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Applied multiple regression/correlation analysis for the behavioral sciences

Author: Cohen, Jacob ; Cohen, Patricia ; West, Stephen G. ; Aiken, Leona S.Publisher: Lawrence Erlbaum Associates, 2003.Edition: 3rd revised ed.Language: EnglishDescription: 703 p. : Graphs/Ill. ; 26 cm. includes CD-ROM / DVDISBN: 0805822232Type of document: BookBibliography/Index: Includes bibliographical references and index and glossaryContents Note: CD avalaible inside back cover
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Book Europe Campus
Main Collection
Print HA33 .C63 2003
(Browse shelf)
001325965
Available 1 CD inside back cover 001325965
Book Europe Campus
Main Collection
Print HA33 .C63 2003
(Browse shelf)
001253795
Available 1 CD inside back cover 001253795
Total holds: 0

Includes bibliographical references and index and glossary

CD avalaible inside back cover

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Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences Contents Preface Chapter 1: Introduction 1.1 Multiple Regression/Correlation 1 as a General Data-Analytic System 1.1.1 Overv iew 1 1.1.2 Testing Hypotheses Using Multiple Regression/Correlation: Some Examples 2 1.1.3 Multiple Regression/Correlation in Prediction Models 3 A Comparison of Multiple Regression/Correlation and Analysis of Variance Approaches 4 1.2.1 Historical Background 4 Hypothesis Testing and Effect Sizes 5 1.2.2 Multiple Regression/Correlation 6 and the Complexity of Behavioral Science 1.3.1 Multiplicity of Influences 6 Correlation Among Research Factors and Partialing 1.3.2 Form of Information 7 1.3.3 Shape of Relationship 8 1.3.4 General and Conditional Relationships 9 1.3.5 10 Orientation of the Book 1.4.1 Nonmathematical 11 Applied 11 1.4.2 Data-Analytic 12 1.4.3 13 1.4.4 Inference Orientation and Specification Error Computation, the Computer, and Numerical Results 14 14 Computation 1.5.1 )0(V 1 1.2 1.3 6 1.4 1.5 1.5.2 1.5.3 1.6 1.7 1.8 Numerical Results: Reporting and Rounding Significance Tests, Confidence Intervals, and Appendu Tables 15 The Spectrum of Behavioral Science 16 Plan for the Book 16 1.7.1 Content 16 1.7.2 Structure: Numbering of Sections, Tables, and Equations 17 Summary 18 14 Chapter 2: Bivariate Correlation and Regression Tabular and Graphic Representations of Relationships 19 The Index of Linear Correlation Between Two Variables: The Pearson Product Moment Correlation Coefficient 23 2.2.1 Standard Scores: Making Units Comparable 23 2.2.2 The Product Moment Correlation as a Function of Differences Between z Scores 26 2.3 Alternative Formulas for the Product Moment Correlation Coefficient 28 2.3.1 r as the Average Product of z Scores 28 2.3.2 Raw Score Formulas for r 29 2.3.3 Point Biserial r 29 2.3.4 Phi (11:·) Coefficient 30 2.3.5 Rank Correlation 31 2.4 Regression Coefficients: Estimating Y From X 32 2.5 Regression Toward the Mean 36 2.6 The Standard Error of Estimate and Measures of the Strength of Association 37 2.7 Summary of Definitions and Interpretations 41 2.8 Statistical Inference With Regression and Correlation Coefficients 2.8.1 Assumptions Underlying Statistical Inference With Byx, Bo, i>,, and rxr 41 2.8.2 Estimation With Confidence Intervals 42 2.8.3 Null Hypothesis Significance Tests (NIISTs) 47 2.8.4 Confidence Limits and Null Hypothesis Significance Testing 50 2.9 Precision and Power 50 2.9.1 Precision of Estimation 50 2.9.2 Power of Null Hypothesis Significance Tests 51 2.10 Factors Affecting the Size of r 53 2.10.1 The Distributions of X and Y 53 2.10.2 The Reliability of the Variables 55 2.10.3 Restriction of Range 57 2.10.4 Part-Whole Correlations 59 2.10.5 Ratio or Index Variables 60 2.10.6 Curvilinear Relationships 62 62 2.11 Summary 2.1 2.2 19 41 Chapter 3: Multiple Regression/Correlation With Two or More Independent Variables 3.1 3.2 3.3 Introduction: Regression and Causal Models 64 3.1.1 What Is a Cause? 