f WILEY ANOVA and ANCOVA A GLM Approach Second Edition ANDREW RUTHERFORD Staffordshire, United Kingdom Keele University School of Psychology

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1 ANOVA and ANCOVA A GLM Approach Second Edition ANDREW RUTHERFORD Keele University School of Psychology Staffordshire, United Kingdom f WILEY A JOHN WILEY & SONS, INC., PUBLICATION

2 Contents Acknowledgments xiii 1 An Introduction to General Linear Models: Regression, Analysis of Variance, and Analysis of Covariance Regression, Analysis of Variance, and Analysis of Covariance A Pocket History of Regression, ANOVA, and ANCOVA An Outline of General Linear Models (GLMs) Regression Analysis of Variance Analysis of Covariance The "General" in GLM The "Linear" in GLM Least Squares Estimates Fixed, Random, and Mixed Effects Analyses The Benefits of a GLM Approach to ANOVA and ANCOVA The GLM Presentation Statistical Packages for Computers 15 2 Traditional and GLM Approaches to Independent Measures Single Factor ANOVA Designs Independent Measures Designs Balanced Data Designs Factors and Independent Variables An Outline of Traditional ANOVA for Single Factor Designs Variance Traditional ANOVA Calculations for Single Factor Designs Confidence Intervals 30 v

3 I) vi CONTENTS 2.8 GLM Approaches to Single Factor ANOVA Experimental Design GLMs 31 Pull and Reduced Estimating Effects by Comparing Experimental Design GLMs Regression GLMs Schemes for Coding Experimental Conditions Dummy Coding 41 Variables Are Used to Why Only (p Represent All Experimental Conditions? Effect Coding Coding Scheme Solutions to the Overparameterization Problem Cell Mean GLMs Experimental Design Regression and Cell Mean GLMs 51 3 Comparing Experimental Condition Means, Multiple Hypothesis Testing, Type 1 Error, and a Basic Data Analysis Strategy Introduction Comparisons Between Experimental Condition Means Linear Contrasts Comparison Sum of Squares Orthogonal Contrasts Testing Multiple Hypotheses Type 1 and Type 2 Errors Type 1 Error Rate Inflation with Multiple Hypothesis Testing Type 1 Error Rate Control and Analysis Power Different Conceptions of Type 1 Error Rate Testwise Type 1 Error Rate Familywise Type 1 Error Rate Experimentwise Type 1 Error Rate False Discovery Rate Identifying the "Family" in Familywise Type 1 Error Rate Control Logical and Empirical Relations Logical Relations Empirical Relations Planned and Unplanned Comparisons 76

4 CONTENTS Vii Direct Assessment of Planned Comparisons Contradictory Results with ANOVA Omnibus F-tests and Direct Planned Comparisons A Basic Data Analysis Strategy ANOVA First? Strong and Weak Type 1 Error Control Stepwise Tests Test Power The Three Basic Stages of Data Analysis Stage Stage Rom's Test Shaffer's R Test Applying Shaffer's R Test After a Significant F-test Stage The Role of the Omnibus F-Test 91 4 Measures of Effect Size and Strength of Association, Power, and Sample Size Introduction Effect Size as a Standardized Mean Difference Effect Size as Strength of Association (SOA) SOA for Specific Comparisons Small, Medium, and Large Effect Sizes Effect Size in Related Measures Designs Overview of Standardized Mean Difference and SOA Measures of Effect Size Power Influences on Power Uses of Power Analysis Determining the Sample Size Needed to Detect the Omnibus Effect Determining the Sample Size Needed to Detect Specific Effects Determining the Power Level of a Planned or Completed Study The Fallacy of Observed Power 110

5 viii CONTENTS 5 GLM Approaches to Independent Measures Factorial Designs Factorial Designs Factor Main Effects and Factor Interactions Estimating Effects by Comparing Full and Reduced Experimental Design GLMs Regression GLMs for Factorial ANOVA Estimating Effects with Incremental Analysis Incremental Regression Analysis Step Step Step Effect Size Estimation SOA for Omnibus Main and Interaction Effects Complete a)2 for Main and Interaction Effects Partial S2 for Main and Interaction Effects Partial S2 for Specific Comparisons Further Analyses Main Effects: Encoding Instructions and Study Time Interaction Effect: Encoding Instructions x Study Time Simple Effects: Comparing the Three Levels of Factor B at al, and at a Simple Effects: Comparing the Two Levels of Factor A at b 1, at b2, and at b Power Determining the Sample Size Needed to Detect Omnibus Main Effects and Interactions Determining the Sample Size Needed to Detect Specific Effects GLM Approaches to Related Measures Designs Introduction Randomized Block Designs Matched Sample Designs Repeated Measures Designs Order Effect Controls in Repeated Measures Designs Randomization Counterbalancing Crossover Designs Latin Square Designs 145

