## Bayesian Item Response Modeling: Theory and ApplicationsThe 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. |

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### Contents

1 | |

2 Bayesian Hierarchical Response Modeling | 31 |

3 Basic Elements of Bayesian Statistics | 45 |

4 Estimation of Bayesian Item Response Models | 67 |

5 Assessment of Bayesian Item Response Models | 107 |

6 Multilevel Item Response Theory Models | 141 |

7 Random Item E ects Models | 193 |

8 Response Time Item Response Models | 227 |

9 Randomized Item Response Models | 255 |

289 | |

309 | |

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ability parameters analysis assumed augmented data Bayes factor cognitive impairment computed conditional distribution conditional posterior corresponding country-speciﬁc covariance matrix cross-national deﬁned denoted diﬀerent diﬃculty parameters discrimination parameters Equation ﬁrst ﬁt ﬁxed function Gibbs sampling given graded response model hierarchical prior hyperparameters identiﬁed independent individual inferences inﬂuence item characteristics item diﬃculty item response data item response model item response theory iterations latent response latent variable level-1 likelihood logistic marginal likelihood marginal posterior MCMC scheme methods mixed eﬀects model MLIRT model model parameters multilevel model multivariate nested normal prior normally distributed p-value parameter values person parameters polytomous population posterior density posterior distribution posterior mean posterior predictive posterior probability prior distributions random eﬀects random item eﬀects randomized response Rasch model restricted sampling school eﬀects scores shrinkage speciﬁc statistical testlet threshold parameters two-parameter variance variation vector WinBUGS Yijk zero Zijk