9781593858698

Table of Contents


	Symbols and Acronyms	xiii
1	Introduction to Measurement	1
	Measurement	1
	Some Measurement Issues	3
	Item Response Theory	4
	Classical Test Theory	5
	Latent Class Analysis	7
	Summary	9
2	The One-Parameter Model	11
	Conceptual Development of the Rasch Model	11
	The One-Parameter Model	16
	The One-Parameter Logistic Model and the Rasch Model	19
	Assumptions Underlying the Model	20
	An Empirical Data Set: The Mathematics Data Set	21
	Conceptually Estimating an Individuals Location	22
	Some Pragmatic Characteristics of Maximum Likelihood Estimates	26
	The Standard Error of Estimate and Information	27
	An Instrument�s Estimation Capacity	31
	Summary	34
3	Joint Maximum Likelihood Parameter Estimation	39
	Joint Maximum Likelihood Eslimation	39
	Indeterminacy of Parameter Estimates	41
	How Large a Calibration Sample?	42
	Example: Application of the Rasch Model to the Mathematics Data, JMLE	43
	Summary	64
4	Marginal Maximum Likelihood Parameter Estimation	68
	Marginal Maximum Likelihood Estimation	68
	Estimating an Individual�s Location: Expected a Posteriori	75
	Example: Application of the Rasch Model to the Mathematics Data, MMLE	80
	Metric Transformation and the Total Characteristic Function	92
	Summary	96
5	The Two-Parameter Model	99
	Conceptual Development of the Two-Parameler Model	99
	Information for the Two-Parameter Model	101
	Conceptual Parameter Estimation for the 2PL Model	103
	How Large a Calibration Sample?	104
	Metric Transformation, 2PL Model	106
	Example: Application of the 2PL Model to the Mathematics Data, MMLE	107
	Fit Assessment: An Alternative Approach for Assessing Invariance	110
	Information and Relative Efficiency	114
	Summary	118
6	The Three-Parameter Model	123
	Conceptual Development of the Three-Parameter Model	123
	Additional Comments About the Pseudo-Guessing Parameter, χ_j	126
	Conceptual Parameter Estimation for the 3PL Model	127
	How Large a Calibration Sample?	130
	Assessing Conditional Independence	131
	Example: Application of the 3PL Model to the Mathematics Data, MMLE	134
	Assessing Person Fit: Appropriateness Measurement	142
	Information for the Three-Parameter Model	144
	Metric Transformation, 3PL Model	147
	Handling Missing Responses	148
	Issues to Consider in Selecting Among the 1PL, 2PL, and 3PL Models	152
	Summary	154
7	Rasch Models for Ordered Polytomous Data	162
	Conceptual Development of the Panial Credit Model	163
	Conceptual Parameter Estimation of the PC Model	169
	Example: Application of the PC Model to a Reasoning Ability Instrument, MMLE	169
	The Rating Scale Model	179
	Conceptual Estimation of the RS Model	184
	Example: Application of the RS Model to an Attitudes Towards Condoms Scale, JMLE	184
	How Large a Calibration Sample?	198
	Information for the PC and RS Models	200
	Metric Transformation, PC and RS Models	201
	Summary	202
8	Non-Rasch Models for Ordered Polytomous Data	209
	The Generalized Partial Credit Model	209
	Example: Application of the GPC Model to a Reasoning Ability Instrument, MMLE	214
	Conceptual Development of the Graded Response Model	217
	How Large a Calibration Sample?	223
	Example: Application of the GR Model to an Attitudes Towards Condoms Scale, MMLE	224
	Information for Graded Data	230
	Metric Transformation, GPC and GR Models	233
	Summary	234
9	Models for Nominal Polytomous Data	237
	Conceptual Development of the Nominal Response Model	238
	How Large a Calibration Sample?	246
	Example: Application of the NR Model to a Science Test, MMLE	248
	Example: Mixed Model Calibration of the Science Test�NR and PC Models, MMLE	251
	Example: NR and PC Mixed Model Calibration of the Science Test, Collapsed Options, MMLE	254
	Information for the NR Model	259
	Metric Transformation, NR Model	261
	Conceptual Development of the Multiple-Choice Model	261
	Example: Application of the MC Model to a Science Test, MMLF	263
	Example: Application of the BS Model to a Science Test, MMLE	269
	Summary	272
10	Models for Multidimensional Data	275
	Conceptual Development of a Multidimensional IRT Model	275
	Multidimensional Item Location and Discrimination	281
	Item Vectors and Vector Graphs	285
	The Multidimensional Three-Parameter Logistic Model	288
	Assumptions of the MIRT Model	288
	Estimation of the M2PL Model	289
	Information for the M2PL Model	290
	Indeterminacy in MIRT	291
	Metric Transformation, M2PL Model	294
	Example: Application of the M2PL Model, Normal-Ogive Harmonic Analysis Robust Method	296
	Obtaining Person Location Estimates	302
	Summary	303
11	Linking and Equating	306
	Equating Defined	306
	Equating: Data Collection Phase	307
	Equating: Transformation Phase	309
	Example: Application of the Total Characteristic Function Equating Method	316
	Summary	318
12	Differential Item Functioning	323
	Differential Item Functioning and Item Bias	324
	Mamel-Haenszel Chi-Square	327
	The TSW Likelihood Ratio Test	330
	Logistic Regression	331
	Example: DIF Analysis	334
	Summary	343
Appendix A	Maximum Likelihood Estimation of Person Locations	347
	Estimating an Individual�s Location: Empirical Maximum Likelihood Estimation	347
	Estimating an Individual�s Location: Newton�s Method for MLE	348
	Revisiting Zero Variance Binary Response Patterns	354
Appendix B	Maximum Likelihood Estimation of Item Locations	356
Appendix C	The Normal Ogive Models	360
	Conceptual Development of the Normal Ogive Model	360
	The Relationship Between IRT Statistics and Traditional Item Analysis Indices	365
	Relationship of the Two-Parameter Normal Ogive and Logistic Models	368
	Extending the Two-Parameter Normal Ogive Model to a Mullidimensional Space	370
Appendix D	Computerized Adaptive Testing	373
	A Brief History	373
	Fixed-Branching Techniques	374
	Variable-Branching Techniques	375
	Advantages of Variable-Branching Over Fixed-Branching Methods	375
	IRT-Based Variable-Branching Adaptive Testing Algorithm	376
Appendix E	Miscellanea	382
	Linear Logistic Test Model (LLTM)	382
	Using Principal Axis for Estimating Item Discrimination	384
	Infinite Item Discrimination Parameter Estimates	385
	Example: NOHARM Unidimensional Calibration	387
	An Approximate Chi-Square Statistic for NOHARM	389
	Mixture Models	391
	Relative Efficiency, Monotonicity, and Information	393
	FORTRAN Formats	395
	Example: Mixed Model Calibration of the Science Test�NR and 2PL Models, MMLE	396
	Example: Mixed Model Calibration of the Science Test�NR and GR Models, MMLE	399
	Odds, Odds Ratios, and Logits	399
	The Person Response Function	403
	Linking: A Temperature Analogy Example	405
	Should DIF Analyses be Based on Latent Classes?	407
	The Separation and Reliability Indices	408
	Dependency in Traditional Item Statistics and Observed Scores	409
	References	419
	Author Index	439
	Subject Index	444
	About the Author	448