Handbook of Item Response Theory: Applications (3)

Chapter 15 of “Markov Chain Monte Carlo for Item Response Models” by Junker et al., deals with applied Bayes inference achieved via MCMC implemented by Gibbs sampling and Metropolis–Hastings algorithms. Theoretical and practical MCMC aspects of single and multiple chain methods are discussed, and applications to IRT simulated and real data are presented with the results of R modeling described in detail. Chapter 16 of “Statistical Optimal Design Theory” by Holling and Schwabe, reviews general properties of planning experiments in applications to IRT modeling. Design objectives and criteria of optimality and the general equivalence theory are outlined, Many IRT models can be embedded into the framework of generalized linear models (GLM), and the optimal designs are arranged adequately to their specifics. Particularly, designs by DG-, and c-optimality conditions are considered for logit, probit, and Poisson regressions. Locally optimal designs, various Bayesian designs by D-optimality criteria, maximum efficient and sequential optimal designs are also discussed. Section IV Model Fit and Comparison covers four works. Chapter 17 of “Frequentist Model-Fit Tests” by Glas, deals with different aspects of the IRT measures of model fit. One of the most important assumptions to be checked concerns the subpopulation invariance which violation is labeled as differential item functioning (DIF). This assumption implies that the item responses can be described by the same parameters in all subpopulations, so the demographic and socio-economic states should not be relevant in a specific test. Another assumption is on the appropriateness of shape of the item response function that describes the relationship between a latent variable and observable responses on an item. This test can be done via comparison of the observed and expected item response frequencies given a measure of the latent trait level. The third assumption of local stochastic independence implies that responses to different items are independent given the latent trait value, so the latent variable completely describes the responses and no additional latent dimension is needed. The tests are presented on the examples of 1PL, 2PL, and 3PL IRT models, and additional likelihood ratio (LR), Wald, uniformly most powerful (UMP), and limited-information goodness-of-fit tests are discussed. Chapter 18 of “Information Criteria” by Cohen and Cho, emphasizes that a more parsimonious model usually demonstrates less inconsistencies, ambiguities, and redundancies in explanation, and it is better to pick a model which provides the best fit to the data but also has the capability for predicting future or different data. The information criteria used in the IRT with ML estimation (MLE) and Bayesian estimation methods are described. They include KL (also known as KL distance, or KLD), Akaike, Takeuchi, and Bayesian information criteria (AIC, TIC, and BIC, respectively). By each of these criteria, for instance, AIC, the differences of the AIC for a candidate model and the minimum AIC of all the considered models can be considered as approximation to expected KLD, and the best model has minimum difference value. Plausibility and weights are also defined for choosing a model of the best fit. The criteria in the IRT special cases are considered for nested and multilevel models with respect to the number of item parameters, models with different number of random effects, comparison of models with fixed and random effects, comparison across different estimation methods, and many other special issues involved in the model selection. Chapter 19 of “Bayesian Model Fit and Model Comparison” by Sinharay, continues with Bayesian estimation research for IRT, giving a detail description of the model-fit and residual analysis, prior and posterior predictive model checking (PPMC). Model-comparison methods include AIC, BIC, and the deviance information criterion (DIC), Bayes factors (BF), cross-validation likelihood and partial BF criteria. Chapter 20 of “Model Fit with Residual Analyses” by Wells and Hambleton, finalizes with investigation on raw and standardized residuals, graphical techniques for assessing model fit, using residuals in evaluation of polytomous IRT models such as graded-response model (GRM), nonparametric IRT in form of kernel smoothing and the difference with parametric models summarized by the root integrated squared error (RISE), evaluating model fit at the test score level, and other discussions. Volume 2 is concluded by the comprehensive Index. Many dozens and hundreds of classical and the most recent references are given in each chapter that makes the handbook convenient in a quick finding for a needed research. Resuming, the handbook presents a huge compendium of models which could be innovative even for specialists in IRT and related applied research. It can definitely be useful for lecturers and graduate students, researchers and practitioners in applied psychosociological projects. Actually, it can be useful in much wider area of research because it describes a large variety of statistical techniques valuable in estimations for many other problems beyond IRT.