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Bayesian Item Response Modeling Theory and Applications

:	Jean-Paul Fox
:	Bayesian Item Response Modeling Theory and Applications
:	Springer-Verlag
:	9781441907424
:	1
:	CHF 132.60
:

:	Methoden der empirischen und qualitativen Sozialforschung
:	English

:	313
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

The modeling of item response data is governed by item response theory, also referred to as modern test theory. The eld of inquiry of item response theory has become very large and shows the enormous progress that has been made. The mainstream literature is focused on frequentist statistical methods for - timating model parameters and evaluating model t. However, the Bayesian methodology has shown great potential, particularly for making further - provements in the statistical modeling process. The Bayesian approach has two important features that make it attractive for modeling item response data. First, it enables the possibility of incorpor- ing nondata information beyond the observed responses into the analysis. The Bayesian methodology is also very clear about how additional information can be used. Second, the Bayesian approach comes with powerful simulation-based estimation methods. These methods make it possible to handle all kinds of priors and data-generating models. One of my motives for writing this book is to give an introduction to the Bayesian methodology for modeling and analyzing item response data. A Bayesian counterpart is presented to the many popular item response theory books (e.g., Baker and Kim 2004; De Boeck and Wilson, 2004; Hambleton and Swaminathan, 1985; van der Linden and Hambleton, 1997) that are mainly or completely focused on frequentist methods. The usefulness of the Bayesian methodology is illustrated by discussing and applying a range of Bayesian item response models.

Jean-Paul Fox is Associate Professor of Measurement and Data Analysis, University of Twente, The Netherlands. His main research activities are in several areas of Bayesian response modeling. Dr. Fox has published numerous articles in the areas of Bayesian item response analysis, statistical methods for analyzing multivariate categorical response data, and nonlinear mixed effects models.

"8 Response Time Item Response Models (p. 227-228)

Response times and responses can be collected via computer adaptive testing or computer-assisted questioning. Inferences about test takers and test items can therefore be based on the response time and response accuracy information. Response times and responses are used to measure a respondents speed of working and ability using a multivariate hierarchical item response model. A multivariate multilevel structural population model is de ned for the person parameters to explain individual and group di erences given background information. An application is presented that illustrates novel features of the model.

8.1 Mixed Multivariate Response Data

Nowadays, response times (RTs) are easily collected via computer adaptive testing or computer-assisted questioning. The RTs can be a valuable source of information on test takers and test items. The RT information can help to improve routine operations in testing such as item calibration, test design, detection of cheating, and adaptive item selection. The collection of multiple item responses and RTs leads to a set of mixed multivariate response data since the individual item responses are often observed on an ordinal scale, whereas the RTs are observed on a continuous scale.

The observed responses are imperfect indicators of a respondents ability. When measuring a construct such as ability, attention is focused on the accuracy of the test results. The observed RTs are indicators of a respondents speed of working, and speed is considered to be a di erent construct. As a result, mixed responses are used to measure the two constructs ability and speed. Although response speed and response accuracy measure di erent con- structs (Schnipke and Scrams, 2002, and references therein), the reaction-time research in psychology indicates that there is a relationship between response speed and response accuracy (Luce, 1986).

This relationship is often characterized as a speed{accuracy trade-o . A person can decide to work faster, but this will lead to a lower accuracy. The trade-o is considered to be a withinperson relationship: a respondent controls the speed of working and accepts the related level of accuracy. It will be assumed that each respondent chooses a xed level of speed, which is related to a xed accuracy. A hierarchical measurement model was proposed by van der Linden (2007) to model RTs and dichotomous responses simultaneously that accounts for di erent levels of dependency.

The di erent stages of the model capture the dependency structure of observations nested within persons at the observational level and the relationship between speed and ability at the individual level. Klein Entink, Fox and van der Linden (2009a), and Fox, Klein Entink and van der Linden (2007) extended the model for measuring accuracy and speed (1) to allow time-discriminating items, (2) to handle individual and/or group characteristics, and (3) to handle the nesting of individuals in groups.

This extension has a multivariate multilevel structural population model for the ability and the speed parameters that can be considered a multivariate extension of the structural part of the MLIRT model of Chapter 6. In this chapter, the complete modeling framework will be discussed, and an extension is made to handle polytomous response data."

