Preface |
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xiii | |
Acknowledgment |
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xv | |
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1 | (96) |
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3 | (6) |
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1.1 The Role of Psychological and Educational Tests |
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3 | (1) |
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1.2 The Rasch Model and Item Response Theory |
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4 | (2) |
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1.3 Where You Will Find What in This Book |
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6 | (3) |
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9 | (28) |
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10 | (2) |
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2.2 The Item Response Function |
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12 | (6) |
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2.2.1 Ability and Difficulty |
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12 | (2) |
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14 | (2) |
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2.2.3 The Logistic Function |
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16 | (2) |
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2.3 Alternative Representations |
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18 | (3) |
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2.3.1 Probability of an Incorrect Response |
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18 | (1) |
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2.3.2 Probability of an Arbitrary Response |
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19 | (1) |
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2.3.3 Alternative Representation of the Logistic Function |
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20 | (1) |
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2.3.4 Multiplicative Form |
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20 | (1) |
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2.4 Properties and Assumptions |
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21 | (13) |
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2.4.1 Sufficient Statistics |
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21 | (2) |
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2.4.2 Local Stochastic Independence |
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23 | (1) |
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24 | (2) |
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26 | (2) |
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2.4.3 Specific Objectivity |
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28 | (4) |
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32 | (1) |
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32 | (2) |
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34 | (3) |
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37 | (20) |
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3.1 Joint Maximum Likelihood Estimation |
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38 | (1) |
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3.2 Conditional Maximum Likelihood Estimation |
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39 | (4) |
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3.3 Marginal Maximum Likelihood Estimation |
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43 | (2) |
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45 | (3) |
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3.5 Person Parameter Estimation |
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48 | (2) |
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3.6 Item and Test Information |
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50 | (3) |
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3.7 Sample Size Requirements |
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53 | (1) |
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54 | (3) |
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57 | (40) |
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59 | (5) |
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59 | (1) |
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60 | (2) |
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62 | (2) |
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4.2 Tests for Item and Person Invariance |
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64 | (8) |
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4.2.1 Andersen's Likelihood Ratio Test |
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65 | (2) |
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4.2.2 Martin-Lof Test and Other Approaches for Detecting Multidimensionality |
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67 | (1) |
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68 | (1) |
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69 | (2) |
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4.2.5 Other Approaches for Detecting DIF |
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71 | (1) |
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4.2.6 How to Proceed with Problematic Items |
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71 | (1) |
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4.3 Goodness-of-Fit Tests and Statistics |
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72 | (13) |
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4.3.1 Χ2 and G2 Goodness-of-Fit Tests |
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72 | (2) |
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4.3.2 M2, RMSEA, and SRMSR |
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74 | (1) |
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4.3.3 Infit and Outfit Statistics |
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75 | (4) |
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4.3.4 Further Fit Statistics for Items |
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79 | (2) |
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4.3.5 Fit Statistics for Item Pairs |
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81 | (1) |
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4.3.6 Fit Statistics for Persons |
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82 | (1) |
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4.3.7 Nonparametric Goodness-of-Fit Tests |
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83 | (1) |
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4.3.8 Posterior Predictive Checks |
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84 | (1) |
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85 | (2) |
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4.4.1 Item Separation Index |
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85 | (1) |
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4.4.2 Person Separation Index |
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86 | (1) |
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4.5 Evaluation Through Model Comparisons |
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87 | (7) |
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4.5.1 Models with Additional Item Parameters |
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87 | (1) |
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4.5.1.1 Two-Parameter Model |
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87 | (2) |
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4.5.1.2 Three-Parameter Model |
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89 | (2) |
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4.5.1.3 Four-Parameter Model |
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91 | (1) |
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4.5.1.4 Sample Size Requirements |
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91 | (1) |
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4.5.2 Likelihood Ratio Tests |
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92 | (1) |
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4.5.3 Information Criteria |
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93 | (1) |
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94 | (3) |
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97 | (98) |
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99 | (12) |
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5.1 Installation of R and Add-On Packages |
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99 | (2) |
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5.2 Code Editors and RStudio |
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101 | (1) |
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5.3 Loading and Importing Data |
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102 | (1) |
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5.4 Getting Information About Persons and Variables |
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103 | (5) |
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5.5 Addressing Elements in Lists |
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108 | (2) |
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110 | (1) |
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111 | (36) |
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6.1 Item Parameter Estimation |
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112 | (7) |
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119 | (18) |
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120 | (1) |
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121 | (1) |
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6.2.3 Andersen's Likelihood Ratio Test and Graphical Test |
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121 | (6) |
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127 | (1) |
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128 | (5) |
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6.2.6 Removing Problematic Items |
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133 | (1) |
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134 | (1) |
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6.2.8 Item and Person Fit |
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135 | (2) |
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6.3 Plots of ICCs, Item and Test Information |
