple Slopes	274	(1)
A Numerical Example	275	(6)
The Uncentered Regression Equation Revisited	281	(1)
First-Order Coefficients in Equations Without and With Interactions	281	(1)
Interpretation and the Range of Data	282	(1)
Standardized Estimates for Equations Containing Interactions	282	(2)
Interactions as Partialed Effects: Building Regression Equations With Interactions	284	(1)
Patterns of First-Order and Interactive Effects	285	(5)
Three Theoretically Meaningful Patterns of First-Order and Interaction Effects	285	(1)
Ordinal Versus Disordinal Interactions	286	(4)
Three-Predictor Interactions in Multiple Regression	290	(2)
Curvilinear by Linear Interactions	292	(3)
Interactions Among Sets of Variables	295	(2)
Issues in the Detection of Interactions: Reliability, Predictor Distributions, Model Specification	297	(3)
Variable Reliability and Power to Detect Interactions	297	(1)
Sampling Designs to Enhance Power to Detect Interactions---Optimal Design	298	(1)
Difficulty in Distinguishing Interactions Versus Curvilinear Effects	299	(1)
Summary	300	(2)
Categorical or Nominal Independent Variables	302	(52)
Introduction	302	(1)
Categories as a Set of Independent Variables	302	(1)
The Representation of Categories or Nominal Scales	302	(1)
Dummy-Variable Coding	303	(17)
Coding the Groups	303	(5)
Pearson Correlations of Dummy Variables With Y	308	(3)
Correlations Among Dummy-Coded Variables	311	(1)
Multiple Correlation of the Dummy-Variable Set With Y	311	(1)
Regression Coefficients for Dummy Variables	312	(4)
Partial and Semipartial Correlations for Dummy Variables	316	(1)
Dummy-Variable Multiple Regression/Correlation and One-Way Analysis of Variance	317	(2)
A Cautionary Note: Dummy-Variable-Like Coding Systems	319	(1)
Dummy-Variable Coding When Groups Are Not Mutually Exclusive	320	(1)
Unweighted Effects Coding	320	(8)
Introduction: Unweighted and Weighted Effects Coding	320	(1)
Constructing Unweighted Effects Codes	321	(3)
The R2 and the ryiS for Unweighted Effects Codes	324	(1)
Regression Coefficients and Other Partial Effects in Unweighted Code Sets	325	(3)
Weighted Effects Coding	328	(4)
Selection Considerations for Weighted Effects Coding	328	(1)
Constructing Weighted Effects	328	(2)
The R2 and R2 for Weighted Effects Codes	330	(1)
Interpretation and Testing of B With Unweighted Codes	331	(1)
Contrast Coding	332	(9)
Considerations in the Selection of a Contrast Coding Scheme	332	(1)
Constructing Contrast Codes	333	(4)
The R2 and R2	337	(1)
Partial Regression Coefficients	337	(3)
Statistical Power and the Choice of Contrast Codes	340	(1)
Nonsense Coding	341	(1)
Coding Schemes in the Context of Other Independent Variables	342	(9)
Combining Nominal and Continuous Independent Variables	342	(1)
Calculating Adjusted Means for Nominal Independent Variables	343	(1)
Adjusted Means for Combinations of Nominal and Quantitative Independent Variables	344	(4)
Adjusted Means for More Than Two Groups and Alternative Coding Methods	348	(2)
Multiple Regression/Correlation With Nominal Independent Variables and the Analysis of Covariance	350	(1)
Summary	351	(3)
Interactions With Categorical Variables	354	(36)
Nominal Scale by Nominal Scale Interactions	354	(12)
The 2 by 2 Design	354	(7)
Regression Analyses of Multiple Sets of Nominal Variables With More Than Two Categories	361	(5)
Interactions Involving More Than Two Nominal Scales	366	(9)
An Example of Three Nominal Scales Coded by Alternative Methods	367	(5)
Interactions Among Nominal Scales in Which Not All Combinations Are Considered	372	(1)
What If the Categories for One or More Nominal ``Scales'' Are Not Mutually Exclusive?	373	(1)
Consideration of pr, β, and Variance Proportions for Nominal Scale Interaction Variables	374	(1)
Summary of Issues and Recommendations for Interactions Among Nominal Scales	374	(1)
Nominal Scale by Continuous Variable Interactions	375	(13)
A Reminder on Centering	375	(1)
Interactions of a Continuous Variable With Dummy-Variable Coded Groups	375	(3)
Interactions Using Weighted or Unweighted Effects Codes	378	(1)
Interactions With a Contrast-Coded Nominal Scale	379	(1)
Interactions Coded to Estimate Simple Slopes of Groups	380	(3)
