Research Design and Statistical Analysis

This book emphasizes the statistical concepts and assumptions necessary to describe and make inferences about real data. Throughout the book the authors encourage the reader to plot and examine their data, find confidence intervals, use power analyses to determine sample size, and calculate effect sizes. The goal is to ensure the reader understands the underlying logic and assumptions of the analysis and what it tells them, the limitations of the analysis, and the possible consequences of violating assumptions. The simpler, less abstract discussion of analysis of variance is presented prior to developing the more general model. A concern for alternatives to standard analyses allows for the integration of non-parametric techniques into relevant design chapters, rather than in a single, isolated chapter. This organization allows for the comparison of the pros and cons of alternative procedures within the research context to which they apply. Basic concepts, such as sampling distributions, expected mean squares, design efficiency, and statistical models are emphasized throughout. This approach provides a stronger conceptual foundation in order to help the reader generalize the concepts to new situations they will encounter in their research and to better understand the advice of statistical consultants and the content of articles using statistical methodology. The second edition features a greater emphasis on graphics, confidence intervals, measures of effect size, power analysis, tests of contrasts, elementary probability, correlation, and regression. A Free CD that contains several real and artificial data sets used in the book in SPSS, SYSTAT, and ASCII formats, is included in the back of the book. An Instructor's Solutions Manual, containing the intermediate steps to all of the text exercises, is available free to adopters. Contents......Page 8 Preface......Page 14 1.1 Variability and the Need for Statistics......Page 22 1.2 Systematic Versus Random Variability......Page 24 1.4 Reducing Error Variance......Page 26 1.6 Concluding Remarks......Page 28 2.1 Introduction......Page 31 2.2 Exploring a Single Sample......Page 32 2.3 Comparing Two Data Sets......Page 39 2.4 Other Measures of Location and Spread: The Mean and Standard Deviation......Page 41 2.5 Standardized (Ζ) Scores......Page 48 2.6 Measures of the Shape of a Distribution......Page 49 2.7 Concluding Remarks......Page 54 3.2 Some Examples......Page 58 3.3 Linear Relations......Page 64 3.4 The Pearson Product-Moment Correlation Coefficient......Page 65 3.5 Linear Regression......Page 72 3.6 The Coefficient of Determination, r2......Page 75 3.7 Influential Data Points and Resistant Measures of Regression......Page 76 3.9 Concluding Remarks......Page 77 4.1 Introduction......Page 82 4.2 Discrete Random Variables......Page 83 4.3 Probability Distributions......Page 84 4.4 Some Elementary Probability......Page 88 4.5 The Binomial Distribution......Page 96 4.6 Means and Variances of Discrete Distributions......Page 100 4.7 Hypothesis Testing......Page 101 4.8 Independence and the Sign Test......Page 107 4.10 Concluding Remarks......Page 110 5.2 Continuous Random Variables......Page 121 5.3 The Normal Distribution......Page 123 5.4 Point Estimates of Population Parameters......Page 125 5.5 Inferences About Population Means: The One-Sample Case......Page 133 5.6 Inferences About Population Means: The Correlated-Samples Case......Page 138 5.7 The Power of the Ζ Test......Page 140 5.8 Hypothesis Tests and CIs......Page 143 5.9 Validity of Assumptions......Page 144 5.10 Comparing Means of Two Independent Populations......Page 146 5.11 The Normal Approximation to the Binomial Distribution......Page 149 5.12 Concluding