6 CONTENTS ix 6.3 The GLM Approach to Single Factor Repeated Measures Designs Estimating Effects by Comparing Full and Reduced Repeated Measures Design GLMs Regression GLMs for Single Factor Repeated Measures Designs Effect Size Estimation A Complete 32 SOA for the Omnibus Effect Comparable Across Repeated and Independent Measures Designs A Partial to2 SOA for the Omnibus Effect Appropriate for Repeated Measures Designs A Partial m2 SOA for Specific Comparisons Appropriate for Repeated Measures Designs Further Analyses Power Determining the Sample Size Needed to Detect the Omnibus Effect Determining the Sample Size Needed to Detect Specific Effects The GLM Approach to Factorial Repeated Measures Designs Factorial Related and Repeated Measures Designs Fully Repeated Measures Factorial Designs Estimating Effects by Comparing Full and Reduced Experimental Design GLMs Regression GLMs for the Fully Repeated Measures Factorial ANOVA Effect Size Estimation A Complete a>2 SOA for Main and Interaction Omnibus Effects Comparable Across Repeated Measures and Independent Designs A Partial S32 SOA for the Main and Interaction Omnibus Effects Appropriate for Repeated Measures Designs A Partial S)2 SOA for Specific Comparisons Appropriate for Repeated Measures Designs Further Analyses Main Effects: Encoding Instructions and Study Time Interaction Effect: Encoding Instructions x Study Time 191

7 X CONTENTS Simple Effects: Comparison of Differences Between the Three Levels of Factor B (Study Time) at Each Level of Factor A (Encoding Instructions) Simple Effects: Comparison of Differences Between the Two Levels of Factor A (Encoding Instructions) at Each Level of Factor B (Study Time) Power GLM Approaches to Factorial Mixed Measures Designs Mixed Measures and Split-Plot Designs Factorial Mixed Measures Designs Estimating Effects by Comparing Full and Reduced Experimental Design GLMs Regression GLM for the Two-Factor Mixed Measures ANOVA Effect Size Estimation Further Analyses Main Effects: Independent Factor Encoding Instructions Main Effects: Related Factor Study Time Interaction Effect: Encoding Instructions x Study Time Simple Effects: Comparing Differences Between the Three Levels of Factor B (Study Time) at Each Level of Factor A (Encoding Instructions) Simple Effects: Comparing Differences Between the Two Levels of Factor A (Encoding Instructions) at Each Level of Factor B (Study Time) Power The GLM Approach to ANCOVA The Nature of ANCOVA Single Factor Independent Measures ANCOVA Designs Estimating Effects by Comparing Full and Reduced ANCOVA GLMs Regression GLMs for the Single Factor, Single-Covariate ANCOVA Further Analyses Effect Size Estimation 231

8 CONTENTS Xi A Partial m2 SOA for the Omnibus Effect A Partial co2 SOA for Specific Comparisons Power Other ANCOVA Designs Single Factor and Fully Repeated Measures Factorial ANCOVA Designs Mixed Measures Factorial ANCOVA Assumptions Underlying ANOVA, Traditional ANCOVA, and GLMs Introduction ANOVA and GLM Assumptions Independent Measures Designs Related Measures Assessing and Dealing with Sphericity Violations Traditional ANCOVA A Strategy for Checking GLM and Traditional ANCOVA Assumptions Assumption Checks and Some Assumption Violation Consequences Independent Measures ANOVA and ANCOVA Designs Random Sampling Independence Normality Homoscedasticity: Homogeneity of Variance Traditional ANCOVA Designs Covariate Independent of Experimental Conditions Linear Regression Homogeneous Regression Should Assumptions be Checked? Some Alternatives to Traditional ANCOVA Alternatives to Traditional ANCOVA The Heterogeneous Regression Problem The Heterogeneous Regression ANCOVA GLM 265

9 Xii CONTENTS 11.4 Single Factor Independent Measures Heterogeneous Regression ANCOVA Estimating Heterogeneous Regression ANCOVA Effects Regression GLMs for Heterogeneous Regression ANCOVA Covariate-Experimental Condition Relations Adjustments Based on the General Covariate Mean Multicolinearity Other Alternatives Stratification (Blocking) Replacing the Experimental Conditions with the Covariate The Role of Heterogeneous Regression ANCOVA Multilevel Analysis for the Single Factor Repeated Measures Design Introduction Review of the Single Factor Repeated Measures Experimental Design GLM and ANOVA The Multilevel Approach to the Single Factor Repeated Measures Experimental Design Parameter Estimation in Multilevel Analysis Applying Multilevel Models with Different Covariance Structures Using SYSTAT to Apply the Multilevel GLM of the Repeated Measures Experimental Design GLM The Linear Mixed Model The Hierarchical Linear Mixed Model Applying Alternative Multilevel GLMs to the Repeated Measures Data Empirically Assessing Different Multilevel Models 303 Appendix A 305 Appendix B 307 Appendix C 315 References 325 Index 339

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