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Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, 3rd Edition

by Jacob Cohen, Patricia Cohen, Stephen G. West, Leona S. Aiken

June 2013

Beginner to intermediate

536 pages

29h 22m

English

Routledge

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Copyright
Dedication
Contents
Preface
Chapter 1: Introduction
1.1 Multiple Regression/Correlation as a General Data-Analytic System1.1.1 Overview1.1.2 Testing Hypotheses Using Multiple Regression/Correlation: Some Examples1.1.3 Multiple Regression/Correlation in Prediction Models1.2 A Comparison of Multiple Regression/Correlation and Analysis of Variance Approaches1.2.1 Historical Background1.2.2 Hypothesis Testing and Effect Sizes1.3 Multiple Regression/Correlation and the Complexity of Behavioral Science1.3.1 Multiplicity of Influences1.3.2 Correlation Among Research Factors and Partialing1.3.3 Form of Information1.3.4 Shape of Relationship1.3.5 General and Conditional Relationships1.4 Orientation of the Book1.4.1 Nonmathematical1.4.2 Applied1.4.3 Data-Analytic1.4.4 Inference Orientation and Specification Error1.5 Computation, the Computer, and Numerical Results1.5.1 Computation1.5.2 Numerical Results: Reporting and Rounding1.5.3 Significance Tests, Confidence Intervals, and Appendix Tables1.6 The Spectrum of Behavioral Science1.7 Plan for the Book1.7.1 Content1.7.2 Structure: Numbering of Sections, Tables, and Equations1.8 Summary
Chapter 2: Bivariate Correlation and Regression
2.1 Tabular and Graphic Representations of Relationships2.2 The Index of Linear Correlation Between Two Variables: The Pearson Product Moment Correlation Coefficient2.2.1 Standard Scores: Making Units Comparable2.2.2 The Product Moment Correlation as a Function of Differences Between z Scores2.3 Alternative Formulas for the Product Moment Correlation Coefficient2.3.1 r as the Average Product of z Scores2.3.2 Raw Score Formulas for r2.3.3 Point Biserial r2.3.4 Phi (Φ) Coefficient2.3.5 Rank Correlation2.4 Regression Coefficients: Estimating Y From X2.5 Regression Toward the Mean2.6 The Standard Error of Estimate and Measures of the Strength of Association2.7 Summary of Definitions and Interpretations2.8 Statistical Inference With Regression and Correlation Coefficients2.8.1 Assumptions Underlying Statistical Inference With BYX, B0, Ŷi, and rXY2.8.2 Estimation With Confidence Intervals2.8.3 Null Hypothesis Significance Tests (NHSTs)2.8.4 Confidence Limits and Null Hypothesis Significance Testing2.9 Precision and Power2.9.1 Precision of Estimation2.9.2 Power of Null Hypothesis Significance Tests2.10 Factors Affecting the Size of r2.10.1 The Distributions of X and Y2.10.2 The Reliability of the Variables2.10.3 Restriction of Range2.10.4 Part-Whole Correlations2.10.5 Ratio or Index Variables2.10.6 Curvilinear Relationships2.11 Summary
Chapter 3: Multiple Regression/Correlation With Two or More Independent Variables
3.1 Introduction: Regression and Causal Models3.1.1 What Is a Cause?3.1.2 Diagrammatic Representation of Causal Models3.2 Regression With Two Independent Variables3.3 Measures of Association With Two Independent Variables3.3.1 Multiple R and R23.3.2 Semipartial Correlation Coefficients and Increments to R23.3.3 Partial Correlation Coefficients3.4 Patterns of Association Between Y and Two Independent Variables3.4.1 Direct and Indirect Effects3.4.2 Partial Redundancy3.4.3 Suppression in Regression Models3.4.4 Spurious Effects and Entirely Indirect Effects3.5 Multiple Regression/Correlation With k Independent Variables3.5.1 Introduction: Components of the Prediction Equation3.5.2 Partial Regression Coefficients3.5.3 R, R2, and Shrunken R23.5.4 sr and sr23.5.5 pr and pr23.5.6 Example of Interpretation of Partial Coefficients3.6 Statistical Inference With k Independent Variables3.6.1 Standard Errors and Confidence Intervals for B and β3.6.2 Confidence Intervals for R23.6.3 Confidence Intervals for Differences Between Independent R2s3.6.4 Statistical Tests on Multiple and Partial Coefficients3.7 Statistical Precision and Power Analysis3.7.1 Introduction: Research Goals and the Null Hypothesis3.7.2 The Precision and Power of R23.7.3 Precision and Power Analysis for Partial Coefficients3.8 Using Multiple Regression Equations in Prediction3.8.1 Prediction of Y for a New Observation3.8.2 Correlation of Individual Variables With Predicted Values3.8.3 Cross-Validation and Unit Weighting3.8.4 Multicollinearity3.9 Summary