Remarks......Page 150 6.1 Introduction......Page 161 6.2 Inferences About a Population Mean......Page 162 6.3 The Standardized Effect Size......Page 166 6.4 Power of the One-Sample t Test......Page 168 6.5 The t Distribution: Two Independent Groups......Page 173 6.6 Standardized Effect Size for Two Independent Means......Page 177 6.7 Power of the Test of Two Independent Means......Page 178 6.8 Assumptions Underlying the Two-Group t Test......Page 179 6.9 Contrasts Involving More than Two Means......Page 182 6.10 Correlated Scores or Independent Groups?......Page 186 6.11 Concluding Remarks......Page 188 7.1 Introduction......Page 194 7.2 The χ2 Distribution......Page 195 7.3 Inferences About the Population Variance......Page 196 7.4 The F Distribution......Page 200 7.5 Inferences About Population Variance Ratios......Page 203 7.6 Relations Among Distributions......Page 206 7.7 Concluding Remarks......Page 207 8.1 Introduction......Page 212 8.2 Exploring the Data......Page 214 8.3 The Analysis of Variance......Page 216 8.4 The Model for the One-Factor Design......Page 222 8.5 Assessing the Importance of the Independent Variable......Page 228 8.6 Power of the F Test......Page 233 8.7 Assumptions Underlying the F Test......Page 237 8.8 Concluding Remarks......Page 248 9.1 Introduction......Page 254 9.2 Definitions and Examples of Contrasts......Page 255 9.3 Calculations of the t Statistic for Testing Hypotheses About Contrasts......Page 256 9.4 The Proper Unit for the Control of Type 1 Error......Page 262 9.5 Planned Versus Post Hoc Contrasts......Page 264 9.6 Controlling the FWE for Families of K Planned Contrasts......Page 265 9.7 Testing All Pairwise Contrasts......Page 268 9.8 Comparing a – 1 Treatment Means with a Control: Dunnett's Test......Page 276 9.9 Controlling the Familywise Error Rate for Post Hoc Contrasts......Page 277 9.10 The Sum of Squares Associated with a Contrast......Page 279 9.11 Concluding Remarks......Page 281 10.1 Introduction......Page 288 10.2 Linear Trend......Page 289 10.3 Testing Nonlinear Trends......Page 295 10.4 Concluding Remarks......Page 301 11.1 Introduction......Page 305 11.2 A First Look at the Data......Page 306 11.3 Two-Factor Designs: The ANOVA......Page 309 11.4 The Structural Model and Expected Mean Squares......Page 316 11.5 Main Effect Contrasts......Page 318 11.6 More About Interaction......Page 319 11.7 Simple Effects......Page 323 11.8 Two-Factor Designs: Trend Analysis......Page 326 11.9 Concluding Remarks......Page 330 12.2 Measures of Effect Size......Page 336 12.3 Power of the F Test......Page 339 12.4 Unequal Cell Frequencies......Page 340 12.5 Three-Factor Designs......Page 345 12.7 Pooling in Factorial Designs......Page 353 12.8 Blocking to Reduce Error Variance......Page 356 12.9 Concluding Remarks......Page 357 13.1 Introduction......Page 363 13.2 The Additive Model and Expected Mean Squares for the S × A Design......Page 366 13.3 The Nonadditive Model for the S × A Design......Page 373 13.4 Hypothesis Tests Assuming Nonadditivity......Page 376 13.6 Multifactor Repeated-Measures Designs......Page 384 13.7 Fixed or Random Effects?......Page 392 13.8 Nonparametric Procedures for Repeated-Measures Designs......Page 393 13.9 Concluding Remarks......Page 398 14.2 One Between-Subjects and One Within-Subjects Factor......Page 407 14.3 Rules for Generating Expected Mean Squares......Page 413 14.4 Measures of Effect Size......Page 415 14.5 Power Calculations......Page 417 14.6 Contrasting Means in Mixed Designs......Page 418 14.7 Testing Simple Effects......Page 422 14.8 Pretest-Posttest Designs......Page 423 14.9 Additional Mixed Designs......Page 424 14.10 Concluding Remarks......Page 428 15.1 Introduction......Page 433 15.2 Example of an ANCOVA......Page 436 15.3 