Categorical Variable Interactions With Nonlinear Effects of Scaled Independent Variables	383	(3)
Interactions of a Scale With Two or More Categorical Variables	386	(2)
Summary	388	(2)
Outliers and Multicollinearity: Diagnosing and Solving Regression Problems II	390	(41)
Introduction	390	(1)
Outliers: Introduction and Illustration	391	(3)
Detecting Outliers: Regression Diagnostics	394	(17)
Extremity on the Independent Variables: Leverage	394	(4)
Extremity on Y: Discrepancy	398	(4)
Influence on the Regression Estimates	402	(4)
Location of Outlying Points and Diagnostic Statistics	406	(3)
Summary and Suggestions	409	(2)
Sources of Outliers and Possible Remedial Actions	411	(8)
Sources of Outliers	411	(4)
Remedial Actions	415	(4)
Multicollinearity	419	(6)
Exact Collinearity	419	(1)
Multicollinearity: A Numerical Illustration	420	(2)
Measures of the Degree of Multicollinearity	422	(3)
Remedies for Multicollinearity	425	(5)
Model Respecification	426	(1)
Collection of Additional Data	427	(1)
Ridge Regression	427	(1)
Principal Components Regression	428	(1)
Summary of Multicollinearity Considerations	429	(1)
Summary	430	(1)
Missing Data	431	(21)
Basic Issues in Handling Missing Data	431	(4)
Minimize Missing Data	431	(1)
Types of Missing Data	432	(1)
Traditional Approaches to Missing Data	433	(2)
Missing Data in Nominal Scales	435	(7)
Coding Nominal Scale X for Missing Data	435	(4)
Missing Data on Two Dichotomies	439	(1)
Estimation Using the EM Algorithm	440	(2)
Missing Data in Quantitative Scales	442	(8)
Available Alternatives	442	(2)
Imputation of Values for Missing Cases	444	(3)
Modeling Solutions to Missing Data in Scaled Variables	447	(1)
An Illustrative Comparison of Alternative Methods	447	(3)
Rules of Thumb	450	(1)
Summary	450	(2)
Multiple Regression/Correlation and Causal Models	452	(27)
Introduction	452	(8)
Limits on the Current Discussion and the Relationship Between Causal Analysis and Analysis of Covariance	452	(2)
Theories and Multiple Regression/Correlation Models That Estimate and Test Them	454	(3)
Kinds of Variables in Causal Models	457	(2)
Regression Models as Causal Models	459	(1)
Models Without Reciprocal Causation	460	(7)
Direct and Indirect Effects	460	(4)
Path Analysis and Path Coefficients	464	(1)
Hierarchical Analysis and Reduced Form Equations	465	(1)
Partial Causal Models and the Hierarchical Analysis of Sets	466	(1)
Testing Model Elements	467	(1)
Models With Reciprocal Causation	467	(1)
Identification and Overidentification	468	(1)
Just Identified Models	468	(1)
Overidentification	468	(1)
Underidentification	469	(1)
Latent Variable Models	469	(6)
An Example of a Latent Variable Model	469	(2)
How Latent Variables Are Estimated	471	(1)
Fixed and Free Estimates in Latent Variable Models	472	(1)
Goodness-of-Fit Tests of Latent Variable Models	472	(1)
Latent Variable Models and the Correction for Attenuation	473	(1)
Characteristics of Data Sets That Make Latent Variable Analysis the Method of Choice	474	(1)
A Review of Causal Model and Statistical Assumptions	475	(1)
Specification Error	475	(1)
Identification Error	475	(1)
Comparisons of Causal Models	476	(1)
Nested Models	476	(1)
Longitudinal Data in Causal Models	476	(1)
Summary	477	(2)
Alternative Regression Models: Logistic, Poisson Regression, and the Generalized Linear Model	479	(57)
Ordinary Least Squares Regression Revisited	479	(3)
Three Characteristics of Ordinary Least Squares Regression	480	(1)
The Generalized Linear Model	480	(1)
Relationship of Dichotomous and Count Dependent Variables Y to a Predictor	481	(1)
Dichotomous Outcomes and Logistic Regression	482	(37)
Extending Linear Regression: The Linear Probability Model and Discriminant Analysis	483	(2)
The Nonlinear Transformation From Predictor to Predicted Scores: Probit and Logistic Transformation	485	(1)
The Logistic Regression Equation	486	(1)
Numerical Example: Three Forms of the Logistic Regression Equation	487	(5)
Understanding the Coefficients for the Predictor in Logistic Regression	492	(1)
Multiple Logistic Regression	493	(1)
Numerical Example	494	(3)
Confidence Intervals on Regression Coefficients and Odds Ratios	497	(1)
Estimation of the Regression Model: Maximum Likelihood	498	(1)
