**A clear and efficient balance between theory and applications of statistical modeling techniques in the social and behavioral sciences**

Written as a general and accessible introduction,* Applied Univariate, Bivariate, and Multivariate Statistics* provides an overview of statistical modeling techniques used in fields in the social and behavioral sciences. Providing a unique balance of statistical theory and methodology, the book surveys both the technical and theoretical aspects of good data analysis.

Featuring applied resources at various levels, the book includes statistical techniques such as *t*–tests and correlation as well as more advanced procedures such as MANOVA, factor analysis, and structural equation modeling. To promote a more in–depth interpretation of statistical techniques across the sciences, the book surveys some of the technical arguments underlying formulas and equations. *Applied Univariate, Bivariate, and Multivariate Statistics* also features:

- Demonstrations of statistical techniques using software packages such as R and SPSS®
- Examples of hypothetical and real data with subsequent statistical analysis
- Historical and philosophical insights into many of the techniques used in modern social science
- A companion website that includes further instructional details, additional data sets, solutions to selected exercises, and multiple programming options

An ideal textbook for courses in statistics and methodology at the upper–undergraduate and graduate–level in psychology, political science, biology, sociology, education, economics, communications, law, and survey research, *Applied Univariate, Bivariate, and Multivariate Statistics* is also a useful reference for practitioners and researchers in their field of application.

**Daniel J. Denis, PhD,** is Associate Professor of Quantitative Psychology in the Department of Psychology at the University of Montana where he teaches courses in univariate and multivariate statistics. He has published a number of articles in peer–reviewed journals and has served as consultant to researchers and practitioners in a variety of fields. He is also founder of *The Mathematical Psychology Lab*, which is a think tank with the mission of advancing the teaching, learning, and application of mathematical and statistical tools. * *

