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A combination of psychometric theory, history, philosophy, and practice, with recent advances in analytical methods, metrology, and design.
Psychometrics, Test Theory, and the Latent Factors Model began as a strong manuscript by Petr Blahuš, based upon his lifelong passion of applying the latent factors model to kinesiology and biomechanics. Before his death he entrusted it to three co-authors who completed the book with in-chapter explanations, accessible mathematical appendices, and computational guides. It was also expanded to include important advances since 2010 in psychometric methods, contemporary developments in metrology, and the science of domain-specific design. Comprehensive in scope, the text contains computational guides for the use of Stata, M-plus, and SPSS in Factor Analysis, Classical Test Theory, and Item-Response Theory. The authors highlight the practicality of software integration in order to successfully produce psychometrically sound research. Written by an international and decades-spanning team of experienced psychometricians, the text anticipates a future psychometric science that could earn its place in the international metrology community.
The book is filled with suggestions, tips, and practical guidance about best practices and efficient strategies for modeling and model selection. In addition, the book includes important cautions and warnings about misuse and misinterpretations of common, but limited, analytical techniques. The book is historically informed, philosophically grounded, mathematically justified, and methodologically current. This important text:
Applies intuitive reasoning and common examples to aid in the understanding of advanced technical concepts.
Includes the conceptual, statistical, and philosophical background of psychometrics.
Features recent advances and opposing views in psychometric theory.
Contains concrete examples from current research, including cognitive tests and neurological data.
Opens the vista for a future of testing with greatly increased use of well-constructed, learner-centered performance scales using computer-adaptive testing with feedback over multiple attempts.
Psychometrics, Test Theory, and the Latent Factors Model is intended for forward-looking students and teachers in the behavioral, educational, health, and social sciences.
Contents
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 Correlated Factors Model 464
6.3 Reliability Index for the Higher Order Factor Model 466
6.4 Reliability Estimation Via the Reliability Index in the Bifactor CFA 467
6.5 Reliability Estimation for Higher-Order Factor Models Using Analytic Formulas 469
7 Omega Reliability Estimation Via Reliability Formulas for Hierarchical Data 471
7.1 Omega Hierarchical 471
7.2 Omega Total 471
7.3 Omega Hierarchical Subscale 472
7.4 Omega Subscale 472
7.5 Calculating Omega Reliability Coefficients from Factor Loadings and Residual Variances 472
8 Expected Common Variance 475
8.1 Expected Common Variance for the General Factor 475
8.2 Expected Common Variance for Subscales 475
8.3 Item Expected Common Variance 475
9 Other Reliability Related Information 477
9.1 Construct Replicability Index 477
9.2 Average Relative Parameter Bias 477
9.3 Factor Determinacy 478
10 Special Situations in Reliability Assessment 480
10.1 Negative Factor Loadings on the Specific Factors 480
10.2 Correlated Indicator Residuals 481
References 482
Online Datasets and Services 483
Appendix 1: Central Tendency: Mean, Median, and Mode 485
Appendix 2: Calculation of Means, Variances, and Correlation Coefficients by the Algebra of Expectations 491
Appendix 3: Matrix Multiplication 505
Appendix 4: Bivariate Regression Analysis 519
Appendix 5: Multiple Regression Analysis using Matrix Algebra and Stata 535
References for the Five Mathematical Appendices 561
Index 563
