Item Response Theory: Parameter Estimation Techniques, Second Edition

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ISBN: 9780824758257 | 0824758250
Cover: Hardcover
Copyright: 7/20/2004

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Offering more than 25 percent new material, this reference describes the mostrecently utilized IRT models and furnishes detailed explanation of algorithmsthat can be used to estimate the item or ability parameters under various IRTmodels.

Preface to the Second Edition

Preface to the First Edition

1. The Item Characteristic Curve: Dichotomous Response

(22)

1.1 Introduction

(1)

1.2 The Item Characteristic Curve

(4)

1.3 Two Item Characteristic Curve Models

(12)

1.3.1 The Normal Ogive Model

(8)

1.3.2 The Logistic Ogive Model

(4)

1.4 Extension of the Item Characteristic Curve Models: Dichotomous Scoring

(4)

1.4.1 Birnbaum's Three-parameter Model

(3)

1.4.2 The One-parameter Logistic Model-The Rasch Model

(1)

1.5 Summary

(1)

2. Estimating the Parameters of an Item Characteristic Curve

(40)

2.1 Introduction

(2)

2.2 Maximum Likelihood Estimation: Normal Ogive Model

(13)

2.3 Maximum Likelihood Estimation: Logistic Model

(6)

2.4 Influence of the Weighting Coefficients

(1)

2.5 The Item Log-Likelihood Surface

(1)

2.6 Maximum Likelihood Estimation: Three-Parameter Model

(9)

2.7 Minimum χ² and Minimum Transform χ² Estimations

(4)

2.7.1 Minimum χ² Estimation

(1)

2.7.2 Minimum Transform χ² Estimation

(3)

2.8 Summary

(4)

3. Maximum Likelihood Estimation of Examinee Ability

(20)

3.1 Introduction

(1)

3.2 Maximum Likelihood Estimation of Ability

(6)

3.2.1 Normal Model

(2)

3.2.2 Logistic Model

(1)

3.2.3 Birnbaum's Three-Parameter (Logistic) Model

(2)

3.3 Information Functions

(11)

3.3.1 Item Information Function

(5)

3.3.2 Samejima's Approach to the Item Information Function

(2)

3.3.3 Test Information Function

(2)

3.4 Summary

(3)

4. Procedures for Estimating Both Ability and Item Parameters

(26)

4.1 Introduction

(1)

4.2 Joint Maximum Likelihood Estimation: The Birnbaum Paradigm

(23)

4.2.1 Some Additional Facets of the Birnbaum Paradigm

(9)

4.2.2 Quality of the Parameter Estimates

101

(6)

4.3 Summary

107

(2)

5. The Rasch Model

109

(48)

5.1 Introduction

109

(1)

5.2 The Rasch Model

110

(5)

5.3 Separation of Parameters

115

(12)

5.4 Specific Objectivity

127

(2)

5.5 Conditional Maximum Likelihood Estimation Procedures

129

(7)

5.6 Application of the JMLE Procedure to the Rasch Model

136

(11)

5.6.1 Implementation of the JMLE Paradigm

142

(3)

5.6.2 Bias of the Parameter Estimates

145

(2)

5.7 Measuring the Goodness of Fit of the Rasch Model

147

(4)

5.7.1 Chi-square Tests for Goodness of Fit

147

(3)

5.7.2 Likelihood Ratio Tests for Goodness of Fit

150

(1)

5.8 The Rasch Model and Additive Conjoint Measurement

151

(1)

5.9 Research Related to the Rasch Model

152

(2)

5.10 Summary

154

(3)

6. Parameter Estimation via MMLE and an EM Algorithm

157

(20)

6.1 Introduction

157

(1)

6.2 Item Parameter Estimation via Marginal Maximum Likelihood

158

(1)

6.3 The Bock and Lieberman Solution

159

(7)

6.3.1 Quadrature Distributions

164

(2)

6.4 The Bock and Aitkin Solution

166

(9)

6.4.1 Some Background on the EM Algorithm

169

(6)

6.5 Summary

175

(2)

7. Bayesian Parameter Estimation Procedures

177

(26)