64 3.1.2 Diagrammatic Representation of Causal Models 65 Regression With Two Independent Variables 66 Measures of Association With Two Independent Variables 69 3.3.1 Multiple R and R269 3.3.2 Semipartial Correlation Coefficients and Increments to R2 72 3.3.3 Partial Correlation Coefficients 74 Patterns of Association Between Y and Two Independent Variables 75 3.4.1 Direct and Indirect Effects 75 3.4.2 Partial Redundancy 76 3.4.3 Suppression in Regression Models 77 3.4.4 Spurious Effects and Entirely Indirect Effects 78 Multiple Regression/Correlation With k Independent Variables 79 3.5.1 Introduction: Components of the Prediction Equation 79 3.5.2 Partial Regression Coefficients 80 3.5.3 R, R2, and Shrunken R282 3.5.4 sr and sr2 84 3.5.5 pr and pr2 85 3.5.6 Example of Interpretation of Partial Coefficients 85 Statistical Inference With k Independent Variables 86 3.6.1 Standard Errors and Confidence Intervals for B and e 86 3.6.2 Confidence Intervals for R2 88 3.6.3 Confidence Intervals for Differences Between Independent R2s 88 3.6.4 Statistical Tests on Multiple and Partial Coefficients 88 Statistical Precision and Power Analysis 90 3.7.1 Introduction: Research Goals and the Null Hypothesis 90 3.7.2 The Precision and Power of R2 91 3.7.3 Precision and Power Analysis for Partial Coefficients 93 Using Multiple Regression Equations in Prediction 95 3.8.1 Prediction of Y for a New Observation 95 3.8.2 Correlation of Individual Variables With Predicted Values 96 3.8.3 Cross-Validation and Unit Weighting 97 3.8.4 Multicollinearity 98 Summary 99 64 3.4 3.5 3.6 3.7 3.8 3.9 Chapter 4: Data Visualization, Exploration, and Assumption Checking: Diagnosing and Solving Regression Problems I 101 .1 4.2 4.3 4.4 4.5 4.6 Introduction 101 Some Useful Graphical Displays of the Original Data 102 4.2.1 Univariate Displays 103 4.2.2 Bivariate Displays 110 4.2.3 Correlation and Scatterplot Matrices 115 Assumptions and Ordinary Least Squares Regression 117 4.3.1 Assumptions Underlying Multiple Linear 117 Regression 4.3.2 Ordinary Least Squares Estimation 124 Detecting Violations of Assumptions 125 4.4.1 Form of the Relationship 125 4.4.2 Omitted Independent Variables 127 4.4.3 Measurement Error 129 4.4.4 Homoscedasticity of Residuals 130 4.4.5 Nonindependence of Residuals 134 4.4.6 Normality of Residuals 137 Remedies: Alternative Approaches When Problems Are Detected 4.5.1 Form of the Relationship 141 4.5.2 Inclusion of Ail Relevant Independent Variables 143 4.5.3 Measurement Error in the Independent Variables 144 4.5.4 Nonconstant Variance 145 4.5.5 Nonindependence of Residuals 147 Summary 150 141 Chapter 5: Data-Analytic Strategies Using Multiple Regression/Correlation Research Questions Answered by Correlations and Their Squares 5.1.1 Net Contribution to Prediction 152 5.1.2 Indices of Differential Validity 152 5.1.3 Comparisons of Predictive Utility 152 5.1.4 Attribution of a Fraction of the XY Relationship to a Third Variable 153 5.1.5 Which of Two Variables Accounts for More of the XY Relationship? 153 5.1.6 Are the Various Squared Correlations in One Population Different From Those in Another Given the Saine Variables? 154 Research Questions Answered by B Or fl 154 5.2.1 Regression Coefficients as Reflections of Causal Effects 154 5.2.2 Alternative Approaches to Making Byx Substantively Meaningful 154 5.2.3 Are the Effects of a Set of Independent Variables on Two Different Outcomes in a Sample Different? 157 151 151 .1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 What Are the Reciprocal Effects of Two Variables on One Another? 