Handbook of item response theory volume one models 1st Edition by Wim Van Der Linden 1466514310 9781466514317
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ISBN 10: 1466514310
ISBN 13: 9781466514317
Author: Wim J. Van Der Linden
Drawing on the work of internationally acclaimed experts in the field, Handbook of Item Response Theory, Volume One: Models presents all major item response models. This first volume in a three-volume set covers many model developments that have occurred in item response theory (IRT) during the last 20 years. It describes models for different response formats or response processes, the need of deeper parameterization due to a multilevel or hierarchical structure of the response data, and other extensions and insights. In Volume One, all chapters have a common format with each chapter focusing on one family of models or modeling approach. An introductory section in every chapter includes some history of the model and a motivation of its relevance. Subsequent sections present the model more formally, treat the estimation of its parameters, show how to evaluate its fit to empirical data, illustrate the use of the model through an empirical example, and discuss further applications and remaining research issues.
Table of contents:
1 Introduction
1.1 Alfred Binet
1.2 Louis Thurstone
1.3 Frederic Lord and George Rasch
1.4 Later Contributions
1.5 Unifying Features of IRT
References
Section I Dichotomous Models
2 Unidimensional Logistic Response Models
2.1 Introduction
2.2 Presentation of the Models
2.2.1 Fixed-Effects Models
2.2.2 Random-Effects Models
2.2.3 Model Identifiability
2.2.4 Parameter Linking
2.2.5 Model Interpretation
2.3 Parameter Estimation
2.4 Model Fit
2.5 Empirical Example
2.6 Discussion
Acknowledgments
References
3 Rasch Model
3.1 Introduction
3.2 Presentation of the Model
3.3 Parameter Estimation
3.4 Model Fit
3.5 Empirical Example
3.6 Discussion
References
Section II Nominal and Ordinal Models
4 Nominal Categories Models
4.1 Introduction
4.2 Presentation of the Model
4.2.1 Bock’s (1972) Original Nominal Categories Model
4.2.2 Thissen et al.’s (2010) General-Purpose Multidimensional Nominal Model
4.3 Parameter Estimation
4.4 Model Fit
4.5 Empirical Examples
4.5.1 Unidimensional Example: Testlet Scores on a Reading Comprehension Test
4.5.2 Multidimensional Example: Response Alternatives on a Quality-of-Life Scale with Item Clusters
4.6 Discussion and Conclusions
4.6.1 Other Uses of the Nominal Model
4.6.2 Current Research
Acknowledgments
References
5 Rasch Rating-Scale Model
5.1 Introduction
5.2 Presentation of the Models
5.2.1 Class of Rasch Models
5.2.2 Noncollapsed Adjacent Categories
5.2.3 Empirical Ordering of Categories
5.2.4 Parameter Estimation
5.3 Model Fit
5.3.1 Item Fit
5.3.2 Person Fit
5.4 Empirical Example
5.4.1 Threshold Order
5.4.2 Interpretation of Response Structure and Threshold Disorder
5.4.3 Interpreting Threshold Distances
5.5 Discussion
References
6 Graded Response Models
6.1 Introduction
6.2 Presentation of the Models
6.2.1 General Graded-Response Model
6.2.2 Homogeneous Case
6.2.3 Heterogeneous Case
6.3 Parameter Estimation
6.4 Goodness of Fit
6.5 Heterogeneous Case Based on the Logistic Positive Exponent Family Models
6.6 Discussion
References
7 Partial Credit Model
7.1 Introduction
7.2 Presentation of the Model
7.2.1 Response Functions
7.2.2 Parameter Interpretation
7.2.3 Relations to Other Models
7.3 Parameter Estimation
7.3.1 CML Estimation
7.3.2 Marginal Maximum-Likelihood Estimation
