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The Analysis of Covariance and Alternatives
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Preface xv PART I BASIC EXPERIMENTAL DESIGN AND ANALYSIS 1 Review of Basic Statistical Methods 3 1.1 Introduction, 3 1.2 Elementary Statistical Inference, 4 1.3 Elementary Statistical Decision Theory, 7 1.4 Effect Size, 10 1.5 Measures of Association, 14 1.6 A Practical Alternative to Effect Sizes and Measures of Association That Is Relevant to the Individual: p(YTx > YControl), 17 1.7 Generalization of Results, 19 1.8 Control of Nuisance Variation, 20 1.9 Software, 22 1.10 Summary, 24 2 Review of Simple Correlated Samples Designs and Associated Analyses 25 2.1 Introduction, 25 2.2 Two-Level Correlated Samples Designs, 25 2.3 Software, 32 2.4 Summary, 32 3 ANOVA Basics for One-Factor Randomized Group, Randomized Block, and Repeated Measurement Designs 35 3.1 Introduction, 35 3.2 One-Factor Randomized Group Design and Analysis, 35 3.3 One-Factor Randomized Block Design and Analysis, 51 3.4 One-Factor Repeated Measurement Design and Analysis, 56 3.5 Summary, 60 PART II ESSENTIALS OF REGRESSION ANALYSIS 4 Simple Linear Regression 63 4.1 Introduction, 63 4.2 Comparison of Simple Regression and ANOVA, 63 4.3 Regression Estimation, Inference, and Interpretation, 68 4.4 Diagnostic Methods: Is the Model Apt?, 80 4.5 Summary, 82 5 Essentials of Multiple Linear Regression 85 5.1 Introduction, 85 5.2 Multiple Regression: Two-Predictor Case, 86 5.3 General Multiple Linear Regression: m Predictors, 105 5.4 Alternatives to OLS Regression, 115 5.5 Summary, 119 PART III ESSENTIALS OF SIMPLE AND MULTIPLE ANCOVA 6 One-Factor Analysis of Covariance 123 6.1 Introduction, 123 6.2 Analysis of Covariance Model, 127 6.3 Computation and Rationale, 128 6.4 Adjusted Means, 133 6.5 ANCOVA Example 1: Training Effects, 140 6.6 Testing Homogeneity of Regression Slopes, 144 6.7 ANCOVA Example 2: Sexual Activity Reduces Lifespan, 148 6.8 Software, 150 6.9 Summary, 157 7 Analysis of Covariance Through Linear Regression 159 7.1 Introduction, 159 7.2 Simple Analysis of Variance Through Linear Regression, 159 7.3 Analysis of Covariance Through Linear Regression, 172 7.4 Computation of Adjusted Means, 177 7.5 Similarity of ANCOVA to Part and Partial Correlation Methods, 177 7.6 Homogeneity of Regression Test Through General Linear Regression, 178 7.7 Summary, 179 8 Assumptions and Design Considerations 181 8.1 Introduction, 181 8.2 Statistical Assumptions, 182 8.3 Design and Data Issues Related to the Interpretation of ANCOVA, 200 8.4 Summary, 213 9 Multiple Comparison Tests and Confidence Intervals 215 9.1 Introduction, 215 9.2 Overview of Four Multiple Comparison Procedures, 215 9.3 Tests on All Pairwise Comparisons: Fisher?Hayter, 216 9.4 All Pairwise Simultaneous Confidence Intervals and Tests: Tukey?Kramer, 219 9.5 Planned Pairwise and Complex Comparisons: Bonferroni, 222 9.6 Any or All Comparisons: Scheff'e, 225 9.7 Ignore Multiple Comparison Procedures?, 227 9.8 Summary, 228 10 Multiple Covariance Analysis 229 10.1 Introduction, 229 10.2 Multiple ANCOVA Through Multiple Regression, 232 10.3 Testing Homogeneity of Regression Planes, 234 10.4 Computation of Adjusted Means, 236 10.5 Multiple Comparison Procedures for Multiple ANCOVA, 237 10.6 Software: Multiple ANCOVA and Associated Tukey?Kramer Multiple Comparison Tests Using Minitab, 243 10.7 