157 Hierarchical Analysis Variables in Multiple Regression/ Correlation 158 5.3.1 Causal Priority and the Removal of Confounding Variables 158 5.3.2 Research Relevance 160 5.3.3 Examination of Alternative Hierarchical Sequences of Independent Variable Sets 160 5.3.4 Stepwise Regression 161 The Analysis of Sets of Independent Variables 162 5.4.1 Types of Sets 162 5.4.2 The Simultaneous and Hierarchical Analyses of Sets 164 5.4.3 Variance Proportions for Sets and the Ballantine Again 166 5.4.4 B and 13 Coefficients for Variables Within Sets 169 Significance Testing for Sets 171 5.5.1 Application in Hierarchical Analysis 172 5.5.2 Application in Simultaneous Analysis 173 5.5.3 Using Computer Output to Determine Statistical Significance 174 5.5.4 An Alternative F Test: Using Model 2 Error Estimate From the Final Model 174 Power Analysis for Sets 176 5.6.1 Determining n* for the F Test of sn with Model 1 or Model 2 Error 177 5.6.2 Estimating the Population sR2 Values 179 5.6.3 Setting Power for n* 180 5.6.4 Reconciling Different n*s 180 5.6.5 Power as a Function of n 181 5.6.6 Tactics of Power Analysis 182 Statistical Inference Strategy in Multiple Regression/ Correlation 182 5.7.1 Controlling and Balancing Type I and Type II Errors in Inference 182 5.7.2 Less Is More 185 5.7.3 Least Is Last 186 5.7.4 Adaptation of Fisher's Protected t Test 187 5.7.5 Statistical Inference and the Stage of Scientific Investigations 190 Summary 190 5.2.4 Chapter 6: Quantitative Sosies, Curvilinear Relationships, and Transformations Introduction 193 6.1.1 What Do We Mean by Linear Regression? 193 193 1 6.1.2 6.2 6.3 6.4 Linearity in the Variables and Linear Multiple Regression 194 6.1.3 Four Approaches to Examining Nonlinear Relationships in Multiple Regression 195 Power Polynomials 196 6.2.1 Method 196 6.2.2 An Example: Quadratic Fit 198 Centering Predictors in Polynomial Equations 201 6.2.3 6.2.4 Relationship of Test of Significance of Highest Ortler Coefficient and Gain in Prediction 204 6.2.5 Interpreting Polynomial Regression Results 205 Another Example: A Cubic Fit 207 6.2.6 6.2.7 Strategy and Limitations 209 6.2.8 More Complex Equations 213 Orthogonal Polynomials 214 6.3.1 The Cubic Example Revisited 216 6.3.2 219 Unequal n and Unequal Intervals 6.3.3 Applications and Discussion 220 Nonlinear Transformations 221 6.4.1 Purposes of Transformation and the Nature of Transformations 221 6.4.2 The Conceptual Basis of Transformations and Model Checking Before and After Transformation--Is It Always Ideal to Transform? 223 6.4.3 Logarithms and Exponents; Additive and Proportional Relationships 223 6.4.4 Linearizing Relationships 225 6.4.5 Linearizing Relationships Based on Strong Theoretical Models 227 6.4.6 Linearizing Relationships Based on Weak Theoretical Models 232 6.4.7 Empirically Driven Transformations in the Absence of Strong or Weak Models 233 Empirically Driven Transformation for Linearization: 6.4.8 The Ladder of Re-expression and the Bulging Rule 233 6.4.9 Empirically Driven Transformation for Linearization in the Absence of Models: Box-Cox Family of Power Transformations on Y 236 6.4.10 Empirically Driven Transformation for Linearization in the Absence of Models: Box-Tidwell Family of Power Transformations on X 239 6.4.11 Linearization of Relationships With Correlations: Fisher 240 z' Transform of r 6.4.12 Transformations That Linearize Relationships for Counts and Proportions 240 6.4.13 Variance Stabilizing Transformations and Alternatives for Treatment of Heteroscedasticity 244 6.4.14 Transformations to Normalize Variables 246 6.4.15 Diagnostics Following Transformation 247 6.5 6.6 6.7 6.4.16 Measuring and Comparing Model Fit 248 6.4.17 Second-Order Polynomial Numerical Example Revisited 248 6.4.18 When to Transform and