Assumptions and Interpretation in an ANCOVA......Page 443 15.4 Testing Homogeneity of Slopes......Page 448 15.5 More About ANCOVA Versus Treatments × Blocks......Page 449 15.6 Estimating Power in an ANCOVA......Page 451 15.8 Some Extensions of the ANCOVA......Page 452 15.9 Concluding Remarks......Page 453 16.1 Introduction......Page 457 16.2 Groups Within Treatments......Page 458 16.3 Groups Versus Individuals......Page 464 16.4 Extensions of the Groups-Within-Treatments Design......Page 466 16.5 Items Within Treatments......Page 470 16.6 Concluding Remarks......Page 473 17.1 Introduction......Page 478 17.2 Selecting a Latin Square......Page 480 17.3 The Single Latin Square......Page 482 17.4 The Replicated Latin Square Design......Page 490 17.5 Balancing Carry-Over Effects......Page 495 17.6 Greco-Latin Squares......Page 497 17.7 Concluding Remarks......Page 498 18.1 Introduction......Page 501 18.2 Further Issues in Understanding the Correlation Coefficient......Page 502 18.3 Inference About Correlation......Page 510 18.4 Partial Correlations......Page 522 18.5 Other Measures of Correlation......Page 525 18.6 Concluding Remarks......Page 532 19.1 Introduction......Page 540 19.2 Regression Toward the Mean......Page 541 19.3 Inference in Linear Regression......Page 543 19.4 An Example: Regressing Cholesterol Level on Age......Page 553 19.5 Checking for Violations of Assumptions......Page 555 19.6 Locating Outliers and Influential Data Points......Page 563 19.7 Testing Independent Slopes for Equality......Page 569 19.8 Repeated-Measures Designs......Page 570 19.10 Concluding Remarks......Page 572 20.1 Introduction......Page 583 20.2 A Regression Example with Several Predictor Variables......Page 584 20.3 The Nature of the Regression Coefficients......Page 593 20.4 The Multiple Correlation Coefficient and the Partitioning of Variability in Multiple Regression......Page 594 20.5 Inference in Multiple Regression......Page 601 20.6 Selecting the Best Regression Equation for Prediction......Page 612 20.7 Explanation Versus Prediction in Regression......Page 614 20.8 Testing for Curvilinearity in Regression......Page 619 20.9 Including Interaction Terms in Multiple Regression......Page 622 20.10 Multiple Regression in Repeated-Measures Designs......Page 628 20.11 Concluding Remarks......Page 629 21.1 Introduction......Page 635 21.2 One-Factor Designs......Page 636 21.3 Regression Analyses and Factorial Designs......Page 642 21.4 Using Categorical and Continuous Variables in the Same Analysis......Page 651 21.5 Coding Designs with Within-Subjects Factors......Page 655 21.6 Concluding Remarks......Page 658 Appendix A: Notation and Summation Operations......Page 662 Appendix B: Expected Values and Their Applications......Page 670 Appendix C: Statistical Tables......Page 674 Answers to Selected Exercises......Page 706 Endnotes......Page 742 References......Page 750 C......Page 764 G......Page 765 L......Page 766 R......Page 767 W......Page 768 Z......Page 769 A......Page 770 B......Page 771 C......Page 772 D......Page 773 E......Page 774 H......Page 775 M......Page 776 P......Page 777 R......Page 778 T......Page 780 Z......Page 781 Contents 8 Preface 14 CHAPTER 1 INTRODUCTION 22 1.1 Variability and the Need for Statistics 22 1.2 Systematic Versus Random Variability 24 1.3 Error Variance Again 26 1.4 Reducing Error Variance 26 1.5 Overview of the Book 28 1.6 Concluding Remarks 28 CHAPTER 2 LOOKING AT DATA: UNIVARIATE DISTRIBUTIONS 31 2.1 Introduction 31 2.2 Exploring a Single Sample 32 2.3 Comparing Two Data Sets 39 2.4 Other Measures of Location and Spread: The Mean and Standard Deviation 41 2.5 Standardized (Ζ) Scores 48 2.6 Measures of the Shape of a Distribution 49 2.7 Concluding Remarks 54 CHAPTER 3 LOOKING AT DATA: RELATIONS