Summary, 246 PART IV ALTERNATIVES FOR ASSUMPTION DEPARTURES 11 Johnson?Neyman and Picked-Points Solutions for Heterogeneous Regression 249 11.1 Introduction, 249 11.2 J?N and PPA Methods for Two Groups, One Covariate, 251 11.3 A Common Method That Should Be Avoided, 269 11.4 Assumptions, 270 11.5 Two Groups, Multiple Covariates, 272 11.6 Multiple Groups, One Covariate, 277 11.7 Any Number of Groups, Any Number of Covariates, 278 11.8 Two-Factor Designs, 278 11.9 Interpretation Problems, 279 11.10 Multiple Dependent Variables, 281 11.11 Nonlinear Johnson-Neyman Analysis, 282 11.12 Correlated Samples, 282 11.13 Robust Methods, 282 11.14 Software, 283 11.15 Summary, 283 12 Nonlinear ANCOVA 285 12.1 Introduction, 285 12.2 Dealing with Nonlinearity, 286 12.3 Computation and Example of Fitting Polynomial Models, 288 12.4 Summary, 295 13 Quasi-ANCOVA: When Treatments Affect Covariates 297 13.1 Introduction, 297 13.2 Quasi-ANCOVA Model, 298 13.3 Computational Example of Quasi-ANCOVA, 300 13.4 Multiple Quasi-ANCOVA, 304 13.5 Computational Example of Multiple Quasi-ANCOVA, 304 13.6 Summary, 308 14 Robust ANCOVA/Robust Picked Points 311 14.1 Introduction, 311 14.2 Rank ANCOVA, 311 14.3 Robust General Linear Model, 314 14.4 Summary, 320 15 ANCOVA for Dichotomous Dependent Variables 321 15.1 Introduction, 321 15.2 Logistic Regression, 323 15.3 Logistic Model, 324 15.4 Dichotomous ANCOVA Through Logistic Regression, 325 15.5 Homogeneity of Within-Group Logistic Regression, 328 15.6 Multiple Covariates, 328 15.7 Multiple Comparison Tests, 330 15.8 Continuous Versus Forced Dichotomy Results, 331 15.9 Summary, 331 16 Designs with Ordered Treatments and No Covariates 333 16.1 Introduction, 333 16.2 Qualitative, Quantitative, and Ordered Treatment Levels, 333 16.3 Parametric Monotone Analysis, 337 16.4 Nonparametric Monotone Analysis, 346 16.5 Reversed Ordinal Logistic Regression, 350 16.6 Summary, 353 17 ANCOVA for Ordered Treatments Designs 355 17.1 Introduction, 355 17.2 Generalization of the Abelson?Tukey Method to Include One Covariate, 355 17.3 Abelson?Tukey: Multiple Covariates, 358 17.4 Rank-Based ANCOVA Monotone Method, 359 17.5 Rank-Based Monotone Method with Multiple Covariates, 362 17.6 Reversed Ordinal Logistic Regression with One or More Covariates, 362 17.7 Robust R-Estimate ANCOVA Monotone Method, 363 17.8 Summary, 364 PART V SINGLE-CASE DESIGNS 18 Simple Interrupted Time-Series Designs 367 18.1 Introduction, 367 18.2 Logic of the Two-Phase Design, 370 18.3 Analysis of the Two-Phase (AB) Design, 371 18.4 Two Strategies for Time-Series Regression Intervention Analysis, 374 18.5 Details of Strategy II, 375 18.6 Effect Sizes, 385 18.7 Sample Size Recommendations, 389 18.8 When the Model Is Too Simple, 393 18.9 Summary, 394 19 Examples of Single-Case AB Analysis 403 19.1 Introduction, 403 19.2 Example I: Cancer Death Rates in the United Kingdom, 403 19.3 Example II: Functional Activity, 411 19.4 Example III: Cereal Sales, 414 19.5 Example IV: Paracetamol Poisoning, 424 19.6 Summary, 430 20 Analysis of Single-Case Reversal Designs 433 20.1 Introduction, 433 20.2 Statistical Analysis of Reversal Designs, 434 20.3 Computational Example: Pharmacy Wait Time, 441 20.4 Summary, 452 21 Analysis of Multiple-Baseline Designs 453 21.1 Introduction, 453 21.2 Case I Analysis: Independence of Errors Within and Between Series, 455 21.3 Case II Analysis: Autocorrelated Errors Within