the Choice of Transformation 249 Nonlinear Regression 251 Nonparametric Regression 252 Summary 253 Chapter 7: Interactions Among Continuous Variables 7.1 Introduction 255 7.1.1 Interactions Versus Additive Effects 256 7.1.2 Conditional First-Order Effects in Equations Containing Interactions 259 Centering Predictors and the Interpretation of Regression Coefficients in Equations Containing Interactions 261 7.2.1 Regression with Centered Predictors 261 7.2.2 Relationship Between Regression Coefficients in the Uncentered and Centered Equations 262 7.2.3 Centered Equations With No Interaction 262 7.2.4 Essential Versus Nonessential Multicollinearity 264 7.2.5 Centered Equations With Interactions 264 7.2.6 The Highest Order Interaction in the Centered Versus Uncentered Equation 266 7.2.7 Do Not Center Y 266 7.2.8 A Recommendation for Centering 266 Simple Regression Equations and Simple Slopes 267 7.3.1 Plotting Interactions 269 7.3.2 Moderator Variables 269 7.3.3 Simple Regression Equations 269 7.3.4 Overall Regression Coefficient and Simple S'ope at the Mean 270 7.3.5 Simple Slopes From Uncentered Versus Centered Equations Are Identical 271 7.3.6 Linear by Linear Interactions 271 7.3.7 Interpreting Interactions in Multiple Regression and Analysis of Variance 272 Post Hoc Probing of Interactions 272 7.4.1 Standard Error of Simple Slopes 272 7.4.2 Equation Dependence of Simple Slopes and Their Standard Errors 273 7.4.3 Tests of Significance of Simple Slopes 273 7.4.4 Confidence Intervals Around Simple Slopes 274 7.4.5 A Numerical Example 275 7.4.6 The Uncentered Regression Equation Revisited 281 7.4.7 First-Order Coefficients in Equations Without and With Interactions 281 7.4.8 Interpretation and the Range of Data 282 255 7.2 7.3 7.4 7.5 7.6 Standardized Estimates for Equations Containing Interactions 282 Interactions as Partialed Effects: Building Regression Equations With Interactions 284 7.7 Patterns of First-Order and Interactive Effects 285 7.7.1 Three Theoretically Meaningful Patterns of First-Order and Interaction Effects 285 7.7.2 Ordinal Versus Disordinal Interactions 286 7.8 Three-Predictor Interactions in Multiple Regression 290 7.9 Curvilinear by Linear Interactions 292 7.10 Interactions Among Sets of Variables 295 7.11 Issues in the Detection of Interactions: Reliability, Predictor Distributions, Model Specification 297 7.11.1 Variable Reliability and Power to Detect Interactions 297 7.11.2 Sampling Designs to Enhance Power to Detect Interactions--Optimal Design 298 7.11.3 Difficulty in Distinguishing Interactions Versus Curvilinear Effects 299 300 7.12 Summary Chapter 8: Categorical or Nominal Independent Variables 8.1 Introduction 302 8.1.1 Categories as a Set of Independent Variables 302 8.1.2 The Representation of Categories or Nominal Scales 302 Dummy-Variable Coding 303 8.2.1 Coding the Groups 303 8.2.2 Pearson Correlations of Dummy Variables With Y 308 8.2.3 Correlations Among Dummy-Coded Variables 311 8.2.4 Multiple Correlation of the Dummy-Variable Set With Y 311 8.2.5 Regression Coefficients for Dummy Variables 312 8.2.6 Partial and Semipartial Correlations for Dummy Variables 316 8.2.7 Dummy-Variable Multiple Regression/Correlation and One-Way Analysis of Variance 317 8.2.8 A Cautionary Note: Dummy-Variable-Like Coding Systems 319 8.2.9 Dummy-Variable Coding When Groups Are Not Mutually Exclusive 320 Unweighted Effects Coding 320 8.3.1 Introduction: Unweighted and Weighted Effects Coding 320 8.3.2 Constructing Unweighted Effects Codes 321 8.3.3 The R2 and the rN s for Unweighted Effects Codes 324 8.3.4 Regression Coefficients and Other Partial Effects in Unweighted Code Sets 325 302 8.2 8.3 8.4 8.5 8.6 8.7 8.8 Weighted Effects Coding 328 8.4.1 Selection Considerations for Weighted Effects