Deviances: Indices of Overall Fit of the Logistic Regression Model	499	(3)
Multiple R2 Analogs in Logistic Regression	502	(2)
Testing Significance of Overall Model Fit: The Likelihood Ratio Test and the Test of Model Deviance	504	(3)
Χ2 Test for the Significance of a Single Predictor in a Multiple Logistic Regression Equation	507	(1)
Hierarchical Logistic Regression: Likelihood Ratio Χ2 Test for the Significance of a Set of Predictors Above and Beyond Another Set	508	(1)
Akaike's Information Criterion and the Bayesian Information Criterion for Model Comparison	509	(1)
Some Treachery in Variable Scaling and Interpretation of the Odds Ratio	509	(3)
Regression Diagnostics in Logistic Regression	512	(4)
Sparseness of Data	516	(1)
Classification of Cases	516	(3)
Extensions of Logistic Regression to Multiple Response Categories: Polytomous Logistic Regression and Ordinal Logistic Regression	519	(6)
Polytomous Logistic Regression	519	(1)
Nested Dichotomies	520	(2)
Ordinal Logistic Regression	522	(3)
Models for Count Data: Poisson Regression and Alternatives	525	(7)
Linear Regression Applied to Count Data	525	(1)
Poisson Probability Distribution	526	(2)
Poisson Regression Analysis	528	(2)
Overdispersion and Alternative Models	530	(2)
Independence of Observations	532	(1)
Sources on Poisson Regression	532	(1)
Full Circle: Parallels Between Logistic and Poisson Regression, and the Generalized Linear Model	532	(3)
Parallels Between Poisson and Logistic Regression	532	(2)
The Generalized Linear Model Revisited	534	(1)
Summary	535	(1)
Random Coefficient Regression and Multilevel Models	536	(32)
Clustering Within Data Sets	536	(3)
Clustering, Alpha Inflation, and the Intraclass Correlation	537	(1)
Estimating the Intraclass Correlation	538	(1)
Analysis of Clustered Data With Ordinary Least Squares Approaches	539	(4)
Numerical Example, Analysis of Clustered Data With Ordinary Least Squares Regression	541	(2)
The Random Coefficient Regression Model	543	(1)
Random Coefficient Regression Model and Multilevel Data Structure	544	(6)
Ordinary Least Squares (Fixed Effects) Regression Revisited	544	(1)
Fixed and Random Variables	544	(1)
Clustering and Hierarchically Structured Data	545	(1)
Structure of the Random Coefficient Regression Model	545	(1)
Level 1 Equations	546	(1)
Level 2 Equations	547	(1)
Mixed Model Equation for Random Coefficient Regression	548	(1)
Variance Components---New Parameters in the Multilevel Model	548	(1)
Variance Components and Random Coefficient Versus Ordinary Least Squares (Fixed Effects) Regression	549	(1)
Parameters of the Random Coefficient Regression Model: Fixed and Random Effects	550	(1)
Numerical Example: Analysis of Clustered Data With Random Coefficient Regression	550	(3)
Unconditional Cell Means Model and the Intraclass Correlation	551	(1)
Testing the Fixed and Random Parts of the Random Coefficient Regression Model	552	(1)
Clustering as a Meaningful Aspect of the Data	553	(1)
Multilevel Modeling With a Predictor at Level 2	553	(2)
Level 1 Equations	553	(1)
Revised Level 2 Equations	554	(1)
Mixed Model Equation With Level 1 Predictor and Level 2 Predictor of Intercept and Slope and the Cross-Level Interaction	554	(1)
An Experimental Design as a Multilevel Data Structure: Combining Experimental Manipulation With Individual Differences	555	(1)
Numerical Example: Multilevel Analysis	556	(4)
Estimation of the Multilevel Model Parameters: Fixed Effects, Variance Components, and Level 1 Equations	560	(3)
Fixed Effects and Variance Components	560	(1)
An Equation for Each Group: Empirical Bayes Estimates of Level 1 Coefficients	560	(3)
Statistical Tests in Multilevel Models	563	(1)
Fixed Effects	563	(1)
Variance Components	563	(1)
Some Model Specification Issues	564	(1)
The Same Variable at Two Levels	564	(1)
Centering in Multilevel Models	564	(1)
Statistical Power of Multilevel Models	565	(1)
Choosing Between the Fixed Effects Model and the Random Coefficient Model	565	(1)
Sources on Multilevel Modeling	566	(1)
Multilevel Models Applied to Repeated Measures Data	566	(1)
Summary	567	(1)
Longitudinal Regression Methods	568	(40)
Introduction	568	(1)
Chapter Goals	568	(1)