Preface xix

About the Companion Website xxxiii

1 Preliminary Considerations 1

1.1 The Philosophical Bases of Knowledge: Rationalistic versus Empiricist Pursuits 1

1.2 What is a Model ? 4

1.3 Social Sciences versus Hard Sciences 6

1.4 Is Complexity a Good Depiction of Reality? Are Multivariate Methods Useful? 8

1.5 Causality 9

1.6 The Nature of Mathematics: Mathematics as a Representation of Concepts 10

1.7 As a Social Scientist How Much Mathematics Do You Need to Know? 11

1.8 Statistics and Relativity 12

1.9 Experimental versus Statistical Control 13

1.10 Statistical versus Physical Effects 14

1.11 Understanding What Applied Statistics Means 15

Review Exercises 15

2 Mathematics and Probability Theory 18

2.1 Set Theory 20

2.2 Cartesian Product A × B 24

2.3 Sets of Numbers 26

2.4 Set Theory Into Practice: Samples Populations and Probability 27

2.5 Probability 28

2.6 Interpretations of Probability: Frequentist versus Subjective 35

2.7 Bayes Theorem: Inverting Conditional Probabilities 39

2.8 Statistical Inference 44

2.9 Essential Mathematics: Precalculus Calculus and Algebra 48

2.10 Chapter Summary and Highlights 72

Review Exercises 74

3 Introductory Statistics 78

3.1 Densities and Distributions 79

3.2 Chi–Square Distributions and Goodness–of–Fit Test 91

3.3 Sensitivity and Specificity 98

3.4 Scales of Measurement: Nominal Ordinal and Interval Ratio 98

3.5 Mathematical Variables versus Random Variables 101

3.6 Moments and Expectations 103

3.7 Estimation and Estimators 106

3.8 Variance 108

3.9 Degrees of Freedom 110

3.10 Skewness and Kurtosis 111

3.11 Sampling Distributions 113

3.12 Central Limit Theorem 116

3.13 Confidence Intervals 117

3.14 Bootstrap and Resampling Techniques 119

3.15 Likelihood Ratio Tests and Penalized Log–Likelihood Statistics 121

3.16 Akaike s Information Criteria 122

3.17 Covariance and Correlation 123

3.18 Other Correlation Coefficients 128

3.19 Student s t Distribution 131

3.20 Statistical Power 139

3.21 Paired Samples t–Test: Statistical Test for Matched Pairs (Elementary Blocking) Designs 146

3.22 Blocking with Several Conditions 149

3.23 Composite Variables: Linear Combinations 149

3.24 Models in Matrix Form 151

3.25 Graphical Approaches 152

3.26 What Makes a p–Value Small? A Critical Overview and Simple Demonstration of Null Hypothesis

Significance Testing 155

3.27 Chapter Summary and Highlights 164

Review Exercises 167

4 Analysis of Variance: Fixed Effects Models 173

4.1 What is Analysis of Variance? Fixed versus Random Effects 174

4.2 How Analysis of Variance Works: A Big Picture Overview 178

4.3 Logic and Theory of ANOVA: A Deeper Look 180

4.4 From Sums of Squares to Unbiased Variance Estimators: Dividing by Degrees of Freedom 189

4.5 Expected Mean Squares for One–Way Fixed Effects Model: Deriving the F–Ratio 190

4.6 The Null Hypothesis in ANOVA 196

4.7 Fixed Effects ANOVA: Model Assumptions 198

4.8 A Word on Experimental Design and Randomization 201

4.9 A Preview of the Concept of Nesting 201

4.10 Balanced versus Unbalanced Data in ANOVA Models 202

4.11 Measures of Association and Effect Size in ANOVA: Measures of Variance Explained 202

4.12 The F–Test and the Independent Samples t–Test 205

4.13 Contrasts and Post–Hocs 205

4.14 Post–Hoc Tests 212

4.15 Sample Size and Power for ANOVA: Estimation with R and G Power 218

4.16 Fixed Effects One–Way Analysis of Variance in R: Mathematics Achievement as a Function of Teacher 222

4.17 Analysis of Variance Via R s lm 226

4.18 Kruskal Wallis Test in R 227

4.19 ANOVA in SPSS: Achievement as a Function of Teacher 228

4.20 Chapter Summary and Highlights 230

Review Exercises 232

5 Factorial Analysis of Variance: Modeling Interactions 237

5.1 What is Factorial Analysis of Variance? 238

5.2 Theory of Factorial ANOVA: A Deeper Look 239

5.3 Comparing One–Way ANOVA to Two–Way ANOVA: Cell Effects in Factorial ANOVA versus Sample

Effects in One–Way ANOVA 245

5.4 Partitioning the Sums of Squares for Factorial ANOVA: The Case of Two Factors 246

5.5 Interpreting Main Effects in the Presence of Interactions 253

5.6 Effect Size Measures 253

5.7 Three–Way Four–Way and Higher–Order Models 254

5.8 Simple Main Effects 254

5.9 Nested Designs 256

5.9.1 Varieties of Nesting: Nesting of Levels versus Subjects 257

5.10 Achievement as a Function of Teacher and Textbook: Example of Factorial ANOVA in R 258

5.11 Interaction Contrasts 266

5.12 Chapter Summary and Highlights 267

Review Exercises 268

6 Introduction to Random Effects and Mixed Models 270

6.1 What is Random Effects Analysis of Variance? 271

6.2 Theory of Random Effects Models 272

6.3 Estimation in Random Effects Models 273

6.4 Defining Null Hypotheses in Random Effects Models 276

6.5 Comparing Null Hypotheses in Fixed versus Random Effects Models: The Importance of Assumptions

278

6.6 Estimating Variance Components in Random Effects Models: ANOVA ML REML Estimators 279

6.7 Is Achievement a Function of Teacher? One–Way Random Effects Model in R 282

6.8 R Analysis Using REML 285

6.9 Analysis in SPSS: Obtaining Variance Components 286

6.10 Factorial Random Effects: A Two–Way Model 287

6.11 Fixed Effects versus Random Effects: A Way of Conceptualizing Their Differences 289

6.12 Conceptualizing the Two–Way Random Effects Model: The Makeup of a Randomly Chosen Observation 289

6.13 Sums of Squares and Expected Mean Squares for Random Effects: The Contaminating Influence of Interaction Effects 291