Coding 328 8.4.2 Constructing Weighted Effects 328 8.4.3 The R2 and R2 for Weighted Effects Codes 330 8.4.4 Interpretation and Testing of B With Unweighted Codes 331 Contrast Coding 332 8.5.1 Considerations in the Selection of a Contrast Coding Scheme 332 8.5.2 Constructing. Contrast Codes 333 8.5.3 The R2 and R2337 8.5.4 Partial Regression Coefficients 337 8.5.5 Statistical Power and the Choice of Contrast Codes 340 Nonsense Coding 341 Coding Schemes in the Context of Other Independent Variables 8.7.1 Combining Nominal and Continuous 342 Independent Variables 8.7.2 Calculating Adjusted Means for Nominal Independent Variables 343 8.7.3 Adjusted Means for Combinations of Nominal and Quantitative Independent Variables 344 8.7.4 Adjusted Means for More Than Two Groups and Alternative Coding Methods 348 8.7.5 Multiple Regression/Correlation With Nominal Independent Variables and the Analysis of Covariance 350 Summary 351 342 Chapter 9: Interactions With Categorical Variables 9.1 Nominal Scale by Nominal Scale Interactions 354 9.1.1 The 2 by 2 Design 354 9.1.2 Regression Analyses of Multiple Sets of Nominal Variables With More Than Two Categories 361 Interactions Involving More Than Two Nominal Scales 366 9.2.1 An Example of Three Nominal Scales Coded by Alternative Methods 367 9.2.2 Interactions Among Nominal Scales in Which Not Ail Combinations Are Considered 372 9.2.3 What If the Categories for One or More Nominal "Scales" Are Not Mutually Exclusive? 373 9.2.4 Consideration of pr, g, and Variance Proportions for Nominal Scale Interaction Variables 374 9.2.5 Summary of Issues and Recommendations for Interactions Among Nominal Scales 374 Nominal Scale by Continuous Variable Interactions 375 9.3.1 A Reminder on Centering 375 354 9.2 9.3 9.3.2 9.4 Interactions of a Continuons Variable With Dummy-Variable Coded Groups 375 9.3.3 Interactions Using Weighted or Unweighted Effects Codes 378 9.3.4 Interactions With a Contrast-Coded Nominal Scale 379 9.3.5 Interactions Coded to Estimate Simple Slopes of Groups 380 Categorical Variable Interactions With Nonlinear Effects 9.3.6 of Scaled Independent Variables 383 Interactions of a Scale With Two or More Categorical 9.3.7 Variables 386 Summary 388 Chapter 10: Outliers and Multicollinearity: Diagnosing and Solving Regression Problems II 10.1 Introduction 390 10.2 Outliers: Introduction and Illustration 391 10.3 Detecting Outliers: Regression Diagnostics 394 10.3.1 Extremity on the Independent Variables: Leverage 394 10.3.2 Extremity on Y: Discrepancy 398 10.3.3 Influence on the Regression Estimates 402 10.3.4 Location of Outlying Points and Diagnostic Statistics 406 10.3.5 Summary and Suggestions 409 10.4 Sources of Outliers and Possible Remedial Actions 411 10.4.1 Sources of Outliers 411 10.4.2 Remedial Actions 415 10.5 Multicollinearity 419 10.5.1 Exact Collineatity 419 10.5.2 Multicollinearity: A Numerical Illustration 420 10.5.3 Measures of the Degree of Multicollinearity 422 10.6 Remedies for Multicollinearity 425 .10.6.1 Model Respecification 426 10.6.2 Collection of Additional Data 427 10.6.3 Ridge Regression 427 10.6.4 Principal Components Regression 428 10.6.5 Summary of Multicollinearity Considerations 429 10.7 Summary 430 390 Chapter 11: Missing Data 11.1 Basic Issues in Handling Missing Data 431 11.1.1 Minimize Missing Data 431 11.1.2 Types of Missing Data 432 11.1.3 Traditional Approaches to Missing Data 431 433 11.2 Missing Data in Nominal Scales 435 11.2.1 Coding Nominal Scale X for Missing Data 435 11.2.2 Missing Data on Two Dichotomies 439 11.2.3 Estimation Using the EM Algorithm 440 11.3 Missing Data in Quantitative Scales 442 11.3.1 Available Alternatives 442 11.3.2 Imputation of Values for Missing Cases 444 11.3.3 Modeling Solutions to Missing Data in