About the Authors xvii Foreword xviii Preface xx Acknowledgements xxvii About the Companion Website xxxii 1 The Psychometric Quest 1 1.1 Quantification in Psychometrics: Historical Origins 3 1.2 Variables, Scales, and Data 9 1.3 Statistical Methods: Location, Dispersion, and Statistical Inference 25 1.4 Summary 56 1.5 Study Questions 56 References 62 2 Scores and Their Distributions 65 2.1 Quantification in Distributions: The Standard Deviation and Standard Scores 65 2.2 Combining and Using Standard Scores 82 2.

3 Summary 91 2.4 Study Questions 93 References 96 3 Validity, Explanation, and Constructs 97 3.1 Validity and Validation 99 3.2 Terms and Tools in External Validity 111 3.3 Explanation Requires More Than Correlation 121 3.4 Internal Construct Validity and Latent Factors 132 3.5 Summary 147 3.6 Study Questions 148 References 150 4 Measurement Error and Classical Test Theory 153 4.

1 Introduction and Overview of "Unfinished Business" from Chapter 3 154 4.2 Measurement Functions that Provide Associative Links Between the Empirical World and the Informational World 160 4.3 Reliability and Classical Test Theory 171 4.4 Summary 182 4.5 Study Questions 185 References 189 5 Content Validity, Item Response Theory, and Latent Variables 191 5.1 Design and Initial Validation of Latent Factors and Domain Models 192 5.2 Content Validity and Item Response Theory 195 5.3 Latent Variable Modeling 207 5.

4 Summary 218 5.5 Study Questions 220 References 222 6 The Latent Common Factor Model 225 6.1 Latent Variables and the Common Factor Model 226 6.2 Factor Scores 243 6.3 Summary 256 6.4 Study Questions 257 References 274 7 Analysis of Multiple Factors 277 7.1 Modeling Multiple and Hierarchical Constructs 277 7.2 Confirmatory and Exploratory 287 7.

3 Summary 306 7.4 Study Questions 307 References 313 8 Metrology: Scientific, Applied, and Legal: An Introduction with Implications for Human and Social Sciences 315 8.1 Purposes of Metrology, the Science of Measurement 316 8.2 Brief History of International Metrology and Its Relevance to Human Sciences 318 8.3 Scientific Metrology 322 8.4 Applied and Legal Metrology 327 8.5 A Manifesto for Establishing International Metrology in All of the Sciences 329 8.6 Properties: Extensive and Intensive Metadomains 332 8.

7 Ostensive Metadomain 334 8.8 Summary and Concluding Thoughts 339 Acknowledgments 342 8.9 Study Questions 343 References 345 9 Epilogue: Aspects of This Psychometrics Book 347 9.1 Improbable and Unusual 347 9.2 Unusual Scientific and Pedagogical Perspectives 348 9.3 The Value of Design and Domain Theory 351 9.4 New Approaches to Measurement, Validation, and Educational Evaluation 353 9.5 Summary 357 References 358 1 Computational Guide 1 Classical Test Theory: A Model-Based Approach 359 1 Classical Test Theory and the True Score Model 360 1.

1 Reliability with Multiple Test Components 360 1.2 True Score Models for Multiple Items 361 1.3 Latent Variable Analysis for True Score Models 361 1.4 True Score Models as Factor Models 363 1.5 Latent Variable Modeling of a Composite Score 365 1.6 Inter-Item and Item-Total Correlations 366 2 Estimating True Score Models as Latent Variable Models 368 2.1 The Congeneric Model 368 2.2 Tau-Equivalent and Parallel Models 370 2.

3 A Strictly Congeneric Model 371 2.4 Item Reliability 371 2.5 The Congeneric Model with Equally Reliable Indicators 372 3 Composite Reliability 374 3.1 Coefficient Alpha 375 3.2 Bootstrapping Coefficient Alpha 376 3.3 The Reliability Index for Congeneric Data 376 3.4 Alternate Estimation of Coefficient Alpha 378 3.5 Coefficient Omega 379 3.

6 Bootstrapping Omega 380 3.7 Jointly Estimating and Comparing Alpha and Omega 380 3.8 Omega If an Item Is Deleted 382 3.9 Coefficient Rho 383 3.10 Maximal Reliability and Optimally Weighted Indicators 384 4 Standardized Reliability 385 4.1 Standardized Alpha 386 4.2 Standardized Omega 387 4.3 Standardized Rho 389 4.

4 Standardized Maximal Reliability (Coefficient H) 390 5 Summary 391 References 397 2 Computational Guide 2 Item Response Theory 399 1 Item Response Theory (IRT) Models for Dichotomous Items 400 2 Logistic (Logit) Item Response Theory Models 400 2.1 The Rasch Model 401 2.2 The One-Parameter Logistic (1PL) Model 401 2.3 The Two-Parameter Logistic (2PL) Model 402 3 Data Example 403 3.1 Analyzing the Compressed Data 403 3.2 Expanding the Compact Data and Analyzing the Expanded Data 404 3.3 Estimating the Rasch Model 405 3.4 Item Response Theory (IRT) and Categorical Item Factor Analysis (CIFA) for Dichotomous Items (1PL) 406 4 Categorical Item Factor Analysis and Item Response Theory Models for Dichotomous Items 407 4.