Chapter 4: Data Visualization, Exploration, and Assumption Checking: Diagnosing and Solving Regression Problems I
4.1 Introduction4.2 Some Useful Graphical Displays of the Original Data4.2.1 Univariate Displays4.2.2 Bivariate Displays4.2.3 Correlation and Scatterplot Matrices4.3 Assumptions and Ordinary Least Squares Regression4.3.1 Assumptions Underlying Multiple Linear Regression4.3.2 Ordinary Least Squares Estimation4.4 Detecting Violations of Assumptions4.4.1 Form of the Relationship4.4.2 Omitted Independent Variables4.4.3 Measurement Error4.4.4 Homoscedasticity of Residuals4.4.5 Nonindependence of Residuals4.4.6 Normality of Residuals4.5 Remedies: Alternative Approaches When Problems Are Detected4.5.1 Form of the Relationship4.5.2 Inclusion of All Relevant Independent Variables4.5.3 Measurement Error in the Independent Variables4.5.4 Nonconstant Variance4.5.5 Nonindependence of Residuals4.6 Summary
Chapter 5: Data-Analytic Strategies Using Multiple Regression/Correlation
5.1 Research Questions Answered by Correlations and Their Squares5.1.1 Net Contribution to Prediction5.1.2 Indices of Differential Validity5.1.3 Comparisons of Predictive Utility5.1.4 Attribution of a Fraction of the XY Relationship to a Third Variable5.1.5 Which of Two Variables Accounts for More of the XY Relationship?5.1.6 Are the Various Squared Correlations in One Population Different From Those in Another Given the Same Variables?5.2 Research Questions Answered by B Or β5.2.1 Regression Coefficients as Reflections of Causal Effects5.2.2 Alternative Approaches to Making BYX Substantively Meaningful5.2.3 Are the Effects of a Set of Independent Variables on Two Different Outcomes in a Sample Different?5.2.4 What Are the Reciprocal Effects of Two Variables on One Another?5.3 Hierarchical Analysis Variables in Multiple Regression/Correlation5.3.1 Causal Priority and the Removal of Confounding Variables5.3.2 Research Relevance5.3.3 Examination of Alternative Hierarchical Sequences of Independent Variable Sets5.3.4 Stepwise Regression5.4 The Analysis of Sets of Independent Variables5.4.1 Types of Sets5.4.2 The Simultaneous and Hierarchical Analyses of Sets5.4.3 Variance Proportions for Sets and the Ballantine Again5.4.4 B and β Coefficients for Variables Within Sets5.5 Significance Testing for Sets5.5.1 Application in Hierarchical Analysis5.5.2 Application in Simultaneous Analysis5.5.3 Using Computer Output to Determine Statistical Significance5.5.4 An Alternative F Test: Using Model 2 Error Estimate From the Final Model5.6 Power Analysis for Sets5.6.1 Determining n* for the F Test of sR2B with Model 1 or Model 2 Error5.6.2 Estimating the Population sR2 Values5.6.3 Setting Power for n*5.6.4 Reconciling Different n* s5.6.5 Power as a Function of n5.6.6 Tactics of Power Analysis5.7 Statistical Inference Strategy in Multiple Regression/Correlation5.7.1 Controlling and Balancing Type I and Type II Errors in Inference5.7.2 Less Is More5.7.3 Least Is Last5.7.4 Adaptation of Fisher’s Protected t Test5.7.5 Statistical Inference and the Stage of Scientific Investigations5.8 Summary
Chapter 6: Quantitative Scales, Curvilinear Relationships, and Transformations
6.1 Introduction6.1.1 What Do We Mean by Linear Regression?6.1.2 Linearity in the Variables and Linear Multiple Regression6.1.3 Four Approaches to Examining Nonlinear Relationships in Multiple Regression6.2 Power Polynomials6.2.1 Method6.2.2 An Example: Quadratic Fit6.2.3 Centering Predictors in Polynomial Equations6.2.4 Relationship of Test of Significance of Highest Order Coefficient and Gain in Prediction6.2.5 Interpreting Polynomial Regression Results6.2.6 Another Example: A Cubic Fit6.2.7 Strategy and Limitations6.2.8 More Complex Equations6.3 Orthogonal Polynomials6.3.1 The Cubic Example Revisited6.3.2 Unequal n and Unequal Intervals6.3.3 Applications and Discussion6.4 Nonlinear Transformations6.4.1 Purposes of