Scaled Variables 447 11.3.4 An Illustrafive Comparison of Alternative Methods 447 11.3.5 Rides of Thumb 450 11.4 Summary 450 Chapter 12: Multiple Regression/Correlation and Causal Models 12.1 Introduction 452 12.1.1 Limits on the Ciment Discussion and the Relationship Between Causal Analysis and Analysis of Covariance 452 12.1.2 Theories and Multiple Regression/Correlation Models That Estimate and Test Them 454 12.1.3 Kinds of Variables in Causal Models 457 12.1.4 Regression Models as Causal Models 459 12.2 Models Without Reciprocal Causation 460 12.2.1 Direct and Indirect Effects 460 12.2.2 Path Analysis and Path Coefficients 464 12.2.3 Hierarchical Analysis and Reduced Form Equations 465 12.2.4 Partial Causal Models and the Hierarchical Analysis of Sets 466 12.2.5 Testing Model Elements 467 12.3 Models With Reciprocal Causation 467 12.4 Identification and Overidentification 468 12.4.1 Just Identified Models 468 12.4.2 Overidentification 468 12.4.3 Underidentification 469 12.5 Latent Variable Models 469 12.5.1 An Example of a Latent Variable Model 469 12.5.2 How Latent Variables Are Estimated 471 12.5.3 Fixed and Free Estimates in Latent Variable Models 472 12.5.4 Goodness-of-Fit Tests of Latent Variable Models 472 12.5.5 Latent Variable Models and the Correction for Attenuation 473 12.5.6 Characteristics of Data Sets That Make Latent Variable Analysis the Method of Choice 474 12.6 A Review of Causal Model and Statistical Assumptions 475 452 475 12.6.1 Specification Error 475 12.6.2 Identification Error 12.7 Comparisons of Causal Models 476 12.7.1 Nested Models 476 12.7.2 Longitudinal Data in Causal Models 12.8 Suminaty 477 476 Chapter 13: Alternative Regression Models: Logistic, Poisson Regression, and the Generalized Linear Model 479 13.1 Ordinary Least Squares Regression Revisited 13.1.1 Three Characteristics of Ordinary Least Squares Regression 480 480 13.1.2 The Generalized Linear Model 13.1.3 Relationship of Dichotomous and Count Dependent 481 Variables Y to a Predictor 482 13.2 Dichotomous Outcomes and Logistic Regression 13.2.1 Extending Linear Regression: The Linear Probability 483 Model and Discriminant Analysis 13.2.2 The Nonlinear Transformation From Predictor to Predicted Scores: Probit and Logistic Transformation 485 486 13.2.3 The Logistic Regression Equation 13.2.4 Numerical Example: Three Forms of die Logistic Regression Equation 487 13.2.5 Understanding the Coefficients for the Predictor 492 in Logistic Regression 493 13.2.6 Multiple Logistic Regression 494 13.2.7 Numerical Example 13.2.8 Confidence Intervals on Regression Coefficients and Odds Ratios 497 13.2.9 Estimation of the Regression Model: Maximum Likelihood 498 13.2.10 Deviances: Indices of Overall Fit of the Logistic Regression Model 499 502 13.2.11 Multiple R2 Analogs in Logistic Regression 13.2.12 Testing Significance of Overall Model Fit: The Likelihood Ratio Test and the Test of Model Deviance 504 13.2.13 1.2 Test for the Significance of a Single Predictor in a 507 Multiple Logistic Regression Equation 13.2.14 Hierarchical Logistic Regression: Likelihood Ratio J(.2 Test for the Significance of a Set of Predictors Above and Beyond Another Set 508 13.2.15 Akaike's Information Criterion and die Bayesian 509 Information Criterion for Model Comparison 13.2.16 Some Treachery in Variable Scaling and Interpretation of the Odds Ratio 509 479 13.2.17 Regression Diagnostics in Logistic Regression 512 13.2.18 Sparseness of Data 516 13.2.19 Classification of Cases 516 13.3 Extensions of Logistic Regression to Multiple Response Categories: Polytomous Logistic Regression and Ordinal Logistic Regression 519 13.3.1 Polytomous Logistic Regression 519 13.3.2 Nested Dichotomies 520 13.3.3 Ordinal Logistic Regression 522 13.4 Models for Count