Series, Independence Between Series, 461 21.4 Case III Analysis: Independent Errors Within Series, Cross-Correlation Between Series, 461 21.5 Intervention Versus Control Series Design, 467 21.6 Summary, 471 PART VI ANCOVA EXTENSIONS 22 Power Estimation 475 22.1 Introduction, 475 22.2 Power Estimation for One-Factor ANOVA, 475 22.3 Power Estimation for ANCOVA, 480 22.4 Power Estimation for Standardized Effect Sizes, 482 22.5 Summary, 482 23 ANCOVA for Randomized-Block Designs 483 23.1 Introduction, 483 23.2 Conventional Design and Analysis Example, 484 23.3 Combined Analysis (ANCOVA and Blocking Factor), 486 23.4 Summary, 488 24 Two-Factor Designs 489 24.1 Introduction, 489 24.2 ANCOVA Model and Computation for Two-Factor Designs, 494 24.3 Multiple Comparison Tests for Adjusted Marginal Means, 512 24.4 Two-Factor ANOVA and ANCOVA for Repeated-Measurement Designs, 519 24.5 Summary, 530 25 Randomized Pretest?Posttest Designs 531 25.1 Introduction, 531 25.2 Comparison of Three ANOVA Methods, 531 25.3 ANCOVA for Pretest?Posttest Designs, 534 25.4 Summary, 539 26 Multiple Dependent Variables 541 26.1 Introduction, 541 26.2 Uncorrected Univariate ANCOVA, 543 26.3 Bonferroni Method, 544 26.4 Multivariate Analysis of Covariance (MANCOVA), 544 26.5 MANCOVA Through Multiple Regression Analysis: Two Groups Only, 553 26.6 Issues Associated with Bonferroni F and MANCOVA, 554 26.7 Alternatives to Bonferroni and MANCOVA, 555 26.8 Example Analyses Using Minitab, 557 26.9 Summary, 564 PART VII QUASI-EXPERIMENTS AND MISCONCEPTIONS 27 Nonrandomized Studies: Measurement Error Correction 567 27.1 Introduction, 567 27.2 Effects of Measurement Error: Randomized-Group Case, 568 27.3 Effects of Measurement Error in Exposure and Covariates: Nonrandomized Design, 569 27.4 Measurement Error Correction Ideas, 570 27.5 Summary, 573 28 Design and Analysis of Observational Studies 575 28.1 Introduction, 575 28.2 Design of Nonequivalent Group/Observational Studies, 579 28.3 Final (Outcome) Analysis, 587 28.4 Propensity Design Advantages, 592 28.5 Evaluations of ANCOVA Versus Propensity-Based Approaches, 594 28.6 Adequacy of Observational Studies, 596 28.7 Summary, 597 29 Common ANCOVA Misconceptions 599 29.1 Introduction, 599 29.2 SSAT Versus SSIntuitive AT: Single Covariate Case, 599 29.3 SSAT Versus SSIntuitive AT: Multiple Covariate Case, 601 29.4 ANCOVA Versus ANOVA on Residuals, 606 29.5 ANCOVA Versus Y/X Ratio, 606 29.6 Other Common Misconceptions, 607 29.7 Summary, 608 30 Uncontrolled Clinical Trials 609 30.1 Introduction, 609 30.2 Internal Validity Threats Other Than Regression, 610 30.3 Problems with Conventional Analyses, 613 30.4 Controlling Regression Effects, 615 30.5 Naranjo?Mckean Dual Effects Model, 616 30.6 Summary, 617 Appendix: Statistical Tables 619 References 643 Index 655

About the Author

Bradley E. Huitema, PhD, is Professor of Psychology in theIndustrial/Organizational Program at Western Michigan University.He also serves as a statistical consultant in the behavioralsciences for Western Michigan University and Children's MemorialHospital, the pediatric training center of the NorthwesternUniversity Feinberg School of Medicine. Dr. Huitema has publishedextensively in his areas of research interest, which includeapplied time series analysis, single-case and quasi-experimentaldesign, and the evaluation of health practices.

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