Transformation and the Nature of Transformations6.4.2 The Conceptual Basis of Transformations and Model Checking Before and After Transformation—Is It Always Ideal to Transform?6.4.3 Logarithms and Exponents; Additive and Proportional Relationships6.4.4 Linearizing Relationships6.4.5 Linearizing Relationships Based on Strong Theoretical Models6.4.6 Linearizing Relationships Based on Weak Theoretical Models6.4.7 Empirically Driven Transformations in the Absence of Strong or Weak Models6.4.8 Empirically Driven Transformation for Linearization: The Ladder of Re-expression and the Bulging Rule6.4.9 Empirically Driven Transformation for Linearization in the Absence of Models: Box-Cox Family of Power Transformations on Y6.4.10 Empirically Driven Transformation for Linearization in the Absence of Models: Box-Tidwell Family of Power Transformations on X6.4.11 Linearization of Relationships With Correlations: Fisher z′ Transform of r6.4.12 Transformations That Linearize Relationships for Counts and Proportions6.4.13 Variance Stabilizing Transformations and Alternatives for Treatment of Heteroscedasticity6.4.14 Transformations to Normalize Variables6.4.15 Diagnostics Following Transformation6.4.16 Measuring and Comparing Model Fit6.4.17 Second-Order Polynomial Numerical Example Revisited6.4.18 When to Transform and the Choice of Transformation6.5 Nonlinear Regression6.6 Nonparametric Regression6.7 Summary
Chapter 7: Interactions Among Continuous Variables
7.1 Introduction7.1.1 Interactions Versus Additive Effects7.1.2 Conditional First-Order Effects in Equations Containing Interactions7.2 Centering Predictors and the Interpretation of Regression Coefficients in Equations Containing Interactions7.2.1 Regression with Centered Predictors7.2.2 Relationship Between Regression Coefficients in the Uncentered and Centered Equations7.2.3 Centered Equations With No Interaction7.2.4 Essential Versus Nonessential Multicollinearity7.2.5 Centered Equations With Interactions7.2.6 The Highest Order Interaction in the Centered Versus Uncentered Equation7.2.7 Do Not Center Y7.2.8 A Recommendation for Centering7.3 Simple Regression Equations and Simple Slopes7.3.1 Plotting Interactions7.3.2 Moderator Variables7.3.3 Simple Regression Equations7.3.4 Overall Regression Coefficient and Simple Slope at the Mean7.3.5 Simple Slopes From Uncentered Versus Centered Equations Are Identical7.3.6 Linear by Linear Interactions7.3.7 Interpreting Interactions in Multiple Regression and Analysis of Variance7.4 Post Hoc Probing of Interactions7.4.1 Standard Error of Simple Slopes7.4.2 Equation Dependence of Simple Slopes and Their Standard Errors7.4.3 Tests of Significance of Simple Slopes7.4.4 Confidence Intervals Around Simple Slopes7.4.5 A Numerical Example7.4.6 The Uncentered Regression Equation Revisited7.4.7 First-Order Coefficients in Equations Without and With Interactions7.4.8 Interpretation and the Range of Data7.5 Standardized Estimates for Equations Containing Interactions7.6 Interactions as Partialed Effects: Building Regression Equations With Interactions7.7 Patterns of First-Order and Interactive Effects7.7.1 Three Theoretically Meaningful Patterns of First-Order and Interaction Effects7.7.2 Ordinal Versus Disordinal Interactions7.8 Three-Predictor Interactions in Multiple Regression7.9 Curvilinear by Linear Interactions7.10 Interactions Among Sets of Variables7.11 Issues in the Detection of Interactions: Reliability, Predictor Distributions, Model Specification7.11.1 Variable Reliability and Power to Detect Interactions7.11.2 Sampling Designs to Enhance Power to Detect Interactions—Optimal Design7.11.3 Difficulty in Distinguishing Interactions Versus Curvilinear Effects7.12 Summary
Chapter 8: Categorical or Nominal Independent Variables