7.1 Introduction

177

(2)

7.2 The Bayesian Approach to Parameter Estimation

179

(2)

7.3 The Marginalized Bayesian Estimation Procedure

181

(2)

7.4 Marginalized Bayesian Item Parameter Estimation in PCBILOG

183

(8)

7.4.1 The Likelihood Component

184

(2)

7.4.2 The Prior Distribution Component

186

(3)

7.4.3 Bayesian Modal Estimation via EM

189

(2)

7.5 Estimation of Ability

191

(3)

7.5.1 Bayesian Modal Estimation

192

(1)

7.5.2 Bayes EAP Estimation

193

(1)

7.6 Role of the Prior Distributions: Shrinkage

194

(2)

7.7 Research on Marginal Bayesian Parameter Estimation

196

(5)

7.8 Summary

201

(2)

8. The Graded Item Response

203

(30)

8.1 Introduction

203

(1)

8.2 Some Fundamentals of the Graded Response Case

203

(7)

8.3 Parameter Estimation Procedures for the Graded Response Case

210

(10)

8.3.1 Maximum Likelihood Estimation of the Item Parameters

210

(8)

8.3.2 Ability Estimation in the Graded Response Case

218

(2)

8.4 The Information Function for the Graded Response Case

220

(6)

8.4.1 Normal ICC Model and Graded Response Information

223

(3)

8.4.2 Logistic ICC Model and Graded Response Information

226

(1)

8.5 Other Models for Polytomous Items

226

(4)

8.6 Research on the Graded Response Model

230

(1)

8.7 Summary

231

(2)

9. Nominally Scored Items

233

(34)

9.1 Introduction

233

(5)

9.2 Maximum Likelihood Estimation of Item Parameters

238

(10)

9.3 Maximum Likelihood Estimation of Ability

248

(3)

9.4 The Information Functions

251

(4)

9.5 Extensions of Bock's Nominal Response Model

255

(3)

9.6 The Relation of Nominally Scored Items and Logit-Linear Models

258

(6)

9.6.1 Maximum Likelihood Estimation of Item Parameters Under a Logit-Linear Approach

260

(4)

9.7 Summary

264

(3)

10. Parameter Estimation for Multiple Group Data

267

(16)

10.1 Introduction

267

(1)

10.2 Estimating Properties of Distributions

268

(1)

10.3 Estimation of a Latent Distribution

269

(4)

10.4 The Multiple Group Model

273

(1)

10.5 Parameter Estimation

274

(6)

10.6 Summary

280

(3)

11. Estimation of Item Parameters of Mixed Models

283

(12)

11.1 Introduction

283

(1)

11.2 A General Model

284

(1)

11.3 Parameter Estimation via Marginal Maximum Likelihood

285

(7)

11.3.1 Estimation of Item Parameters

285

(6)

11.3.2 Estimation of Ability Parameters

291

(1)

11.4 Research on Mixed Item Types

292

(1)

11.5 Summary

293

(2)

12. Parameter Estimation via Gibbs Sampler

295

(18)

12.1 Introduction

295

(2)

12.2 Albert's Gibbs Sampler

297

(4)

12.2.1 The Logic of Gibbs Sampler

297

(1)

12.2.2 Albert's Implementation of Gibbs Sampler

297

(3)

12.2.3 Continuing the Markov Chain

300

(1)

12.3 Gibbs Sampler: Another Approach

301

(5)

12.3.1 Gibbs Sampler for Item Response Models

301

(1)

12.3.2 Steps of Gibbs Sampler

302

(1)

12.3.3 Model Specifications

303

(2)

12.3.4 Starting Values

305

(1)

12.3.5 Output Monitoring

305

(1)

12.3.6 Summary Statistics

306

(1)

12.4 Extensions of Gibbs Sampler within Item Response Theory

306

(3)

12.5 Empirical Studies of Gibbs Sampler

309

(2)

12.6 Summary

311

(2)

A. Implementation of Maximum Likelihood Estimation of Item Parameters

313

(6)

A.1 Introduction

313

(1)

A.2 Implementation

313

(4)