BETWEEN QUANTITATIVE VARIABLES 58 3.1 Introduction 58 3.2 Some Examples 58 3.3 Linear Relations 64 3.4 The Pearson Product-Moment Correlation Coefficient 65 3.5 Linear Regression 72 3.6 The Coefficient of Determination, r2 75 3.7 Influential Data Points and Resistant Measures of Regression 76 3.8 Describing Nonlinear Relations 77 3.9 Concluding Remarks 77 CHAPTER 4 PROBABILITY AND THE BINOMIAL DISTRIBUTION 82 4.1 Introduction 82 4.2 Discrete Random Variables 83 4.3 Probability Distributions 84 4.4 Some Elementary Probability 88 4.5 The Binomial Distribution 96 4.6 Means and Variances of Discrete Distributions 100 4.7 Hypothesis Testing 101 4.8 Independence and the Sign Test 107 4.9 More About Assumptions and Statistical Tests 110 4.10 Concluding Remarks 110 CHAPTER 5 ESTIMATION AND HYPOTHESIS TESTS: THE NORMAL DISTRIBUTION 121 5.1 Introduction 121 5.2 Continuous Random Variables 121 5.3 The Normal Distribution 123 5.4 Point Estimates of Population Parameters 125 5.5 Inferences About Population Means: The One-Sample Case 133 5.6 Inferences About Population Means: The Correlated-Samples Case 138 5.7 The Power of the Ζ Test 140 5.8 Hypothesis Tests and CIs 143 5.9 Validity of Assumptions 144 5.10 Comparing Means of Two Independent Populations 146 5.11 The Normal Approximation to the Binomial Distribution 149 5.12 Concluding Remarks 150 CHAPTER 6 ESTIMATION, HYPOTHESIS TESTS, AND EFFECT SIZE: THE t DISTRIBUTION 161 6.1 Introduction 161 6.2 Inferences About a Population Mean 162 6.3 The Standardized Effect Size 166 6.4 Power of the One-Sample t Test 168 6.5 The t Distribution: Two Independent Groups 173 6.6 Standardized Effect Size for Two Independent Means 177 6.7 Power of the Test of Two Independent Means 178 6.8 Assumptions Underlying the Two-Group t Test 179 6.9 Contrasts Involving More than Two Means 182 6.10 Correlated Scores or Independent Groups? 186 6.11 Concluding Remarks 188 CHAPTER 7 THE CHI-SQUARE AND F DISTRIBUTIONS 194 7.1 Introduction 194 7.2 The χ2 Distribution 195 7.3 Inferences About the Population Variance 196 7.4 The F Distribution 200 7.5 Inferences About Population Variance Ratios 203 7.6 Relations Among Distributions 206 7.7 Concluding Remarks 207 CHAPTER 8 BETWEEN-SUBJECTS DESIGNS: ONE FACTOR 212 8.1 Introduction 212 8.2 Exploring the Data 214 8.3 The Analysis of Variance 216 8.4 The Model for the One-Factor Design 222 8.5 Assessing the Importance of the Independent Variable 228 8.6 Power of the F Test 233 8.7 Assumptions Underlying the F Test 237 8.8 Concluding Remarks 248 CHAPTER 9 CONTRASTS AMONG MEANS 254 9.1 Introduction 254 9.2 Definitions and Examples of Contrasts 255 9.3 Calculations of the t Statistic for Testing Hypotheses About Contrasts 256 9.4 The Proper Unit for the Control of Type 1 Error 262 9.5 Planned Versus Post Hoc Contrasts 264 9.6 Controlling the FWE for Families of K Planned Contrasts 265 9.7 Testing All Pairwise Contrasts 268 9.8 Comparing a – 1 Treatment Means with a Control: Dunnett's Test 276 9.9 Controlling the Familywise Error Rate for Post Hoc Contrasts 277 9.10 The Sum of Squares Associated with a Contrast 279 9.11 Concluding Remarks 281 CHAPTER 10 TREND ANALYSIS 288 10.1 Introduction 288 10.2 Linear Trend 289 10.3 Testing Nonlinear Trends 295 10.4 Concluding Remarks 301 CHAPTER 11 MULTIFACTOR BETWEEN-SUBJECTS DESIGNS: SIGNIFICANCE TESTS IN THE TWO-WAY CASE 305 11.1 Introduction 305 11.2 A First Look at the Data 306 11.3 Two-Factor Designs: The ANOVA 309 11.4 The Structural Model and Expected Mean Squares 316 11.5 Main Effect Contrasts 318 11.6 More About Interaction 319 11.7 Simple Effects 323 11.8 Two-Factor Designs: Trend Analysis 326 11.9 Concluding Remarks 330 CHAPTER 12 MULTIFACTOR BETWEEN-SUBJECTS DESIGNS: FURTHER DEVELOPMENTS 336 12.1 Introduction 336 