8.1 Introduction8.1.1 Categories as a Set of Independent Variables8.1.2 The Representation of Categories or Nominal Scales8.2 Dummy-Variable Coding8.2.1 Coding the Groups8.2.2 Pearson Correlations of Dummy Variables With Y8.2.3 Correlations Among Dummy-Coded Variables8.2.4 Multiple Correlation of the Dummy-Variable Set With Y8.2.5 Regression Coefficients for Dummy Variables8.2.6 Partial and Semipartial Correlations for Dummy Variables8.2.7 Dummy-Variable Multiple Regression/Correlation and One-Way Analysis of Variance8.2.8 A Cautionary Note: Dummy-Variable-Like Coding Systems8.2.9 Dummy-Variable Coding When Groups Are Not Mutually Exclusive8.3 Unweighted Effects Coding8.3.1 Introduction: Unweighted and Weighted Effects Coding8.3.2 Constructing Unweighted Effects Codes8.3.3 The R2 and the rYis for Unweighted Effects Codes8.3.4 Regression Coefficients and Other Partial Effects in Unweighted Code Sets8.4 Weighted Effects Coding8.4.1 Selection Considerations for Weighted Effects Coding8.4.2 Constructing Weighted Effects8.4.3 The R2 and R2 for Weighted Effects Codes8.4.4 Interpretation and Testing of B With Unweighted Codes8.5 Contrast Coding8.5.1 Considerations in the Selection of a Contrast Coding Scheme8.5.2 Constructing Contrast Codes8.5.3 The R2 and R28.5.4 Partial Regression Coefficients8.5.5 Statistical Power and the Choice of Contrast Codes8.6 Nonsense Coding8.7 Coding Schemes in the Context of Other Independent Variables8.7.1 Combining Nominal and Continuous Independent Variables8.7.2 Calculating Adjusted Means for Nominal Independent Variables8.7.3 Adjusted Means for Combinations of Nominal and Quantitative Independent Variables8.7.4 Adjusted Means for More Than Two Groups and Alternative Coding Methods8.7.5 Multiple Regression/Correlation With Nominal Independent Variables and the Analysis of Covariance8.8 Summary
Chapter 9: Interactions With Categorical Variables
9.1 Nominal Scale by Nominal Scale Interactions9.1.1 The 2 by 2 Design9.1.2 Regression Analyses of Multiple Sets of Nominal Variables With More Than Two Categories9.2 Interactions Involving More Than Two Nominal Scales9.2.1 An Example of Three Nominal Scales Coded by Alternative Methods9.2.2 Interactions Among Nominal Scales in Which Not All Combinations Are Considered9.2.3 What If the Categories for One or More Nominal “Scales” Are Not Mutually Exclusive?9.2.4 Consideration of pr, β, and Variance Proportions for Nominal Scale Interaction Variables9.2.5 Summary of Issues and Recommendations for Interactions Among Nominal Scales9.3 Nominal Scale by Continuous Variable Interactions9.3.1 A Reminder on Centering9.3.2 Interactions of a Continuous Variable With Dummy-Variable Coded Groups9.3.3 Interactions Using Weighted or Unweighted Effects Codes9.3.4 Interactions With a Contrast-Coded Nominal Scale9.3.5 Interactions Coded to Estimate Simple Slopes of Groups9.3.6 Categorical Variable Interactions With Nonlinear Effects of Scaled Independent Variables9.3.7 Interactions of a Scale With Two or More Categorical Variables9.4 Summary
Chapter 10: Outliers and Multicollinearity: Diagnosing and Solving Regression Problems II
10.1 Introduction10.2 Outliers: Introduction and Illustration10.3 Detecting Outliers: Regression Diagnostics10.3.1 Extremity on the Independent Variables: Leverage10.3.2 Extremity on Y: Discrepancy10.3.3 Influence on the Regression Estimates10.3.4 Location of Outlying Points and Diagnostic Statistics10.3.5 Summary and Suggestions10.4 Sources of Outliers and Possible Remedial Actions10.4.1 Sources of Outliers10.4.2 Remedial Actions10.5 Multicollinearity10.5.1 Exact Collinearity10.5.2 Multicollinearity: A Numerical Illustration10.5.3 Measures of the Degree of Multicollinearity10.6 Remedies for Multicollinearity10.6.1 Model Respecification10.6.2 Collection of Additional Data10.6.3 Ridge Regression10.6.4 Principal Components Regression10.6.5 Summary of Multicollinearity Considerations10.7 Summary
Chapter 11: Missing Data
