In this brief note, we introduce a Bayesian approach to estimating parameters for IRT using a freeware called WinBUGS. We use simple Rasch model below to illustrate such an approach and summarize its benefits at the end, as compared with the use of proprietary software (e.g. WINSTEPS and BILOG).
Simple Dichotomous Rasch Model
A student i will score 1 from answering an item k correctly; 0 otherwise. Let yik be the score. Using Simple Rasch Model, we have
yik ~ Bernoulli(pik)
logit(pik) = θi - dk
where θi is the ability of student i
dk is the difficulty of item k.
Formulation of the Rasch Model in WinBUGS
The BUGS (Bayesian inference Using Gibbs Sampling) project is concerned with flexible software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods. WinBUGS is a freeware, which provides graphical interface to access all these modeling utilities.
The first step using WinBUGS is to specify the model concerned and the prior distributions for the unknown parameters. For the simple Rasch model, this is shown in the box below.
The posterior distribution of the unknown parameters can then be obtained by running the model in WinBUGS with the response data.
Bayesian Graphical Modeling of the Rasch Model
In Bayesian graphical modeling, the simple Rasch model is represented in Figure 1.
Figure 1. Bayesian Graph of the Rasch Model
Empirical Results and Model Checking
We illustrate our approach using the classical example in educational testing - the Law School Admission Test (LSAT) data, which is available in the R package called ltm (Latent Trait Model). The data contain responses of 1000 individuals to five items which were designed to measure a single latent ability. Here are the results obtained using WinBUGS. "ltm" are the R statistics as estimates for reference.
|Estimates of Item difficulty|
We can see that the estimated values from WinBUGS are close to the ones from ltm which uses a Marginal Maximum Likelihood (MMLE) approach. As the observed data are discrete, one common method of model checking in Bayesian approach is to draw samples from posterior predictive distribution and compare the simulated frequencies of different possible outcomes with the observed ones. Here are the results of model checking.
The model checking statistics are displayed in the graph below. The observed frequencies are shown by a dashed line. The expected frequencies are shown by vertical bars. We can conclude that the observed outcomes are very close to the predicted ones.
Figure 2. Observed and Expected Frequencies
|Rasch Measurement Transactions (free, online)||Rasch Measurement research papers (free, online)||Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch||Applying the Rasch Model 3rd. Ed., Bond & Fox||Best Test Design, Wright & Stone|
|Rating Scale Analysis, Wright & Masters||Introduction to Rasch Measurement, E. Smith & R. Smith||Introduction to Many-Facet Rasch Measurement, Thomas Eckes||Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr.||Statistical Analyses for Language Testers, Rita Green|
|Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar||Journal of Applied Measurement||Rasch models for measurement, David Andrich||Constructing Measures, Mark Wilson||Rasch Analysis in the Human Sciences, Boone, Stave, Yale|
|in Spanish:||Análisis de Rasch para todos, Agustín Tristán||Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez|
|Forum||Rasch Measurement Forum to discuss any Rasch-related topic|
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|Oct. 7 - Nov. 4, 2022, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 23 - July 21, 2023, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
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