12.2 Measures of Effect Size 336 12.3 Power of the F Test 339 12.4 Unequal Cell Frequencies 340 12.5 Three-Factor Designs 345 12.6 More than Three Independent Variables 353 12.7 Pooling in Factorial Designs 353 12.8 Blocking to Reduce Error Variance 356 12.9 Concluding Remarks 357 CHAPTER 13 REPEATED-MEASURES DESIGNS 363 13.1 Introduction 363 13.2 The Additive Model and Expected Mean Squares for the S × A Design 366 13.3 The Nonadditive Model for the S × A Design 373 13.4 Hypothesis Tests Assuming Nonadditivity 376 13.5 Power of the F Test 384 13.6 Multifactor Repeated-Measures Designs 384 13.7 Fixed or Random Effects? 392 13.8 Nonparametric Procedures for Repeated-Measures Designs 393 13.9 Concluding Remarks 398 CHAPTER 14 MIXED DESIGNS: BETWEEN-SUBJECTS AND WITHIN-SUBJECTS FACTORS 407 14.1 Introduction 407 14.2 One Between-Subjects and One Within-Subjects Factor 407 14.3 Rules for Generating Expected Mean Squares 413 14.4 Measures of Effect Size 415 14.5 Power Calculations 417 14.6 Contrasting Means in Mixed Designs 418 14.7 Testing Simple Effects 422 14.8 Pretest-Posttest Designs 423 14.9 Additional Mixed Designs 424 14.10 Concluding Remarks 428 CHAPTER 15 USING CONCOMITANT VARIABLES TO INCREASE POWER: BLOCKING AND ANALYSIS OF COVARIANCE 433 15.1 Introduction 433 15.2 Example of an ANCOVA 436 15.3 Assumptions and Interpretation in an ANCOVA 443 15.4 Testing Homogeneity of Slopes 448 15.5 More About ANCOVA Versus Treatments × Blocks 449 15.6 Estimating Power in an ANCOVA 451 15.7 ANCOVA in Higher-Order Designs 452 15.8 Some Extensions of the ANCOVA 452 15.9 Concluding Remarks 453 CHAPTER 16 HIERARCHICAL DESIGNS 457 16.1 Introduction 457 16.2 Groups Within Treatments 458 16.3 Groups Versus Individuals 464 16.4 Extensions of the Groups-Within-Treatments Design 466 16.5 Items Within Treatments 470 16.6 Concluding Remarks 473 CHAPTER 17 LATIN SQUARES AND RELATED DESIGNS 478 17.1 Introduction 478 17.2 Selecting a Latin Square 480 17.3 The Single Latin Square 482 17.4 The Replicated Latin Square Design 490 17.5 Balancing Carry-Over Effects 495 17.6 Greco-Latin Squares 497 17.7 Concluding Remarks 498 CHAPTER 18 MORE ABOUT CORRELATION 501 18.1 Introduction 501 18.2 Further Issues in Understanding the Correlation Coefficient 502 18.3 Inference About Correlation 510 18.4 Partial Correlations 522 18.5 Other Measures of Correlation 525 18.6 Concluding Remarks 532 CHAPTER 19 MORE ABOUT BIVARIATE REGRESSION 540 19.1 Introduction 540 19.2 Regression Toward the Mean 541 19.3 Inference in Linear Regression 543 19.4 An Example: Regressing Cholesterol Level on Age 553 19.5 Checking for Violations of Assumptions 555 19.6 Locating Outliers and Influential Data Points 563 19.7 Testing Independent Slopes for Equality 569 19.8 Repeated-Measures Designs 570 19.9 Multilevel Modeling 572 19.10 Concluding Remarks 572 CHAPTER 20 MULTIPLE REGRESSION 583 20.1 Introduction 583 20.2 A Regression Example with Several Predictor Variables 584 20.3 The Nature of the Regression Coefficients 593 20.4 The Multiple Correlation Coefficient and the Partitioning of Variability in Multiple Regression 594 20.5 Inference in Multiple Regression 601 20.6 Selecting the Best Regression Equation for Prediction 612 20.7 Explanation Versus Prediction in Regression 614 20.8 Testing for Curvilinearity in Regression 619 20.9 Including Interaction Terms in Multiple Regression 622 20.10 Multiple Regression in Repeated-Measures Designs 628 20.11 Concluding Remarks 629 CHAPTER 21 REGRESSION WITH CATEGORICAL AND QUANTITATIVE VARIARLES: THE GENERAL LINEAR MODEL 635 21.1 Introduction 635 21.2 One-Factor Designs 636 21.3 Regression Analyses and Factorial Designs 642 21.4 Using Categorical and Continuous Variables in the Same Analysis 651 21.5 Coding Designs with Within-Subjects