Data: Poisson Regression and Alternatives 525 13.4.1 Linear Regression Applied to Count Data 525 13.4.2 Poisson Probability Distribution 526 13.4.3 Poisson Regression Analysis 528 13.4.4 Overdispersion and Alternative Models 530 13.4.5 Independence of Observations 532 13.4.6 Sources on Poisson Regression 532 13.5 Full Circle: Parallels Between Logistic and Poisson Regression, and the Generalized Linear Model 532 13.5.1 Parallels Between Poisson and Logistic Regression 532 13.5.2 The Generalized Linear Model Revisited 534 13.6 Summary 535 Chapter 14: Random Coefficient Regression and Multilevel Models 14.1 Clustering Within Data Sets 536 14.1.1 Clustering, Alpha Inflation, and the Intraclass Correlation 537 14.1.2 Estimating the Intraclass Correlation 538 14.2 Analysis of Clustered Data With Ordinary Least Squares Approaches 14.2.1 Numerical Example, Analysis of Clustered Data With Ordinary Least Squares Regression 541 14.3 The Random Coefficient Regression Model 543 14.4 Random Coefficient Regression Model and Multilevel Data Structure 14.4.1 Ordinary Least Squares (Fixed Effects) Regression Revisited 544 14.4.2 Fixed and Random Variables 544 14.4.3 Clustering and Hierarchically Structured Data 545 14.4.4 Structure of the Random Coefficient Regression Model 545 14.4.5 Level 1 Equations 546 14.4.6 Level 2 Equations 547 14.4.7 Mixed Model Equation for Random Coefficient Regression 548 14.4.8 Variance Components--New Parameters in the Multilevel Model 548 14.4.9 Variance Components and Random Coefficient Versus Ordinary Least Squares (Fixed Effects) Regression 549 536 539 544 14.4.10 Parameters of the Random Coefficient Regression Model: Fixed and Random Effects 550 14.5 Numerical Example: Analysis of Clustered Data With Random Coefficient Regression 550 14.5.1 Unconditional cen Means Model and the Intraclass Correlation 551 14.5.2 Testing die Fixed and Random Parts of the Random Coefficient Regression Model 552 14.6 Clustering as a Meaningful Aspect of the Data 553 14.7 Multilevel Modeling With a Predictor at Level 2 553 14.7.1 Level 1 Equations 553 14.7.2 Revised Level 2 Equations 554 14.7.3 Mixed Model Equation With Level 1 Predictor and Level 2 Predictor of Intercept and Slope and die Cross-Level Interaction 554 14.8 An Experimental Design as a Multilevel Data Structure: Combining Experimental Manipulation With Individual Differences 555 14.9 Numerical Example: Multilevel Analysis 556 14.10 Estimation of the Multilevel Model Parameters: Fixed Effects, Variance Components, and Level 1 Equations 560 14.10.1 Fixed Effects and Variance Components 560 14.10.2 An Equation for Each Group: Empirical Bayes Estimates of Level 1 Coefficients 560 14.11 Statistical Tests in Multilevel Models 563 14.11.1 Fixed Effects 563 14.11.2 Variance Components 563 14.12 Some Model Specification Issues 564 14.12.1 The Same Variable at Two Levels 564 14.12.2 Centering in Multilevel Models 564 14.13 Statistical Power of Multilevel Models 565 14.14 Choosing Between die Fixed Effects Model and the Random Coefficient Model 565 14.15 Sources on Multilevel Modeling 566 14.16 Multilevel Models Applied to Repeated Measures Data 566 567 14.17 Surnmary Chapter 15: Longitudinal Regression Methods 15.1 Introduction 568 15.1.1 Chapter Goals 568 15.1.2 Purposes of Gathering Data on Multiple Occasions 569 15.2 Analyses of Two-Time-Point Data 569 15.2.1 Change or Regressed Change? 