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137 | (2) |
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6.4 Person Parameter Estimation |
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139 | (3) |
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6.5 Test Evaluation in Small Data Sets |
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142 | (3) |
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145 | (2) |
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147 | (18) |
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148 | (2) |
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7.2 Item Parameter Estimates |
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150 | (6) |
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7.2.1 Illustration via Expected ICCs |
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150 | (2) |
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7.2.2 Displaying the Estimates |
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152 | (4) |
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7.3 Evaluating Goodness-of-Fit |
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156 | (4) |
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160 | (3) |
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163 | (2) |
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165 | (8) |
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8.1 Item Parameter Estimation |
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166 | (2) |
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8.2 Evaluating Goodness-of-Fit |
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168 | (2) |
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8.3 Person Parameter Estimation |
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170 | (2) |
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172 | (1) |
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173 | (22) |
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174 | (4) |
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174 | (2) |
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9.1.2 The parameters Block |
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176 | (1) |
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9.1.3 The transformed parameters Block |
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176 | (1) |
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177 | (1) |
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9.2 Sampling the Posterior Using RStan |
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178 | (5) |
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9.3 Evaluating Goodness-of-Fit |
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183 | (6) |
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189 | (6) |
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Summary of R Commands for eRm, mirt, and TAM |
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191 | (4) |
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III Beyond the Rasch Model |
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195 | (32) |
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10 Extensions to the Rasch Model |
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197 | (8) |
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10.1 The Linear-Logistic Test Model |
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197 | (2) |
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10.2 Modeling Differences Between People |
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199 | (4) |
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10.2.1 The Mixture Rasch Model |
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199 | (1) |
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10.2.2 Model-Based Recursive Partitioning |
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200 | (1) |
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10.2.3 Explanatory IRT -- The Rasch Model as a Mixed Model |
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201 | (2) |
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10.3 Multidimensional IRT Models |
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203 | (1) |
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203 | (2) |
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11 Models for Polytomous Responses |
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205 | (18) |
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11.1 The Partial Credit Model |
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206 | (9) |
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11.1.1 CCCs and Threshold Parameters |
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209 | (1) |
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11.1.2 Alternative Parameterizations |
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210 | (4) |
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11.1.3 Disordered Thresholds |
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214 | (1) |
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11.2 The Rating Scale Model |
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215 | (2) |
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11.3 The Generalized Partial Credit and the Nominal Response Model |
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217 | (1) |
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11.4 The Graded Response Model |
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217 | (1) |
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11.5 The Sequential Model |
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218 | (1) |
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11.6 Sample Size Requirements |
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219 | (1) |
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219 | (1) |
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11.8 Derivations for the Partial Credit Model |
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220 | (3) |
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12 Outlook on Special Applications |
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223 | (4) |
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12.1 Computerized Adaptive Testing |
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223 | (1) |
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12.2 Test Linking and Equating |
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224 | (1) |
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12.3 Longitudinal IRT Models |
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225 | (1) |
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226 | (1) |
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227 | (62) |
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A Useful Mathematical Formulas |
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229 | (4) |
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229 | (1) |
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230 | (1) |
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230 | (1) |
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A.4 Differentiation Rules |
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231 | (1) |
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231 | (1) |
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232 | (1) |
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233 | (22) |
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B.1 Statistical Estimation |
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233 | (1) |
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B.1.1 The Binomial Distribution |
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233 | (2) |
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B.1.2 Maximum Likelihood Estimation |
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235 | (3) |
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B.1.3 Likelihood for Multiple Observations |
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238 | (3) |
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241 | (1) |
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B.1.4.1 Bayes' Rule by Example |
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241 | (2) |
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B.1.4.2 Coin Flipping with a Uniform Prior |
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243 | (3) |
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B.1.4.3 Informative Priors and Beta-Binomial Model |
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246 | (2) |
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248 | (1) |
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B.2.1 Tests Based on the Χ2 Distribution |
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249 | (1) |
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B.2.1.1 Χ2 Test for Independence |
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249 | (3) |
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B.2.1.2 Goodness-of-Fit and Other Χ2 Tests |
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252 | (1) |
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B.2.2 Tests Based on the Normal Distribution |
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253 | (2) |
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C Answers to the End of Chapter Questions |
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255 | (34) |
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255 | (1) |
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256 | (3) |
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259 | (3) |
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262 | (3) |
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265 | (9) |
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274 | (7) |
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281 | (2) |
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283 | (3) |
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C.10 Answers for Chapter 10 |
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286 | (1) |
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C.11 Answers for Chapter 11 |
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287 | (1) |
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C.12 Answers for Chapter 12 |
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288 | (1) |
References |
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289 | (12) |
Author Index |
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301 | (4) |
Index |
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305 | |