An IRT model with a generalized Student-t link function (original) (raw)
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Bayesian Item Response model: a generalised approach for the abilities' distribution using mixtures
arXiv: Applications, 2017
Traditional Item Response Theory models assume the distribution of the abilities of the population in study to be Gaussian. However, this may not always be a reasonable assumption, which motivates the development of more general models. This paper presents a generalised approach for the distribution of the abilities in dichotomous 3-parameter Item Response models. A mixture of normal distributions is considered, allowing for features like skewness, multimodality and heavy tails. A solution is proposed to deal with model identifiability issues without compromising the flexibility and practical interpretation of the model. Inference is carried out under the Bayesian Paradigm through a novel MCMC algorithm. The algorithm is designed in a way to favour good mixing and convergence properties and is also suitable for inference in traditional IRT models. The efficiency and applicability of our methodology is illustrated in simulated and real examples.
Bayesian item response model: a generalized approach for the abilities' distribution using mixtures
Journal of Statistical Computation and Simulation, 2017
Traditional Item Response Theory models assume the distribution of the abilities of the population in study to be Gaussian. However, this may not always be a reasonable assumption, which motivates the development of more general models. This paper presents a generalised approach for the distribution of the abilities in dichotomous 3-parameter Item Response models. A mixture of normal distributions is considered, allowing for features like skewness, multimodality and heavy tails. A solution is proposed to deal with model identifiability issues without compromising the flexibility and practical interpretation of the model. Inference is carried out under the Bayesian Paradigm through a novel MCMC algorithm. The algorithm is designed in a way to favour good mixing and convergence properties and is also suitable for inference in traditional IRT models. The efficiency and applicability of our methodology is illustrated in simulated and real examples.
2014
For the past several years, high-stakes testing has been the predominant indicator used to assess students' academic ability. School systems, teachers, parents, and students are dependent upon the accuracy of academic ability estimates designated, θs, by item response theory (IRT) computer programs. In this study, the accuracy of 3 parameter logistic (3PL) IRT estimates of academic ability were obtained from the BILOG-MG and WinBUGS computer programs which were employed to compare the use of noninformative and informative priors in θ estimation. The rationale for comparing the output of these two computer programs is that the underlying statistical theory employed in these two computer programs is different, and there may be a notable difference in the accuracy of θ estimation when an informative prior is used by WinBUGS in analyzing skewed populations. In particular, the θ parameter estimates of BILOG-MG using traditional IRT analysis with non-informative priors in each situation and the θ parameter estimates of WinBUGS using Recursive Bayesian Analysis (RBA) with informative priors are compared to the true simulated θ value using Root Mean Square Errors (RMSEs). To make this comparison, Monte Carlo computer simulation is used across three occasions within three conditions giving nine comparison situations. For the priors and data generated, results show similar θ estimation accuracy for a normally distributed latent trait (RMSE = 0.35), a more accurate θ estimation process using RBA compared to traditional analysis (RMSEs of 0.36 compared to 0.76) when using latent trait distributions skewed in a similar direction, and less accurate θ estimation using RBA compared to traditional analysis (RMSEs of 1.48 compared to 0.80) when using extremely skewed negative then positive distributions in a longitudinal setting.
The Uniform Prior for Bayesian Estimation of Ability in Item Response Theory Models
International Journal of Assessment Tools in Education, 2019
Item Response Theory (IRT) models traditionally assume a normal distribution for ability. Although normality is often a reasonable assumption for ability, it is rarely met for observed scores in educational and psychological measurement. Assumptions regarding ability distribution were previously shown to have an effect on IRT parameter estimation. In this study, the normal and uniform distribution prior assumptions for ability were compared for IRT parameter estimation when the actual distribution was either normal or uniform. A simulation study that included a short test with a small sample size and a long test with a large sample size was conducted for this purpose. The results suggested using a uniform distribution prior for ability to achieve more accurate estimates of the ability parameter in the 2PL and 3PL models when the true distribution of ability is not known. For the Rasch model, an explicit pattern that could be used to obtain more accurate item parameter estimates was not found.