7.4 Goodness of Fit
7.4.1 Unweighted Mean-Square (“Outfit”)
7.4.2 Weighted Mean-Square (“Infit”)
7.5 Empirical Example
7.5.1 Measuring Essay Quality
7.5.2 Calibrating Markers
7.6 Discussion
References
8 Generalized Partial Credit Model
8.1 Introduction
8.2 Presentation of the Model
8.3 Parameter Estimation
8.3.1 E-Step
8.3.2 M-Step
8.4 Goodness of Fit
8.5 Empirical Example
8.6 Discussion
References
9 Sequential Models for Ordered Responses
9.1 Introduction
9.2 Presentation of the Model
9.2.1 Item Steps and Response: The Sequential Mechanism
9.2.2 Modeling of Steps
9.2.3 An Alternative Concept of Steps: Partial Credit Model
9.2.4 The Sequential Model and the Graded Response Model
9.3 Parameter Estimation
9.3.1 Joint Maximum Likelihood
9.3.2 Marginal Maximum Likelihood
9.4 Model Fit
9.5 Empirical Example
9.6 Discussion
References
10 Models for Continuous Responses
10.1 Introduction
10.2 Presentation of the Models
10.3 Parameter Estimation
10.3.1 Item Parameters
10.3.2 Person Parameters
10.3.3 Precision of Estimates
10.3.4 Observed Test-Score Precision
10.4 Model Fit
10.5 Empirical Example
10.6 Discussion
References
Section III Multidimensional and Multicomponent Models
11 Normal-Ogive Multidimensional Models
11.1 Introduction
11.2 Presentation of the Model
11.3 Parameter Estimation
11.3.1 Asymptotic Distribution of Estimators
11.4 Goodness of Fit
11.5 Empirical Example
11.6 Conclusion
References
12 Logistic Multidimensional Models
12.1 Introduction
12.2 Presentation of the Models
12.3 Parameter Estimation
12.4 Model Fit
12.5 Empirical Example
12.6 Discussion
References
13 Linear Logistic Models
13.1 Introduction
13.2 Presentation of the Models
13.2.1 Lltm
13.2.2 Incorporating Random Item Variation
13.2.3 Estimating the Item Characteristics from Subtask Data
13.2.4 Including Local Dependencies between Items
13.2.5 Incorporating Individual Differences in the Effect of Item Features
13.2.6 Incorporating Group Differences in the Effects of Item Features
13.3 Parameter Estimation
13.4 Model Fit
13.5 Empirical Example
13.5.1 Model and Data Set
13.5.2 Method
13.5.3 Results
13.6 Discussion
References
14 Multicomponent Models
14.1 Introduction
14.1.1 History of the Models
14.2 Presentation of the Models
14.2.1 Multicomponent Models for Subtask Data
14.2.2 Multicomponent Models for Varying Components
14.3 Parameter Estimation
14.4 Model Fit
14.5 Empirical Examples
14.5.1 Standards-Based Mathematical Skills
14.5.2 Cognitive Components in Mathematical Achievement Items
14.6 Discussion
Acknowledgment
References
Section IV Models for Response Times
15 Poisson and Gamma Models for Reading Speed and Error
15.1 Introduction
15.2 Rasch Poisson Counts Model
15.2.1 Derivation of the Model
15.3 Parameter Estimation
15.3.1 Joint Maximum Likelihood Estimation
15.3.2 Maximum Marginal Likelihood Estimation
15.4 Rasch Model for Speed
15.4.1 Derivation of the Model
15.4.2 Parameter Estimation: MML
15.4.3 Model Extensions
15.4.4 Model Fit
15.4.5 Examples
15.5 Discussion
References
16 Lognormal Response-Time Model
16.1 Introduction
16.2 Presentation of the Model
16.2.1 Assumptions
16.2.2 Formal Model
16.2.3 Parameter Interpretation
16.2.4 Parameter Identifiability and Linking
16.2.5 Moments of RT Distributions
16.2.6 Relationships with Other Models
16.3 Parameter Estimation
16.3.1 Estimating Both Item and Person Parameters
16.3.2 Estimating the Speed Parameters
16.4 Model Fit
16.5 Empirical Example
16.6 Discussion
References
17 Diffusion-Based Response-Time Models