Factors 655 21.6 Concluding Remarks 658 APPENDIXES 662 Appendix A: Notation and Summation Operations 662 Appendix B: Expected Values and Their Applications 670 Appendix C: Statistical Tables 674 Answers to Selected Exercises 706 Endnotes 742 References 750 Author Index 764 A 764 B 764 C 764 D 765 E 765 F 765 G 765 H 766 I 766 J 766 K 766 L 766 M 767 N 767 O 767 P 767 R 767 S 768 T 768 V 768 W 768 Y 769 Z 769 Subject Index 770 A 770 B 771 C 772 D 773 E 774 F 775 G 775 H 775 I 776 J 776 K 776 L 776 M 776 N 777 O 777 P 777 Q 778 R 778 S 780 T 780 U 781 V 781 W 781 Y 781 Z 781

this Book Emphasizes The Statistical Concepts And Assumptions Necessary To Describe And Make Inferences About Real Data. Throughout The Book The Authors Encourage The Reader To Plot And Examine Their Data, Find Confidence Intervals, Use Power Analyses To Determine Sample Size, And Calculate Effect Sizes. The Goal Is To Ensure The Reader Understands The Underlying Logic And Assumptions Of The Analysis And What It Tells Them, The Limitations Of The Analysis, And The Possible Consequences Of Violating Assumptions.

the Simpler, Less Abstract Discussion Of Analysis Of Variance Is Presented Prior To Developing The More General Model. A Concern For Alternatives To Standard Analyses Allows For The Integration Of Non-parametric Techniques Into Relevant Design Chapters, Rather Than In A Single, Isolated Chapter. This Organization Allows For The Comparison Of The Pros And Cons Of Alternative Procedures Within The Research Context To Which They Apply.

basic Concepts, Such As Sampling Distributions, Expected Mean Squares, Design Efficiency, And Statistical Models Are Emphasized Throughout. This Approach Provides A Stronger Conceptual Foundation In Order To Help The Reader Generalize The Concepts To New Situations They Will Encounter In Their Research And To Better Understand The Advice Of Statistical Consultants And The Content Of Articles Using Statistical Methodology.

the Second Edition Features A Greater Emphasis On Graphics, Confidence Intervals, Measures Of Effect Size, Power Analysis, Tests Of Contrasts, Elementary Probability, Correlation, And Regression. a Free Cd That Contains Several Real And Artificial Data Sets Used In The Book In Spss, Systat, And Ascii Formats, Is Included In The Back Of The Book. An instructor's Solutions Manual, Containing The Intermediate Steps To All Of The Text Exercises, Is Available Free To Adopters.

"Intended both as a textbook for students and as a resource for researchers, this book emphasizes the statistical concepts and assumptions necessary to describe and make inferences about real data. Throughout the book the authors encourage readers to plot and examine their data find confidence intervals, use power analyses to determine sample size, and calculate effect sizes.". "Using an intuitive, informal style, the authors adopt a "bottom-up" approach - a simpler, less abstract discussion of analysis of variance is presented prior to developing the more general model. A concern for alternatives to standard analyses allows for the integration of non-parametric techniques into relevant design chapter, rather than in a single, isolated chapter. This organization allows for the comparison of the pros and cons of alternative procedures within the research context to which they apply.". "Basic concepts such as sampling distribution, expected mean squares, design efficiency, and statistical models are emphasized throughout. This approach provides a stronger conceptual foundation in order to help readers generalize the concepts to new situations they will encounter in their research and to better understand the advice of statistical consultants and the content of article using statistical methodology."--BOOK JACKET.