Preface xviii About the Companion Website xxi 1 Preliminary Considerations 1 1.1 The Philosophical Bases of Knowledge: Rationalistic Versus Empiricist Pursuits 1 1.2 What is a "Model"? 3 1.3 Social Sciences Versus Hard Sciences 5 1.4 Is Complexity a Good Depiction of Reality? Are Multivariate Methods Useful? 7 1.5 Causality 8 1.6 The Nature of Mathematics: Mathematics as a Representation of Concepts 8 1.7 As a Scientist How Much Mathematics Do You Need to Know? 10 1.
8 Statistics and Relativity 11 1.9 Experimental Versus Statistical Control 12 1.10 Statistical Versus Physical Effects 12 1.11 Understanding What "Applied Statistics" Means 13 Review Exercises 14 Further Discussion and Activities 14 2 Introductory Statistics 16 2.1 Densities and Distributions 17 2.1.1 Plotting Normal Distributions 19 2.1.
2 Binomial Distributions 21 2.1.3 Normal Approximation 23 2.1.4 Joint Probability Densities: Bivariate and Multivariate Distributions 24 2.2 Chi-Square Distributions and Goodness-of-Fit Test 27 2.2.1 Power for Chi-Square Test of Independence 30 2.
3 Sensitivity and Specificity 31 2.4 Scales of Measurement: Nominal, Ordinal, Interval, Ratio 31 2.4.1 Nominal Scale 32 2.4.2 Ordinal Scale 32 2.4.3 Interval Scale 33 2.
4.4 Ratio Scale 33 2.5 Mathematical Variables Versus Random Variables 34 2.6 Moments and Expectations 35 2.6.1 Sample and Population Mean Vectors 36 2.7 Estimation and Estimators 38 2.8 Variance 39 2.
9 Degrees of Freedom 41 2.10 Skewness and Kurtosis 42 2.11 Sampling Distributions 44 2.11.1 Sampling Distribution of the Mean 44 2.12 Central Limit Theorem 47 2.13 Confidence Intervals 47 2.14 Maximum Likelihood 49 2.
15 Akaike''s Information Criteria 50 2.16 Covariance and Correlation 50 2.17 Psychometric Validity Reliability: A Common Use of Correlation Coefficients 54 2.18 Covariance and Correlation Matrices 57 2.19 Other Correlation Coefficients 58 2.20 Student''s t Distribution 61 2.20.1 t -Tests for One Sample 61 2.
20.2 t -Tests for Two Samples 65 2.20.3 Two-Sample t -Tests in R 65 2.21 Statistical Power 67 2.21.1 Visualizing Power 69 2.22 Power Estimation Using R and GâPower 69 2.
22.1 Estimating Sample Size and Power for Independent Samples t -Test 71 2.23 Paired-Samples t- Test: Statistical Test for Matched-Pairs (Elementary Blocking) Designs 73 2.24 Blocking With Several Conditions 76 2.25 Composite Variables: Linear Combinations 76 2.26 Models in Matrix Form 77 2.27 Graphical Approaches 79 2.27.
1 Box-and-Whisker Plots 79 2.28 What Makes a p -Value Small? A Critical Overview and Practical Demonstration of Null Hypothesis Significance Testing 82 2.28.1 Null Hypothesis Significance Testing (NHST): A Legacy of Criticism 82 2.28.2 The Make-Up of a p -Value: A Brief Recap and Summary 85 2.28.3 The Issue of Standardized Testing: Are Students in Your School Achieving More than the National Average? 85 2.
28.4 Other Test Statistics 86 2.28.5 The Solution 87 2.28.6 Statistical Distance: Cohen''s d 87 2.28.7 What Does Cohen''s d Actually Tell Us? 88 2.
28.8 Why and Where the Significance Test Still Makes Sense 89 2.29 Chapter Summary and Highlights 89 Review Exercises 92 Further Discussion and Activities 95 3 Analysis of Variance: Fixed Effects Models 97 3.1 What is Analysis of Variance? Fixed Versus Random Effects 98 3.1.1 Small Sample Example: Achievement as a Function of Teacher 99 3.1.2 Is Achievement a Function of Teacher? 100 3.
2 How Analysis of Variance Works: A Big Picture Overview 101 3.2.1 Is the Observed Difference Likely? ANOVA as a Comparison (Ratio) of Variances 102 3.3 Logic and Theory of ANOVA: A Deeper Look 103 3.3.1 Independent-Samples t -Tests Versus Analysis of Variance 104 3.3.2 The ANOVA Model: Explaining Variation 105 3.
3.3 Breaking Down a Deviation 106 3.3.4 Naming the Deviations 107 3.3.5 The Sums of Squares of ANOVA 108 3.4 From Sums of Squares to Unbiased Variance Estimators: Dividing by Degrees of Freedom 109 3.5 Expected Mean Squares for One-Way Fixed Effects Model: Deriving the F -ratio 110 3.
6 The Null Hypothesis in ANOVA 112 3.7 Fixed Effects ANOVA: Model Assumptions 113 3.8 A Word on Experimental Design and Randomization 115 3.9 A Preview of the Concept of Nesting 116 3.10 Balanced Versus Unbalanced Data in ANOVA Models 116 3.11 Measures of Association and Effect Size in ANOVA: Measures of Variance Explained 117 3.11.1 η 2 Eta-Squared 117 3.