11.1 Basic Issues in Handling Missing Data11.1.1 Minimize Missing Data11.1.2 Types of Missing Data11.1.3 Traditional Approaches to Missing Data11.2 Missing Data in Nominal Scales11.2.1 Coding Nominal Scale X for Missing Data11.2.2 Missing Data on Two Dichotomies11.2.3 Estimation Using the EM Algorithm11.3 Missing Data in Quantitative Scales11.3.1 Available Alternatives11.3.2 Imputation of Values for Missing Cases11.3.3 Modeling Solutions to Missing Data in Scaled Variables11.3.4 An Illustrative Comparison of Alternative Methods11.3.5 Rules of Thumb11.4 Summary
Chapter 12: Multiple Regression/Correlation and Causal Models
12.1 Introduction12.1.1 Limits on the Current Discussion and the Relationship Between Causal Analysis and Analysis of Covariance12.1.2 Theories and Multiple Regression/Correlation Models That Estimate and Test Them12.1.3 Kinds of Variables in Causal Models12.1.4 Regression Models as Causal Models12.2 Models Without Reciprocal Causation12.2.1 Direct and Indirect Effects12.2.2 Path Analysis and Path Coefficients12.2.3 Hierarchical Analysis and Reduced Form Equations12.2.4 Partial Causal Models and the Hierarchical Analysis of Sets12.2.5 Testing Model Elements12.3 Models With Reciprocal Causation12.4 Identification and Overidentification12.4.1 Just Identified Models12.4.2 Overidentification12.4.3 Underidentification12.5 Latent Variable Models12.5.1 An Example of a Latent Variable Model12.5.2 How Latent Variables Are Estimated12.5.3 Fixed and Free Estimates in Latent Variable Models12.5.4 Goodness-of-Fit Tests of Latent Variable Models12.5.5 Latent Variable Models and the Correction for Attenuation12.5.6 Characteristics of Data Sets That Make Latent Variable Analysis the Method of Choice12.6 A Review of Causal Model and Statistical Assumptions12.6.1 Specification Error12.6.2 Identification Error12.7 Comparisons of Causal Models12.7.1 Nested Models12.7.2 Longitudinal Data in Causal Models12.8 Summary
Chapter 13: Alternative Regression Models: Logistic, Poisson Regression, and the Generalized Linear Model
13.1 Ordinary Least Squares Regression Revisited13.1.1 Three Characteristics of Ordinary Least Squares Regression13.1.2 The Generalized Linear Model13.1.3 Relationship of Dichotomous and Count Dependent Variables Y to a Predictor13.2 Dichotomous Outcomes and Logistic Regression13.2.1 Extending Linear Regression: The Linear Probability Model and Discriminant Analysis13.2.2 The Nonlinear Transformation From Predictor to Predicted Scores: Probit and Logistic Transformation13.2.3 The Logistic Regression Equation13.2.4 Numerical Example: Three Forms of the Logistic Regression Equation13.2.5 Understanding the Coefficients for the Predictor in Logistic Regression13.2.6 Multiple Logistic Regression13.2.7 Numerical Example13.2.8 Confidence Intervals on Regression Coefficients and Odds Ratios13.2.9 Estimation of the Regression Model: Maximum Likelihood13.2.10 Deviances: Indices of Overall Fit of the Logistic Regression Model13.2.11 Multiple R2 Analogs in Logistic Regression13.2.12 Testing Significance of Overall Model Fit: The Likelihood Ratio Test and the Test of Model Deviance13.2.13 χ2 Test for the Significance of a Single Predictor in a Multiple Logistic Regression Equation13.2.14 Hierarchical Logistic Regression: Likelihood Ratio χ2 Test for the Significance of a Set of Predictors Above and Beyond Another Set13.2.15 Akaike’s Information Criterion and the Bayesian Information Criterion for Model Comparison13.2.16 Some Treachery in Variable Scaling and Interpretation of the Odds Ratio13.2.17 Regression Diagnostics in Logistic Regression13.2.18 Sparseness of Data13.2.19 Classification of Cases13.3 Extensions of Logistic Regression to Multiple Response Categories: Polytomous Logistic Regression and Ordinal Logistic Regression13.3.1 Polytomous Logistic Regression13.3.2 Nested Dichotomies13.3.3 Ordinal Logistic Regression13.4 Models for Count Data: Poisson Regression and Alternatives13.4.1 Linear Regression Applied to Count Data13.4.2 Poisson Probability Distribution13.4.3 Poisson Regression Analysis13.4.4 Overdispersion and Alternative Models13.4.5 Independence of Observations13.4.6 Sources on Poisson Regression13.5 Full Circle: Parallels Between Logistic and Poisson Regression, and the Generalized Linear Model13.5.1 Parallels Between Poisson and Logistic Regression13.5.2 The Generalized Linear Model Revisited13.6 Summary