Purposes of Gathering Data on Multiple Occasions	569	(1)
Analyses of Two-Time-Point Data	569	(4)
Change or Regressed Change?	570	(1)
Alternative Regression Models for Effects Over a Single Unit of Time	571	(2)
Three- or Four-Time-Point Data	573	(1)
Repeated Measure Analysis of Variance	573	(5)
Multiple Error Terms in Repeated Measure Analysis of Variance	574	(1)
Trend Analysis in Analysis of Variance	575	(1)
Repeated Measure Analysis of Variance in Which Time Is Not the Issue	576	(2)
Multilevel Regression of Individual Changes Over Time	578	(10)
Patterns of Individual Change Over Time	578	(4)
Adding Other Fixed Predictors to the Model	582	(1)
Individual Differences in Variation Around Individual Slopes	583	(1)
Alternative Developmental Models and Error Structures	584	(2)
Alternative Link Functions for Predicting Y From Time	586	(1)
Unbalanced Data: Variable Timing and Missing Data	587	(1)
Latent Growth Models: Structural Equation Model Representation of Multilevel Data	588	(7)
Estimation of Changes in True Scores	589	(1)
Representation of Latent Growth Models in Structural Equation Model Diagrams	589	(5)
Comparison of Multilevel Regression and Structural Equation Model Analysis of Change	594	(1)
Time Varying Independent Variables	595	(1)
Survival Analysis	596	(4)
Regression Analysis of Time Until Outcome and the Problem of Censoring	596	(3)
Extension to Time-Varying Independent Variables	599	(1)
Extension to Multiple Episode Data	599	(1)
Extension to a Categorical Outcome: Event-History Analysis	600	(1)
Time Series Analysis	600	(2)
Units of Observation in Time Series Analyses	601	(1)
Time Series Analyses Applications	601	(1)
Time Effects in Time Series	602	(1)
Extension of Time Series Analyses to Multiple Units or Subjects	602	(1)
Dynamic System Analysis	602	(2)
Statistical Inference and Power Analysis in Longitudinal Analyses	604	(1)
Summary	605	(3)
Multiple Dependent Variables: Set Correlation	608	(35)
Introduction to Ordinary Least Squares Treatment of Multiple Dependent Variables	608	(2)
Set Correlation Analysis	608	(1)
Canonical Analysis	609	(1)
Elements of Set Correlation	610	(1)
Measures of Multivariate Association	610	(3)
R2Y, X, the Proportion of Generalized Variance	610	(1)
T2Y, X and P2Y, X, Proportions of Additive Variance	611	(2)
Partialing in Set Correlation	613	(2)
Frequent Reasons for Partialing Variable Sets From the Basic Sets	613	(1)
The Five Types of Association Between Basic Y and X Sets	614	(1)
Tests of Statistical Significance and Statistical Power	615	(2)
Testing the Null Hypothesis	615	(1)
Estimators of the Population R2Y, X, T2Y, X, and P2Y, X	616	(1)
Guarding Against Type I Error Inflation	617	(1)
Statistical Power Analysis in Set Correlation	617	(2)
Comparison of Set Correlation With Multiple Analysis of Variance	619	(1)
New Analytic Possibilities With Set Correlation	620	(1)
Illustrative Examples	621	(6)
A Simple Whole Association	621	(1)
A Multivariate Analysis of Partial Variance	622	(1)
A Hierarchical Analysis of a Quantitative Set and Its Unique Components	623	(2)
Bipartial Association Among Three Sets	625	(2)
Summary	627	(4)
APPENDICES
Appendix 1: The Mathematical Basis for Multiple Regression/Correlation and Identification of the Inverse Matrix Elements	631	(5)
A1.1 Alternative Matrix Methods	634	(1)
A1.2 Determinants	634	(2)
Appendix 2: Determination of the Inverse Matrix and Applications Thereof	636	(7)
A2.1 Hand Calculation of the Multiple Regression/Correlation Problem	636	(4)
A2.2 Testing the Difference Between Partial βs and Bs From the Same Sample	640	(2)
A2.3 Testing the Difference Between βs for Different Dependent Variables From a Single Sample	642	(1)
Appendix Tables	643	(12)
Table A t Values for α = .01, .05 (Two Tailed)	643	(1)
Table B z' Transformation of r	644	(1)
Table C Normal Distribution	645	(1)
Table D F Values for α = .01, .05	646	(4)
Table E L Values for α = .01, .05	650	(2)
Table F Power of Significance Test of r at α = .01, .05 (Two Tailed)	652	(2)
Table G n* to Detect r by t Test at α = .01, .05 (Two Tailed)	654	(1)
References	655	(16)
Glossary	671	(12)
Statistical Symbols and Abbreviations	683	(4)
Author Index	687	(4)
Subject Index	691