Bayesian Item Response Modeling: Theory and Applications by Jean-Paul Fox
International Statistical Review, 2011
Jean-Paul Fox Springer, 2010, xiv + 313 pages, £53.99/€59.95/$69.95, hardcover ISBN: 978-1-4419-0741-7 Table of contents 1. Introduction to Bayesian item response modeling 2. Bayesian hierarchical response modeling 3. Basic elements of Bayesian statistics 4. Estimation of Bayesian item response models 5. Assessment of Bayesian item response models 6. Multilevel item response theory models 7. Random item effects models 8. Response time item response models 9. Randomized item response models Readership: Researchers (including applied statisticians, psychometricians, and social scientists) as well as graduate students interested in Bayesian Item Response Theory.
Journal of Modern Applied Statistical Methods, 2012
The Mixture Item Response Theory (MixIRT) can be used to identify latent classes of examinees in data as well as to estimate item parameters such as difficulty and discrimination for each of the groups. Parameter estimation via maximum likelihood (MLE) and Bayesian estimation based on the Markov Chain Monte Carlo (MCMC) are compared for classification accuracy and parameter estimation bias for difficulty and discrimination. Standard error magnitude and coverage rates were compared across number of items, number of latent groups, group size ratio, total sample size and underlying item response model. Results show that MCMC provides more accurate group membership recovery across conditions and more accurate parameter estimates for smaller samples and fewer items. MLE produces narrower confidence intervals than MCMC and more accurate parameter estimates for larger samples and more items. Implications of these results for research and practice are discussed.
Bayesian Analysis of Item Response Theory and its Applications to Longitudinal Education Data
2016
Inferences on ability in item response theory (IRT) have been mainly based on item responses while response time is often ignored. This is a loss of information especially with the advent of computerized tests. Most of the IRT models may not apply to these modern computerized tests as they still suffer from at least one of the three problems, local independence, randomized item and individually varying test dates, due to the flexibility and complex designs of computerized (adaptive) tests. In Chapter 2, we propose a new class of state space models, namely dynamic item responses and response times models (DIR-RT models), which conjointly model response time with time series of dichotomous responses. It aims to improve the accuracy of ability estimation via auxilary information from response time. A simulation study is conducted to ensure correctness of proposed sampling schemes to estimate parameters, whereas an empirical study is conducted using MetaMetrics datasets to demonstrate i...
Extensions of the skew-normal ogive item response model
Brazilian Journal of Probability and Statistics, 2014
We introduce new applications of the skew-probit IRT model by considering a flexible skew-normal distribution for the latent variables and by extending this model to include an additional random effect for modeling dependence between items within the same testlet. A Bayesian hierarchical structure is presented using a double data augmentation approach. This can be easily implemented in WinBUGS or SAS by considering MCMC algorithms. Several Bayesian model selection criteria, such as DIC, EAIC and EBIC, have been considered; in addition, we use posterior sum of squares of the latent residuals as a global discrepancy measure to model comparison. Two applications illustrate the methodology, one data set related to a mathematical test and another related to reading comprehension, both applied to Peruvian students. Results indicate better performance of the more flexible models proposed in this paper.
Bias and the Effect of Priors in Bayesian Estimation of Parameters of Item Response Models
Applied Psychological Measurement, 1990
The effectiveness of a Bayesian approach to the estimation problem in item response models has been sufficiently documented in recent years. Although research has indicated that Bayesian estimates, in general, are more accurate than joint maximum likelihood (JML) estimates, the effect of choice of priors on the Bayesian estimates is not well known. Moreover, the extent to which the Bayesian estimates are biased in comparison with JML estimates is not known. The ef fect of priors and the amount of bias in Bayesian estimates is examined in this paper through simulation studies. It is shown that different specifications of prior information have relatively modest effects on the Bayesian estimates. For small samples, it is shown that the Bayesian estimates are less biased than their JML counterparts.