11.2 Omega-Squared 118 3.12 The F -Test and the Independent Samples t -Test 118 3.13 Contrasts and Post-Hocs 119 3.13.1 Independence of Contrasts 122 3.13.2 Independent Samples t -Test as a Linear Contrast 123 3.
14 Post-Hoc Tests 124 3.14.1 Newman-Keuls and Tukey HSD 126 3.14.2 Tukey HSD 127 3.14.3 Scheffé Test 128 3.14.
4 Other Post-Hoc Tests 129 3.14.5 Contrast Versus Post-Hoc? Which Should I be Doing? 129 3.15 Sample Size and Power for ANOVA: Estimation With R and GâPower 130 3.15.1 Power for ANOVA in R and GâPower 130 3.15.2 Computing f 130 3.
16 Fixed effects One-Way Analysis of Variance in R: Mathematics Achievement as a Function of Teacher 133 3.16.1 Evaluating Assumptions 134 3.16.2 Post-Hoc Tests on Teacher 137 3.17 Analysis of Variance Via R''s lm 138 3.18 Kruskal-Wallis Test in R and the Motivation Behind Nonparametric Tests 138 3.19 ANOVA in SPSS: Achievement as a Function of Teacher 140 3.
20 Chapter Summary and Highlights 142 Review Exercises 143 Further Discussion and Activities 145 4 Factorial Analysis of Variance: Modeling Interactions 146 4.1 What is Factorial Analysis of Variance? 146 4.2 Theory of Factorial ANOVA: A Deeper Look 148 4.2.1 Deriving the Model for Two-Way Factorial ANOVA 149 4.2.2 Cell Effects 150 4.2.
3 Interaction Effects 151 4.2.4 Cell Effects Versus Interaction Effects 152 4.2.5 A Model for the Two-Way Fixed Effects ANOVA 152 4.3 Comparing One-Way ANOVA to Two-Way ANOVA: Cell Effects in Factorial ANOVA Versus Sample Effects in One-Way ANOVA 153 4.4 Partitioning the Sums of Squares for Factorial ANOVA: The Case of Two Factors 153 4.4.
1 SS Total: A Measure of Total Variation 154 4.4.2 Model Assumptions: Two-Way Factorial Model 155 4.4.3 Expected Mean Squares for Factorial Design 156 4.4.4 Recap of Expected Mean Squares 159 4.5 Interpreting Main Effects in the Presence of Interactions 159 4.
6 Effect Size Measures 160 4.7 Three-Way, Four-Way, and Higher Models 161 4.8 Simple Main Effects 161 4.9 Nested Designs 162 4.9.1 Varieties of Nesting: Nesting of Levels Versus Subjects 163 4.10 Achievement as a Function of Teacher and Textbook: Example of Factorial ANOVA in R 164 4.10.
1 Comparing Models Through AIC 167 4.10.2 Visualizing Main Effects and Interaction Effects Simultaneously 169 4.10.3 Simple Main Effects for Achievement Data: Breaking Down Interaction Effects 170 4.11 Interaction Contrasts 171 4.12 Chapter Summary and Highlights 172 Review Exercises 173 5 Introduction to Random Effects and Mixed Models 175 5.1 What is Random Effects Analysis of Variance? 176 5.
2 Theory of Random Effects Models 177 5.3 Estimation in Random Effects Models 178 5.3.1 Transitioning from Fixed Effects to Random Effects 178 5.3.2 Expected Mean Squares for MS Between and MS Within 179 5.4 Defining Null Hypotheses in Random Effects Models 180 5.4.
1 F -Ratio for Testing H 0 181 5.5 Comparing Null Hypotheses in Fixed Versus Random Effects Models: The Importance of Assumptions 182 5.6 Estimating Variance Components in Random Effects Models: ANOVA ML REML Estimators 183 5.6.1 ANOVA Estimators of Variance Components 183 5.6.2 Maximum Likelihood and Restricted Maximum Likelihood 184 5.7 Is Achievement a Function of Teacher? One-Way Random Effects Model in R 185 5.
7.1 Proportion of Variance Accounted for by Teacher 187 5.8 R Analysis Using REML 188 5.9 Analysis in SPSS: Obtaining Variance Components 188 5.10 Factorial Random Effects: A Two-Way Model 190 5.11 Fixed Effects Versus Random Effects: A Way of Conceptualizing Their Differences 191 5.12 Conceptualizing the Two-Way Random Effects Model: The Make-Up of a Randomly Chosen Observation 192 5.13 Sums of Squares and Expected Mean Squares for Random Effects: The Contaminating Influence of Interaction Effects 193 5.
13.1 Testing Null Hypotheses 194 5.14 You Get What You Go In With: The Importance of Model Assumptions and Model Selection 195 5.15 Mixed Model Analysis of Variance: Incorporating Fixed and Random Effects 196 5.15.1 Mixed Model in R 199 5.16 Mixed Models in Matrices 199 5.17 Multilevel Modeling as a Special Case of the Mixed Model: Incorporating Nesting and Clustering 200 5.
18 Chapter Summary and Highlights 201 Review Exercises 202 6 Randomized Blocks and Repeated Measures 204 6.1 What is a Randomized Block Design? 205.