6.14 You Get What You Go in with: The Importance of Model Assumptions and Model Selection 293

6.15 Mixed Model Analysis of Variance: Incorporating Fixed and Random Effects 294

6.16 Mixed Models in Matrices 298

6.17 Multilevel Modeling as a Special Case of the Mixed Model: Incorporating Nesting and Clustering 299

6.18 Chapter Summary and Highlights 300

Review Exercises 301

7 Randomized Blocks and Repeated Measures 303

7.1 What Is a Randomized Block Design? 304

7.2 Randomized Block Designs: Subjects Nested Within Blocks 304

7.3 Theory of Randomized Block Designs 306

7.4 Tukey Test for Nonadditivity 311

7.5 Assumptions for the Variance Covariance Matrix 311

7.6 Intraclass Correlation 313

7.7 Repeated Measures Models: A Special Case of Randomized Block Designs 314

7.8 Independent versus Paired Samples t–Test 315

7.9 The Subject Factor: Fixed or Random Effect? 316

7.10 Model for One–Way Repeated Measures Design 317

7.11 Analysis Using R: One–Way Repeated Measures: Learning as a Function of Trial 318

7.12 Analysis Using SPSS: One–Way Repeated Measures: Learning as a Function of Trial 322

7.13 SPSS: Two–Way Repeated Measures Analysis of Variance: Mixed Design: One Between Factor One

Within Factor 326

7.14 Chapter Summary and Highlights 330

Review Exercises 331

8 Linear Regression 333

8.1 Brief History of Regression 334

8.2 Regression Analysis and Science: Experimental versus Correlational Distinctions 336

8.3 A Motivating Example: Can Offspring Height Be Predicted? 337

8.4 Theory of Regression Analysis: A Deeper Look 339

8.5 Multilevel Yearnings 342

8.6 The Least–Squares Line 342

8.7 Making Predictions Without Regression 343

8.8 More About i 345

8.9 Model Assumptions for Linear Regression 346

8.10 Estimation of Model Parameters in Regression 349

8.11 Null Hypotheses for Regression 351

8.12 Significance Tests and Confidence Intervals for Model Parameters 353

8.13 Other Formulations of the Regression Model 355

8.14 The Regression Model in Matrices: Allowing for More Complex Multivariable Models 356

8.15 Ordinary Least–Squares in Matrices 359

8.16 Analysis of Variance for Regression 360

8.17 Measures of Model Fit for Regression: How Well Does the Linear Equation Fit? 363

8.18 Adjusted R2 364

8.19 What Explained Variance Means: And More Importantly What It Does Not Mean 364

8.20 Values Fit by Regression 365

8.21 Least–Squares Regression in R: Using Matrix Operations 365

8.22 Linear Regression Using R 368

8.23 Regression Diagnostics: A Check on Model Assumptions 370

8.24 Regression in SPSS: Predicting Quantitative from Verbal 379

8.25 Power Analysis for Linear Regression in R 383

8.26 Chapter Summary and Highlights 384

Review Exercises 385

9 Multiple Linear Regression 389

9.1 Theory of Partial Correlation and Multiple Regression 390

9.2 Semipartial Correlations 392

9.3 Multiple Regression 393

9.4 Some Perspective on Regression Coefficients: Experimental Coefficients ? 394

9.5 Multiple Regression Model in Matrices 395

9.6 Estimation of Parameters 396

9.7 Conceptualizing Multiple R 396

9.8 Interpreting Regression Coefficients: The Case of Uncorrelated Predictors 397

9.9 Anderson s IRIS Data: Predicting Sepal Length from Petal Length and Petal Width 397

9.10 Fitting Other Functional Forms: A Brief Look at Polynomial Regression 402

9.11 Measures of Collinearity in Regression: Variance Inflation Factor and Tolerance 403

9.12 R–Squared as a Function of Partial and Semipartial Correlations: The Stepping Stones to Forward and Stepwise Regression 405