570 15.2.2 Alternative Regression Models for Effects Over a Single Unit of Time 571 15.2.3 Three- or Four-Time-Point Data 573 15.3 Repeated Measure Analysis of Variance 573 568 15.3.1 Multiple Error Ternis in Repeated Measure Analysis 574 of Variance 15.3.2 Trend Analysis in Analysis of Variance 575 15.3.3 Repeated Measure Analysis of Variance in Which Time Is Not the Issue 576 578 15.4 Multilevel Regression of Individual Changes Over Time 578 15.4.1 Patterns of Individual Change Over lime 15.4.2 Adding Other Fixed Predictors to the Model 582 15.4.3 Individual Differences in Variation Around Individual 583 Slopes 15.4.4 Alternative Developmental Models and Error Structures 584 15.4.5 Alternative Link Functions for Predicting Y From Time 586 15.4.6 Unbalanced Data: Variable Timing and Missing 587 Data 15.5 Latent Growth Models: Structural Equation Model Representation of Multilevel Data 588 15.5.1 Estimation of Changes in True Scores 589 15.5.2 Representation of Latent Growth Models in Structural Equation Model Diagrams 589 15.5.3 Comparison of Multilevel Regression and Structural Équation Model Analysis of Change 594 15.6 Time Varying Independent Variables 595 15.7 Survival Analysis 596 15.7.1 Regression Analysis of Time Until Outcome and the Problem of Censoring 596 15.7.2 Extension to Time-Varying Independent 599 Variables 15.7.3 Extension to Multiple Episode Data 599 15.7.4 Extension to a Categorical Outcome: Event-History 600 Analysis 15.8 Time Series Analysis 600 601 15.8.1 Units of Observation in Time Series Analyses 601 15.8.2 Time Series Analyses Applications 15.8.3 Time Effects in lime Series 602 15.8.4 Extension of lime Series Analyses to Multiple Units 602 or Subjects 15.9 Dynamic System Analysis 602 15.10 Statistical Inference and Power Analysis in Longitudinal Analyses 604 15.11 Summary 605 Chapter 16: Multiple Dependent Variables: Set Correlation 16.1 Introduction to Ordinary Least Squares Treatment of Multiple Dependent Variables 608 16.1.1 Set Correlation Analysis 608 608 16.1.2 Canonical Analysis 609 610 16.1.3 Elements of Set Correlation 610 16.2 Measures of Multivariate Association 610 16.2.1 le, j, the Proportion of Generalized Variance 611 16.2.2 Tîar and Pît,x, Proportions of Additive Variance 16.3 Partialing in Set Correlation 613 16.3.1 Frequent Reasons for Partialing Variable Sets From the Basic Sets 613 16.3.2 The Five Types of Association Between Basic Y and X Sets 614 16.4 Tests of Statistical Significance and Statistical Power 615 615 16.4.1 Testing the Null Hypothesis 616 16.4.2 Estimators of the Population Inu,71.,,,x, and Ity 16.4.3 Guarding Against Type I Error Inflation 617 617 16.5 Statistical Power Analysis in Set Correlation 16.6 Comparison of Set Correlation With Multiple Analysis of Variance 16.7 New Analytic Possibilities With Set Correlation 620 16.8 Illustrative Examples 621 621 16.8.1 A Simple Whole Association 16.8.2 A Multivariate Analysis of Partial Variance 622 16.8.3 A Hierarchical Analysis of a Quantitative Set and Its Unique Components 623 625 16.8.4 Bipartial Association Among Three Sets 16.9 Summary 627 619 APPENDICES Appendix 1: The Mathematical Basis for Multiple Regression/Correlation and Identification of the Inverse Matrix Elements A1.1 Alternative Matrix Methods A1.2 Determinants 634 634 631 Appendix 2: Determination of the Inverse Matrix and Applications Thereof A2.1 Hand Calculation of the Multiple Regression/Correlation Problem 636 A2.2 Testing the Difference Between Partial 13s and Bs From the Saine Sample 640 A2.3 Testing the Difference Between 13s for Different Dependent Variables From a Single Sample 642 636 Appendix Tables 643 Table A t Values for a = .01, .05 (Two Tailed) Table B z' Transformation of r 644 Table C Normal Distribution 645 Table D F Values for a = .01, .05 646 650 Table E L Values for a = .01, .05 Table F Power of Significance Test of r at a = .01, .05 (Two Tailed) 652 Table G nt to Detect r by t Test at a = .01, .05 654 (Two Tailed) 643 References Glossary Statistical Symbols and Abbreviations Author Index Subject Index 655 671 683 687 691

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