1 The Rasch Model 410 4.2 Estimated Ability Scores with the Rasch Model 411 4.3 Relative Model Fit of the 1PL and 2PL Models 412 4.4 Item Characteristic Curves (ICCs) and Item Information Functions (IIFs) 412 5 Normal Ogive (Probit) IRT Models 414 6 IRT Models for Dichotomous Items Using Mplus 415 6.1 The Rasch Model 415 6.2 The 1PL Model 416 6.3 The 2PL Model 417 6.4 IRT Parameterization for the 1PL and Logistic Rasch Models 417 7 IRT Models for Ordered Categorical (Polytomous) Items 417 7.

1 Category Comparisons 417 7.2 Model Specification 418 7.3 Categorical Item Factor Analysis (CIFA) for Ordered Polytomous Categorical Items 418 7.4 Model Equivalence, Nesting, and Comparisons 419 8 Four Ordinal Polytomous IRT Models 421 8.1 Model A: Item Discrimination Parameters and Step Thresholds 421 8.2 Model B: Common Discrimination Parameter and Step Thresholds 421 8.3 Model C: Item Discrimination Parameters with Item Location and Common Threshold Offsets 422 8.4 Model D: Common Discrimination Parameter with Item Location and Common Threshold Offsets 422 9 Software and Example Analyses 422 10 Cumulative Probability Models Using Stata 423 10.

1 Model A: The Graded Response Model (GRM) 423 10.2 Model B: The Graded Response Model with Common Discrimination (GRM-C) 424 11 Adjacent Category Models Using Stata 424 11.1 Model A: The Generalized Partial Credit Model (GPCM) 424 11.2 Model B: The Partial Credit Model (PCM) 425 11.3 Model C: The Generalized Rating Scale Model (GRSM) 425 11.4 Model D: The Rating Scale Model (RSM) 425 11.5 Summary 425 12 Cumulative Probability Models Using Mplus 426 12.1 Model A: The Graded Response Model (GRM) 426 12.

2 Model B: The Graded Response Model with Common Discrimination 427 12.3 Model C: The Modified Graded Response Model (MGRM) 427 12.4 Model D: The Modified Graded Response Model with Common Threshold Offsets (mgrm-c) 428 13 Adjacent Category Models with Mplus 429 13.1 Model A: The Generalized Partial Credit Model (GPCM) 429 13.2 Model B: The Partial Credit Model (PCM) 430 13.3 Model C: The Generalized Rating Scale Model (GRSM) 431 13.4 Model d 431 14 Model Implementation 432 14.1 Cumulative Probability Models 432 14.

2 Adjacent Category Models 432 15 Model Fit for the Example Data 433 A. Appendix 434 References 435 3 Computational Guide 3 Factor Analysis 437 1 Unidimensional and Multidimensional Factor Analysis 438 1.1 Factor Analysis with Multiple Factors 438 1.2 Exploratory and Confirmatory Factor Analysis 439 1.3 The Independent Clusters Correlated Factors CFA Model 440 1.4 The Correlated Factors Exploratory Factor Analysis (EFA) Model 441 1.5 Higher-Order Factor Models 441 1.6 Bifactor Models 442 1.

7 Data Example 444 2 Correlated Factors Model 445 2.1 The Correlated Factors Exploratory Factor Analysis (EFA) Model 445 2.2 Communalities and Exploratory Structural Equation Modeling in Mplus 446 2.3 The Correlated Factors Confirmatory Factor Analysis (CFA) Model 448 3 Higher-Order Factor Models 448 3.1 The Higher-Order Factor CFA 448 3.2 The Higher Order CFA Model with Direct Indicators of the Higher Order Factor 450 3.3 The Higher Order EFA/ESEM Model with Direct Indicators of the Higher Order Factor 451 4 Bifactor Models 452 4.1 The Symmetric or Complete Bifactor CFA 452 4.

2 The Bifactor CFA with a Reference Factor 453 4.3 The Bifactor CFA Model with Direct Indicators of the General Factor 454 4.4 The Orthogonal Bifactor EFA/ESEM with Direct Indicators of the General Factor 455 4.5 Model Fit of Multidimensional Factor Analysis Models 456 5 The Relationship Between the Higher Order and Bifactor Models 457 5.1 The Higher Order Model as a Constrained Bifactor Model 457 5.2 Loadings of the Indicators on the Higher-Order Factor Using the Higher Order Model 458 5.3 The Bifactor Model as an Extended Higher Order Factor Model 461 6 Reliability Estimation for Multidimensional Constructs 462 6.1 Stratified Alpha 462 6.

2 Reliability Index with the Correla.