9.13 Model–Building Strategies: Simultaneous Hierarchichal Forward and Stepwise 406

9.14 Power Analysis for Multiple Regression 410

9.15 Introduction to Statistical Mediation: Concepts and Controversy 411

9.16 Chapter Summary and Highlights 414

Review Exercises 415

10 Interactions in Multiple Linear Regression: Dichotomous Polytomous and Continuous Moderators 418

10.1 The Additive Regression Model with Two Predictors 420

10.2 Why the Interaction is the Product Term xizi: Drawing an Analogy to Factorial ANOVA 420

10.3 A Motivating Example of Interaction in Regression: Crossing a Continuous Predictor with a

Dichotomous Predictor 421

10.4 Theory of Interactions in Regression 424

10.5 Simple Slopes for Continuous Moderators 427

10.6 A Simple Numerical Example: How Slopes Can Change as a Function of the Moderator 428

10.7 Calculating Simple Slopes: A Useful Algebraic Derivation 430

10.8 Summing Up the Idea of Interactions in Regression 432

10.9 Do Moderators Really Moderate Anything? Some Philosophical Considerations 432

10.10 Interpreting Model Coefficients in the Context of Moderators 433

10.11 Mean–Centering Predictors: Improving the Interpretability of Simple Slopes 434

10.12 The Issue of Multicollinearity: A Second Reason to Like Mean–Centering 435

10.13 Interaction of Continuous and Polytomous Predictors in R 436

10.14 Multilevel Regression: Another Special Case of the Mixed Model 440

10.15 Chapter Summary and Highlights 441

Review Exercises 441

11 Logistic Regression and the Generalized Linear Model 443

11.1 Nonlinear Models 445

11.2 Generalized Linear Models 447

11.3 Canonical Links 450

11.4 Distributions and Generalized Linear Models 451

11.5 Dispersion Parameters and Deviance 453

11.6 Logistic Regression: A Generalized Linear Model for Binary Responses 454

11.7 Exponential and Logarithmic Functions 456

11.8 Odds Odds Ratio and the Logit 461

11.9 Putting It All Together: The Logistic Regression Model 462

11.10 Logistic Regression in R: Challenger O–Ring Data 466

11.11 Challenger Analysis in SPSS 469

11.12 Sample Size Effect Size and Power 473

11.13 Further Directions 474

11.14 Chapter Summary and Highlights 475

Review Exercises 476

12 Multivariate Analysis of Variance 479

12.1 A Motivating Example: Quantitative and Verbal Ability as a Variate 480

12.2 Constructing the Composite 482

12.3 Theory of MANOVA 482

12.4 Is the Linear Combination Meaningful? 483

12.5 Multivariate Hypotheses 487

12.6 Assumptions of MANOVA 488

12.7 Hotelling s T2: The Case of Generalizing from Univariate to Multivariate 489

12.8 The Variance Covariance Matrix S 492

12.9 From Sums of Squares and Cross–Products to Variances and Covariances 494

12.10 Hypothesis and Error Matrices of MANOVA 495

12.11 Multivariate Test Statistics 495

12.12 Equality of Variance Covariance Matrices 500

12.13 Multivariate Contrasts 501

12.14 MANOVA in R and SPSS 502

12.15 MANOVA of Fisher s Iris Data 508

12.16 Power Analysis and Sample Size for MANOVA 509

12.17 Multivariate Analysis of Covariance and Multivariate Models: A Bird s Eye View of Linear Models 511

12.18 Chapter Summary and Highlights 512

Review Exercises 513

13 Discriminant Analysis 517

13.1 What is Discriminant Analysis? The Big Picture on the Iris Data 518

13.2 Theory of Discriminant Analysis 520

13.3 LDA in R and SPSS 523

13.4 Discriminant Analysis for Several Populations 529

13.5 Discriminating Species of Iris: Discriminant Analyses for Three Populations 532

13.6 A Note on Classification and Error Rates 535

13.7 Discriminant Analysis and Beyond 537

13.8 Canonical Correlation 538

13.9 Motivating Example for Canonical Correlation: Hotelling s 1936 Data 539

13.10 Canonical Correlation as a General Linear Model 540

13.11 Theory of Canonical Correlation 541

13.12 Canonical Correlation of Hotelling s Data 544

13.13 Canonical Correlation on the Iris Data: Extracting Canonical Correlation from Regression MANOVA

LDA 546

13.14 Chapter Summary and Highlights 547

