Generalized Kernel Equating with Applications in R

This new book focuses on test equating as a statistical process that considers the most recent research in continuizing the discrete test score distributions that go beyond the Gaussian kernel. New developments within the equating field are planned to be included. More specifically, the main focus of this volume is to provide an integrated theory of the kernel equating using several kernels and IRT methods, to study their theoretical properties, and to provide sufficient data applications and software code to facilitate the widespread use of these methods. The generalized kernel equating framework and its features will be illustrated using real empirical data with R. Cover Half Title Series Page Title Page Copyright Page Dedication Contents Foreword Preface Acronyms Symbols I. Test Equating and Kernel Equating Overview 1. Introduction 1.1. Test Score Equating 1.2. Test Equating as a Statistical Modeling Problem 1.2.1. Notation 1.2.2. Equating as a Statistical Model 1.3. The Target Population and Data Collection Designs 1.3.1. The Target Population 1.3.2. Data Collection Designs 1.3.3. Equivalent Groups Design 1.3.4. Single Group Design 1.3.5. Counterbalanced Design 1.3.6. Non-Equivalent Groups with Anchor Test Design 1.3.7. Non-Equivalent Groups with Covariates Design 1.4. Equating Transformations 1.4.1. The Kernel Equating Function 1.5. Evaluating the Equating Transformation 1.6. R Packages Used in This Book 1.7. Empirical Examples 1.8. Summary and Overview of the Book 2. Kernel Equating 2.1. Introduction 2.1.1. The Five Steps in KE 2.2. The Generalized Kernel Equating Framework 2.2.1. Presmoothing 2.2.2. Estimating Score Probabilities 2.2.3. Continuization 2.2.3.1. Bandwidth Selection 2.2.4. The Equating Transformation 2.2.5. Evaluating the Equating Transformation 2.3. Summary II. Generalized Kernel Equating Framework 3. Presmoothing 3.1. Presmoothing the Data 3.2. Parametric Statistical Models for Presmoothing 3.2.1. Polynomial Log-Linear Models 3.3. Nonparametric Methods for Presmoothing 3.3.1. Nonparametric Discrete Kernel Estimators 3.4. Smoothing using Mixture Distributions 3.4.1. Beta4 Models 3.4.2. Item Response Theory Models 3.4.2.1. Binary Item Response Theory Models 3.4.2.2. Polytomous Item Response Theory Models 3.5. Evaluation of Presmoothing Methods 3.5.1. Log-Linear Models 3.5.2. Discrete Kernel Estimators 3.5.3. Beta4 Models 3.5.4. IRT Models 3.5.5. Choosing a Presmoothing Model 3.6. Summary 4. Estimating Score Probabilities 4.1. Design Functions 4.1.1. EG Design 4.1.2. SG Design 4.1.3. CB Design 4.1.4. NEAT Design 4.1.5. NEC Design 4.1.6. Comparison of Designs 4.2. Estimated Probabilities 4.3. Estimated Probabilities from IRT Models 4.3.1. The Lord-Wingersky (LW) Algorithm 4.3.2. The Poisson’s Binomial Distribution 4.3.3. Other Approximate and Exact Methods 4.3.4. The LW Algorithm for Polytomous Data 4.3.5. Conditional Probabilities 4.4. Summary 5. Continuization 5.1. Kernel Density and Distribution Estimation 5.2. Convolutions 5.3. General Kernel Continuization 5.4. Different Kernels 5.4.1. Gaussian Kernel 5.4.2. Logistic Kernel 5.4.3. Uniform Kernel 5.4.4. Epanechnikov Kernel 5.4.5. Adaptive Kernels 5.4.6. The Percentile Rank Method 5.5. Cumulants 5.6. Summary 6. Bandwidth Selection 6.1. Bandwidth