17.1 Introduction
17.2 Models
17.2.1 The D-Diffusion Model
17.2.2 The Q-Diffusion Model
17.2.3 Relations to Other Models and Old Ideas
17.3 Parameter Estimation
17.4 Model Fit
17.5 Empirical Example
17.6 Discussion
References
Section V Nonparametric Models
18 Mokken Models
18.1 Introduction
18.2 Presentation of the Models
18.3 Parameter Estimation
18.4 Model Fit
18.4.1 Conditional Association and Local Independence
18.4.2 Manifest Monotonicity and Latent Monotonicity
18.4.3 Manifest and Latent Invariant Item Ordering
18.4.4 Item Selection
18.4.5 Person Fit and Reliability
18.5 Empirical Example
18.6 Discussion
References
19 Bayesian Nonparametric Response Models
19.1 Introduction
19.2 Mixture IRT and BNPs
19.3 Presentation of the Model
19.4 Parameter Estimation
19.5 Model Fit
19.6 Empirical Example
19.7 Discussion
References
20 Functional Approaches to Modeling Response Data
20.1 Introduction
20.2 Modeling Item Response Manifolds
20.2.1 Performance-Indexing Systems
20.2.2 Uniform Measure or Rank
20.3 Estimating Performance Manifolds P
20.3.1 Improving Sum Score by Weighting
20.3.2 A Functional Test Analysis Algorithm
20.4 Plotting Performance Manifolds
20.5 Conclusion
Acknowledgment
References
Section VI Models for Nonmonotone Items
21 Hyperbolic Cosine Model for Unfolding Responses
21.1 Introduction
21.2 Presentation of the Model
21.2.1 Latitude of Acceptance
21.3 Parameter Estimation
21.3.1 Solution Algorithm
21.3.2 Initial Estimates
21.3.3 Inconsistency of Parameter Estimates and a Constraint on Parameter ρi
21.4 Goodness of Fit
21.5 Example
21.6 Discussion
References
22 Generalized Graded Unfolding Model
22.1 Introduction
22.2 Presentation of the Model
22.3 Parameter Estimation
22.4 Model Fit
22.5 Empirical Example
22.6 Discussion
Appendix 22A
References
Section VII Hierarchical Response Models
23 Logistic Mixture-Distribution Response Models
23.1 Introduction
23.2 Presentation of the Models
23.3 Parameter Estimation
23.4 Model Fit
23.5 Empirical Example
23.6 Discussion
References
24 Multilevel Response Models with Covariates and Multiple Groups
24.1 Introduction
24.1.1 Multilevel Modeling Perspective on IRT
24.2 Bayesian Multilevel IRT Modeling
24.3 Presentation of the Models
24.3.1 Multilevel IRT Model
24.3.2 GLMM Presentation
24.3.3 Multiple-Group IRT Model
24.3.4 Mixture IRT Model
24.3.5 Multilevel IRT with Random Item Parameters
24.4 Parameter Estimation
24.5 Model Fit
24.6 Empirical Example
24.6.1 Data
24.6.2 Model Specification
24.6.3 Results
24.7 Discussion
References
25 Two-Tier Item Factor Analysis Modeling
25.1 Introduction
25.2 Presentation of the Model
25.3 Parameter Estimation
25.4 Model Fit
25.5 Empirical Example
25.6 Discussion
Acknowledgments
References
26 Item-Family Models
26.1 Introduction
26.2 Presentation of the Models
26.2.1 General Model Formulation
26.2.2 Possible Restrictions on the Model
26.3 Parameter Estimation
26.4 Model Fit
26.5 Optimal Test Design
26.6 Empirical Example
26.6.1 Setup of the Study
26.6.2 Results
26.7 Discussion
References
27 Hierarchical Rater Models
27.1 Introduction
27.2 The HRM
27.2.1 IRT Model for Item Responses
27.2.2 Model for Rater Accuracy
27.2.3 Rater Covariates
27.3 Estimation
27.4 Assessing Model Fit
27.5 Example
27.6 Discussion
References
28 Randomized Response Models for Sensitive Measurements
28.1 Introduction
28.2 Presentation of the Models
28.2.1 Randomized IRT Models
28.2.2 Noncompliant Behavior
28.2.3 Structural Models for Sensitive Constructs
28.3 Parameter Estimation
28.4 Model Fit