Review Exercises 548

14 Principal Components Analysis 551

14.1 History of Principal Components Analysis 552

14.2 Hotelling 1933 555

14.3 Theory of Principal Components Analysis 556

14.4 Eigenvalues as Variance 557

14.5 Principal Components as Linear Combinations 558

14.6 Extracting the First Component 558

14.7 Extracting the Second Component 560

14.8 Extracting Third and Remaining Components 561

14.9 The Eigenvalue as the Variance of a Linear Combination Relative to Its Length 561

14.10 Demonstrating Principal Components Analysis: Pearson s 1901 Illustration 562

14.11 Scree Plots 566

14.12 Principal Components versus Least–Squares Regression Lines 569

14.13 Covariance versus Correlation Matrices: Principal Components and Scaling 570

14.14 Principal Components Analysis Using SPSS 570

14.15 Chapter Summary and Highlights 575

Review Exercises 576

15 Factor Analysis 579

15.1 History of Factor Analysis 580

15.2 Factor Analysis: At a Glance 580

15.3 Exploratory versus Confirmatory Factor Analysis 581

15.4 Theory of Factor Analysis: The Exploratory Factor–Analytic Model 582

15.5 The Common Factor–Analytic Model 583

15.6 Assumptions of the Factor–Analytic Model 585

15.7 Why Model Assumptions Are Important 587

15.8 The Factor Model as an Implication for the Covariance Matrix 587

15.9 Again Why is ´ so Important a Result? 589

15.10 The Major Critique Against Factor Analysis: Indeterminacy and the Nonuniqueness of Solutions 589

15.11 Has Your Factor Analysis Been Successful? 591

15.12 Estimation of Parameters in Exploratory Factor Analysis 592

15.13 Estimation of Factor Scores 593

15.14 Principal Factor 593

15.15 Maximum Likelihood 595

15.16 The Concepts (and Criticisms) of Factor Rotation 596

15.17 Varimax and Quartimax Rotation 599

15.18 Should Factors Be Rotated? Is That Not Cheating? 600

15.19 Sample Size for Factor Analysis 601

15.20 Principal Components Analysis versus Factor Analysis: Two Key Differences 602

15.21 Principal Factor in SPSS: Principal Axis Factoring 604

15.22 Bartlett Test of Sphericity and Kaiser Meyer Olkin Measure of Sampling Adequacy (MSA) 612

15.23 Factor Analysis in R: Holzinger and Swineford (1939) 613

15.24 Cluster Analysis 616

15.25 What Is Cluster Analysis? The Big Picture 617

15.26 Measuring Proximity 619

15.27 Hierarchical Clustering Approaches 623

15.28 Nonhierarchical Clustering Approaches 625

15.29 K–Means Cluster Analysis in R 626

15.30 Guidelines and Warnings About Cluster Analysis 630

15.31 Chapter Summary and Highlights 630

Review Exercises 632

16 Path Analysis and Structural Equation Modeling 636

16.1 Path Analysis: A Motivating Example Predicting IQ Across Generations 637

16.2 Path Analysis and Causal Modeling 639

16.3 Early Post–Wright Path Analysis: Predicting Child s IQ (Burks 1928) 641

16.4 Decomposing Path Coefficients 642

16.5 Path Coefficients and Wright s Contribution 644

16.6 Path Analysis in R: A Quick Overview Modeling Galton s Data 644

16.7 Confirmatory Factor Analysis: The Measurement Model 648

16.8 Structural Equation Models 650

16.9 Direct Indirect and Total Effects 652

16.10 Theory of Statistical Modeling: A Deeper Look into Covariance Structures and General Modeling 653

16.11 Other Discrepancy Functions 655

16.12 The Discrepancy Function and Chi–Square 656

16.13 Identification 657

16.14 Disturbance Variables 659

16.15 Measures and Indicators of Model Fit 660

16.16 Overall Measures of Model Fit 660

16.17 Model Comparison Measures: Incremental Fit Indices 662

16.18 Which Indicator of Model Fit Is Best? 665

16.19 Structural Equation Model in R 666

16.20 How All Variables Are Latent: A Suggestion for Resolving the Manifest Latent Distinction 668

16.21 The Structural Equation Model as a General Model: Some Concluding Thoughts on Statistics and

Science 669

16.22 Chapter Summary and Highlights 670

Review Exercises 671

Appendix A: Matrix Algebra 675

References 705

Index 721

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