Selection in Kernel Density Estimation 6.2. Selecting Bandwidths 6.3. Minimizing a Penalty Function 6.4. Leave-One-Out Cross-Validation 6.5. Likelihood Cross-Validation 6.6. Double Smoothing 6.7. Rule-Based Bandwidth Selection 6.8. A CDF-Based Bandwidth Selection Method 6.9. Summary 7. Equating 7.1. Assumptions 7.2. Score Comparability through Scale Transformation 7.3. Equating Transformations for the OSE Framework 7.4. Linear Equating Transformation 7.5. Equating Transformation According to Equating Design 7.5.1. Chained Equating 7.6. IRT Calibration 7.6.1. Concurrent Calibration 7.7. Equating Transformations for the Local Equating Framework 7.8. Graphical Representation of the Equating Transformation 7.9. Summary 8. Evaluating the Equating Transformation 8.1. Equating-Specific Measures 8.1.1. Difference That Matters 8.1.2. Percent Relative Error 8.1.3. First-Order and Second-Order Equity 8.1.4. Standard Error of Equating 8.1.4.1. The Delta Method to Obtain the SEE 8.1.4.2. Bahadur Representation to Obtain the SEE 8.1.5. Bootstrap Standard Error of Equating 8.1.6. Indices to Compare Equating Transformations 8.1.6.1. Mean Signed Difference 8.1.6.2. Mean Absolute Difference 8.1.6.3. Root Mean Squared Difference 8.1.6.4. Standard Error of Equating Difference 8.2. Statistical Measures 8.2.1. Bias 8.2.2. Mean Squared Error 8.2.3. Root Mean Squared Error 8.2.4. Standard Error 8.2.5. Cumulants 8.3. Simulating Test Scores 8.3.1. Simulated Equating-Specific Measures 8.3.2. Simulated Comparison Indices 8.3.3. Examples of Statistical and Simulated Measures 8.4. Choice of Evaluation Measure 8.5. Summary III. Applications 9. Examples under the EG design 9.1. Software Choice 9.2. SEPA data 9.2.1. Preparing the SEPA Data for SNSequate 9.3. Step 1: Presmoothing 9.3.1. Beta4 Models 9.3.2. Discrete Kernel Estimators 9.4. Step 2: Estimating Score Probabilities 9.4.1. Beta4 Models 9.4.2. Discrete Kernel Estimators 9.5. Step 3: Continuization 9.5.1. Bandwidth Selection 9.5.2. Kernel Selection 9.6. Step 4: Equating 9.7. Step 5: Evaluating the Equating Transformation 9.7.1. PRE 9.7.2. Bootstrap Standard Error of Equating 9.7.3. Freeman-Tukey Residuals 9.8. Summary 10. Examples under the NEAT design 10.1. Software Choice 10.2. ADM Data 10.2.1. Preparing the ADM Data for kequate 10.3. Simulated Polytomous Data 10.4. Step 1: Presmoothing 10.4.1. Log-Linear Models 10.4.1.1. Modeling Complexities 10.4.1.2. Log-Linear Model Fit 10.4.2. Binary IRT Models 10.4.2.1. IRT Model Fit 10.4.3. Polytomous IRT Models 10.4.3.1. IRT Model Fit 10.5. Step 2: Estimating Score Probabilities 10.5.1. Log-Linear Models 10.5.2. IRT Models 10.6. Step 3: Continuization 10.6.1. Bandwidth Selection and Kernel Selection 10.7. Step 4: Equating 10.7.1. Log-Linear Presmoothed Data 10.7.2. IRT Presmoothed Data 10.8. Step 5: Evaluating the Equating Transformation 10.8.1. PRE 10.8.2. SEE 10.8.3. Bootstrap Standard Error of Equating 10.8.4. SEED 10.8.5. MSD, MAD, and RMSD 10.9. Summary IV. Appendix A. Installing R and Reading in Data A.1. Installing R for the First Time A.1.1. Rstudio A.2. Installing and Using R packages A.3. Loading Data B. R packages for GKE B.1. Presmoothing B.2. Estimating Score Probabilities B.3. Continuization B.4. Bandwidth Selection B.5. Equating B.6. Evaluating the Equating Transformation Bibliography Index