	Preface	8
	Contents	12
	1 Introduction to Bayesian Response Modeling	16
	1.1 Introduction	16
	1.1.1 Item Response Data Structures	18
	Hierarchically Structured Data	18
	1.1.2 Latent Variables	20
	1.2 Traditional Item Response Models	21
	1.2.1 Binary Item Response Models	22
	The Rasch Model	22
	Two-Parameter Model	24
	Three-Parameter Model	26
	1.2.2 Polytomous Item Response Models	27
	1.2.3 Multidimensional Item Response Models	29
	1.3 The Bayesian Approach	30
	1.3.1 Bayes' Theorem	31
	Constructing the Posterior	33
	Updating the Posterior	33
	1.3.2 Posterior Inference	35
	The Role of Prior Information	30
	1.4 A Motivating Example Using WinBUGS	36
	1.4.1 Modeling Examinees' Test Results	36
	WinBUGS	37
	1.5 Computation and Software	39
	Computer Code Developed for This Book	41
	1.6 Exercises	42
	2 Bayesian Hierarchical Response Modeling	45
	2.1 Pooling Strength	45
	2.2 From Beliefs to Prior Distributions	47
	A Hierarchical Prior for Item Parameters	48
	A Hierarchical Prior for Person Parameters	52
	2.2.1 Improper Priors	52
	2.2.2 A Hierarchical Bayes Response Model	53
	Posterior Computation	55
	2.3 Further Reading	56
	2.4 Exercises	57
	3 Basic Elements of Bayesian Statistics	59
	3.1 Bayesian Computational Methods	59
	3.1.1 Markov Chain Monte Carlo Methods	60
	Gibbs Sampling	60
	Metropolis-Hastings	61
	Issues in MCMC	62
	Single Chain Analysis	63
	Multiple Chain Analysis	64
	3.2 Bayesian Hypothesis Testing	65
	3.2.1 Computing the Bayes Factor	68
	Importance Sampling	69
	Using Identities and MCMC Output	70
	Bayes Factor for Item Response Models	71
	3.2.2 HPD Region Testing	72
	3.2.3 Bayesian Model Choice	73
	3.3 Discussion and Further Reading	75
	3.4 Exercises	76
	4 Estimation of Bayesian Item Response Models	81
	4.1 Marginal Estimation and Integrals	81
	4.2 MCMC Estimation	85
	4.3 Exploiting Data Augmentation Techniques	87
	4.3.1 Latent Variables and Latent Responses	88
	4.3.2 Binary Data Augmentation	89
	4.3.3 TIMMS 2007: Dutch Sixth-Graders' Math Achievement	95
	4.3.4 Ordinal Data Augmentation	97
	4.4 Identification of Item Response Models	100
	4.4.1 Data Augmentation and Identifying Assumptions	101
	4.4.2 Rescaling and Priors with Identifying Restrictions	102
	4.5 Performance MCMC Schemes	103
	4.5.1 Item Parameter Recovery	103
	4.5.2 Hierarchical Priors and Shrinkage	106
	4.6 European Social Survey: Measuring Political Interest	109
	4.7 Discussion and Further Reading	112
	4.8 Exercises	113
	5 Assessment of Bayesian Item Response Models	121
	5.1 Bayesian Model Investigation	121
	5.2 Bayesian Residual Analysis	122
	5.2.1 Bayesian Latent Residuals	123
	5.2.2 Computation of Bayesian Latent Residuals	123
	5.2.3 Detection of Outliers	124
	5.2.4 Residual Analysis: Dutch Primary School Mathematics Test	125
	5.3 HPD Region Testing and Bayesian Residuals	126
	5.3.1 Measuring Alcohol Dependence: Graded Response Analysis	130
	Item and Person Fit	126
	Detecting Discriminating Items	128
	5.4 Predictive Assessment	131
	5.4.1 Prior Predictive Assessment	133
	5.4.2 Posterior Predictive Assessment	136
	Overview of Posterior Predictive Model Checks	138
	5.5 Illustrations of Predictive Assessment	140
	5.5.1 The Observed Score Distribution	140
	5.5.2 Detecting Testle