A.3 BASIC Computer Program

317

(2)

B. Implementation of Maximum Likelihood Estimation of Examinee's Ability

319

(6)

B.1 Introduction

319

(1)

B.2 Implementation

319

(2)

B.3 BASIC Computer Program

321

(4)

C. Implementation of JMLE Procedure for the Rasch Model

325

(8)

C.1 Introduction

325

(1)

C.2 Implementation

325

(4)

C.3 BASIC Program

329

(4)

D. Implementation of Item Parameter Estimation via MMLE/EM

333

(12)

D.1 Introduction

333

(2)

D.2 Implementation

335

(4)

D.3 BASIC Computer Program

339

(6)

E. Implementing The Bayesian Approach

345

(20)

E.1 Introduction

345

(1)

E.2 Marginal Bayesian Modal Item Parameter Estimation

345

(3)

E.3 Implementation of Marginalized Bayesian Modal Item Parameter Estimation

348

(3)

E.4 Implementation of Bayesian Estimation of an Examinee's Ability

351

(5)

E.4.1 Bayesian Modal Estimation

351

(3)

E.4.2 Bayesian "Expected A Posteriori" Estimation

354

(2)

E.5 BASIC Computer Programs

356

(9)

E.5.1 Program for Marginalized Bayesian Item Parameter Estimation

356

(4)

E.5.2 Program for Bayesian Modal Estimation of an Examinee's Ability

360

(2)

E.5.3 Program for Bayesian Expected A Posteriori Estimation of an Examinee's Ability

362

(3)

F. Implementation of Parameter Estimation Under the Graded Response Model

365

(16)

F.1 Introduction

365

(1)

F.2 Implementation of Item Parameter Estimation

365

(6)

F.3 Implementation of Ability Estimation

371

(3)

F.4 BASIC Computer Programs

374

(7)

F.4.1 Item Parameter Estimation

374

(4)

F.4.2 Ability Estimation

378

(3)

G. Implementation of MLE Under Nominal Response Scoring

381

(16)

G.1 Introduction

381

(1)

G.2 Implementation of Item Parameter Estimation

381

(5)

G.3 Implementation of Ability Estimation

386

(3)

G.3.1 Introduction

386

(1)

G.3.2 Implementation of Ability Estimation

386

(3)

G.4 BASIC Computer Programs

389

(8)

G.4.1 Item Parameter Estimation

389

(4)

G.4.2 Ability Estimation

393

(4)

H. Implementation of MMLE/EM for the Rasch Model

397

(16)

H.1 Introduction

397

(1)

H.2 Implementation

398

(4)

H.3 BASIC Computer Programs

402

(11)

H.3.1 Marginal Maximum Likelihood Estimation for the Rasch Model

402

(6)

H.3.2 Marginal Maximum Likelihood Estimation for the Rasch Model: Simpler Relaxation Solution

408

(5)

I. Implementation of Multiple Groups Estimation

413

(26)

I.1 Introduction

413

(1)

I.2 Estimation of Latent Distribution Parameters

413

(2)

I.3 Estimation of Both Item and Latent Distribution Parameters

415

(5)

I.4 BASIC Computer Programs

420

(19)

I.4.1 Estimation of Latent Distribution Parameters

420

(3)

I.4.2 Estimation of Both Item and Latent Distribution Parameters: The Compact Model

423

(7)

I.4.3 Estimation of Both Item and Latent Distribution Parameters: The Augmented Model

430

(9)

J. Implementation of Estimation for Mixed Models

439

(16)

J.1 Introduction

439

(1)

J.2 Implementation

439

(4)

J.3 BASIC Computer Program

443

(12)

K. Implementation of Gibbs Sampler

455

(9)

K.1 Introduction

455

(1)

K.2 Implementation

455

(3)

K.3 BASIC Computer Program

458

(6)

References

464

(19)

Index

483

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Item Response Theory: Parameter Estimation Techniques, Second Edition

Summary

Author Biography

Table of Contents

Supplemental Materials

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Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Item Response Theory: Parameter Estimation Techniques, Second Edition

Summary

Author Biography

Table of Contents

Supplemental Materials

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