Chapter 14: Random Coefficient Regression and Multilevel Models
14.1 Clustering Within Data Sets14.1.1 Clustering, Alpha Inflation, and the Intraclass Correlation14.1.2 Estimating the Intraclass Correlation14.2 Analysis of Clustered Data With Ordinary Least Squares Approaches14.2.1 Numerical Example, Analysis of Clustered Data With Ordinary Least Squares Regression14.3 The Random Coefficient Regression Model14.4 Random Coefficient Regression Model and Multilevel Data Structure14.4.1 Ordinary Least Squares (Fixed Effects) Regression Revisited14.4.2 Fixed and Random Variables14.4.3 Clustering and Hierarchically Structured Data14.4.4 Structure of the Random Coefficient Regression Model14.4.5 Level 1 Equations14.4.6 Level 2 Equations14.4.7 Mixed Model Equation for Random Coefficient Regression14.4.8 Variance Components—New Parameters in the Multilevel Model14.4.9 Variance Components and Random Coefficient Versus Ordinary Least Squares (Fixed Effects) Regression14.4.10 Parameters of the Random Coefficient Regression Model: Fixed and Random Effects14.5 Numerical Example: Analysis of Clustered Data With Random Coefficient Regression14.5.1 Unconditional Cell Means Model and the Intraclass Correlation14.5.2 Testing the Fixed and Random Parts of the Random Coefficient Regression Model14.6 Clustering as a Meaningful Aspect of the Data14.7 Multilevel Modeling With a Predictor at Level14.7.1 Level 1 Equations14.7.2 Revised Level 2 Equations14.7.3 Mixed Model Equation With Level 1 Predictor and Level 2 Predictor of Intercept and Slope and the Cross-Level Interaction14.8 An Experimental Design as a Multilevel Data Structure: Combining Experimental Manipulation With Individual Differences14.9 Numerical Example: Multilevel Analysis14.10 Estimation of the Multilevel Model Parameters: Fixed Effects, Variance Components, and Level 1 Equations14.10.1 Fixed Effects and Variance Components14.10.2 An Equation for Each Group: Empirical Bayes Estimates of Level 1 Coefficients14.11 Statistical Tests in Multilevel Models14.11.1 Fixed Effects14.11.2 Variance Components14.12 Some Model Specification Issues14.12.1 The Same Variable at Two Levels14.12.2 Centering in Multilevel Models14.13 Statistical Power of Multilevel Models14.14 Choosing Between the Fixed Effects Model and the Random Coefficient Model14.15 Sources on Multilevel Modeling14.16 Multilevel Models Applied to Repeated Measures Data14.17 Summary
Chapter 15: Longitudinal Regression Methods
15.1 Introduction15.1.1 Chapter Goals15.1.2 Purposes of Gathering Data on Multiple Occasions15.2 Analyses of Two-Time-Point Data15.2.1 Change or Regressed Change?15.2.2 Alternative Regression Models for Effects Over a Single Unit of Time15.2.3 Three- or Four-Time-Point Data15.3 Repeated Measure Analysis of Variance15.3.1 Multiple Error Terms in Repeated Measure Analysis of Variance15.3.2 Trend Analysis in Analysis of Variance15.3.3 Repeated Measure Analysis of Variance in Which Time Is Not the Issue15.4 Multilevel Regression of Individual Changes Over Time15.4.1 Patterns of Individual Change Over Time15.4.2 Adding Other Fixed Predictors to the Model15.4.3 Individual Differences in Variation Around Individual Slopes15.4.4 Alternative Developmental Models and Error Structures15.4.5 Alternative Link Functions for Predicting Y From Time15.4.6 Unbalanced Data: Variable Timing and Missing Data15.5 Latent Growth Models: Structural Equation Model Representation of Multilevel Data15.5.1 Estimation of Changes in True Scores15.5.2 