28.5 Empirical Example
28.5.1 College Alcohol Problem Scale and Alcohol Expectancy Questionnaire
28.5.2 Data
28.5.3 Model Specification
28.5.4 Results
28.6 Discussion
Acknowledgments
Appendix 28A CAPS-AEQ Questionnaire
References
29 Joint Hierarchical Modeling of Responses and Response Times
29.1 Introduction
29.1.1 Levels of Modeling
29.2 Presentation of the Model
29.2.1 First-Level Models
29.2.2 Second-Level Models
29.2.3 Higher-Level Models
29.2.4 Identifiability
29.2.5 Alternative Plug-in Models
29.2.6 Dependency Structure of the Data
29.3 Parameter Estimation
29.4 Model Fit
29.4.1 Person Fit of the Response Model
29.4.2 Person Fit of RT Model
29.4.3 Item Fit
29.5 Empirical Example
29.6 Discussion
References
Section VIII Generalized Modeling Approaches
30 Generalized Linear Latent and Mixed Modeling
30.1 Introduction
30.2 The GLLAMM Framework
30.2.1 Response Model
30.2.2 Structural Model
30.3 Response Processes
30.3.1 Dichotomous Responses
30.3.2 Ordinal Responses
30.3.3 Comparative Responses
30.3.4 Counts
30.3.5 Continuous Responses and Response Times
30.3.6 Some Other Response Processes
30.4 Person and Item Covariates
30.5 Multidimensional Models
30.5.1 Conventional MIRT
30.5.2 Random Coefficients of Item Characteristics
30.5.3 Growth IRT
30.6 Multilevel Models
30.6.1 Multilevel IRT
30.6.2 Multilevel SEM
30.7 Parameter Estimation
30.8 Model Fit
30.9 Example
30.9.1 PIRLS Data
30.9.2 Model
30.9.3 Results and Interpretation
30.10 Discussion
References
31 Multidimensional, Multilevel, and Multi-Timepoint Item Response Modeling
31.1 Introduction
31.2 Modeling
31.2.1 Single-Level Modeling
31.2.2 Two-Level Modeling
31.3 Estimation
31.3.1 Weighted Least-Squares (WLSMV)
31.3.2 Bayesian Estimation
31.4 Empirical Examples
31.4.1 Item Bi-Factor Exploratory Factor Analysis
31.4.2 Two-Level Item Bi-Factor Exploratory Factor Analysis
31.4.3 Two-Level Item Bi-Factor Confirmatory Factor Analysis
31.4.4 Two-Level Item Bi-Factor Confirmatory Factor Analysis with Random Factor Loadings
31.4.5 Longitudinal Two-Level Item Bi-Factor Confirmatory Factor Analysis
31.5 Conclusions
References
32 Mixed-Coefficients Multinomial Logit Models
32.1 Introduction
32.2 Presentation of the Models
32.2.1 Extended Mixed Coefficient Multinomial Logit Model
32.2.2 Simple Logistic Model
32.2.3 Models with Scoring Parameters
32.2.4 Population Model
32.2.5 Combined Model
32.2.6 Model Identification
32.3 Parameter Estimation
32.3.1 Maximum-Likelihood Estimation
32.3.2 Quadrature and Monte Carlo Approximations
32.3.3 Conditional Maximum-Likelihood Estimation
32.3.4 Latent Ability Estimation and Prediction
32.3.5 Estimating Functionals of the Population Distributions
32.4 Model Fit
32.4.1 Generalized Fit Test
32.4.2 Customized Fit Tests
32.4.3 Tests of Relative Fit
32.5 Empirical Example
32.5.1 Bundle Independence
32.5.2 Saturated Bundle Model
32.5.3 PCM Bundles
32.5.4 SLM Bundles
32.5.5 Customized Approaches
32.5.6 Empirical Comparisons
32.6 Discussion
References
33 Explanatory Response Models
33.1 Introduction
33.1.1 Measurement and Explanation
33.1.2 Explanatory and Descriptive Measurement
33.1.3 Review of Literature
33.2 Presentation of the Model
33.2.1 Long Data Form
33.2.2 Initial Model Formulation
33.3 General Model
33.4 Types of Covariates
33.5 Parameter Estimation
33.6 Model Fit
33.7 Empirical Example
33.7.1 Data
33.7.2 Dendrification of the Responses
33.7.3 Model Description
33.8 Parameter Estimates
33.8.1 Model Fit
33.9 Discussion
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