Representation of Latent Growth Models in Structural Equation Model Diagrams15.5.3 Comparison of Multilevel Regression and Structural Equation Model Analysis of Change15.6 Time Varying Independent Variables15.7 Survival Analysis15.7.1 Regression Analysis of Time Until Outcome and the Problem of Censoring15.7.2 Extension to Time-Varying Independent Variables15.7.3 Extension to Multiple Episode Data15.7.4 Extension to a Categorical Outcome: Event-History Analysis15.8 Time Series Analysis15.8.1 Units of Observation in Time Series Analyses15.8.2 Time Series Analyses Applications15.8.3 Time Effects in Time Series15.8.4 Extension of Time Series Analyses to Multiple Units or Subjects15.9 Dynamic System Analysis15.10 Statistical Inference and Power Analysis in Longitudinal Analyses15.11 Summary
Chapter 16: Multiple Dependent Variables: Set Correlation
16.1 Introduction to Ordinary Least Squares Treatment of Multiple Dependent Variables16.1.1 Set Correlation Analysis16.1.2 Canonical Analysis16.1.3 Elements of Set Correlation16.2 Measures of Multivariate Association16.2.1 R2Y,X, the Proportion of Generalized Variance16.2.2 T2Y,X and P2Y,X, Proportions of Additive Variance16.3 Partialing in Set Correlation16.3.1 Frequent Reasons for Partialing Variable Sets From the Basic Sets16.3.2 The Five Types of Association Between Basic Y and X Sets16.4 Tests of Statistical Significance and Statistical Power16.4.1 Testing the Null Hypothesis16.4.2 Estimators of the Population R2Y,X, T2Y,X and P2Y,X16.4.3 Guarding Against Type I Error Inflation16.5 Statistical Power Analysis in Set Correlation16.6 Comparison of Set Correlation With Multiple Analysis of Variance16.7 New Analytic Possibilities With Set Correlation16.8 Illustrative Examples16.8.1 A Simple Whole Association16.8.2 A Multivariate Analysis of Partial Variance16.8.3 A Hierarchical Analysis of a Quantitative Set and Its Unique Components16.8.4 Bipartial Association Among Three Sets16.9 Summary
Appendices
Appendix 1: The Mathematical Basis for Multiple Regression/Correlation and Identification of the Inverse Matrix Elements
A1.1 Alternative Matrix MethodsA1.2 Determinants
Appendix 2: Determination of the Inverse Matrix and Applications Thereof
A2.1 Hand Calculation of the Multiple Regression/Correlation ProblemA2.2 Testing the Difference Between Partial βs and Bs From the Same SampleA2.3 Testing the Difference Between βs for Different Dependent Variables From a Single Sample
Appendix Tables
Table A t Values for α = .01, .05 (Two Tailed)Table B z′ Transformation of rTable C Normal DistributionTable D F Values for α = .01, .05Table E L Values for α = .01, .05Table F Power of Significance Test of r at α = .01, .05 (Two Tailed)Table G n* to Detect r by t Test at α = .01, .05 (Two Tailed)
References
Glossary
Statistical Symbols and Abbreviations
Author Index
Subject Index

Content preview from Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, 3rd Edition

Random Coefficient Regression and Multilevel Models

OLS regression and the regression approaches subsumed under the generalized linear model, including logistic and Poisson regression, all assume that observations are independent of one another. If observations on two individuals are independent of one another, then knowledge of scores on one individual provides no information whatever about scores on the other individual. Put another way, there is no relationship between the measures on one individual and measures on any other individual. Should there be repeated observations on the same individuals, standard OLS regression analyses assume that these observations are independent across time.

14.1 CLUSTERING WITHIN DATA SETS

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Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, 3rd Edition

by Jacob Cohen, Patricia Cohen, Stephen G. West, Leona S. Aiken

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