Bayesian Estimation for the Rasch Model using WinBUGS

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

The known data response[i,j] is represented in rectangular form. The unknown parameters (θ[i], d[i,j], τ) are represented in circular form. The dependency amongst the data and parameters are shown using directed arrows.

Such a graphical illustration can enhance understanding of the model by others; especially for a more complex model.

WinBUGS specification of the Rasch dichotomous model
model { # Simple Rasch Model in WinBUGS
for (i in 1 : N) { # Total number of students: N
for (k in 1 : T) { # Total number of items: T
response[i, k] ~ dbern(p[i, k]) # Response follows a Bernoulli distribution
logit(p[i, k]) <- theta[i] - d[k] } # The transformed prob. equals to difference between
} # student ability and # item difficulty
 
# Prior distributions for unknown parameters
for (i in 1:N) {theta[i] ~ dnorm(0, tau)} # prior distribution for student abilities
for (k in 1:T) {d[k] ~ dnorm(0, 0.001)} # prior distribution for item difficulties
tau ~ dgamma(0,001, 0.001) # prior distribution for precision of student abilities
sigma<-1/sqrt(tau) # calculate the standard derivation from precision
}

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
Itemmeansd2.5%median97.5%ltm
1-2.740.13-3.00-2.74-2.49-2.87
2-10.08-1.15-1-0.84-1.06
3-0.240.07-0.38-0.24-0.1-0.26
4-1.310.08-1.47-1.31-1.14-1.39
5-2.10.11-2.31-2.1-1.9-2.22

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.

 Obs
Freq
Expected Frequency
Scoremeansd2.5%median97.5%
032.41.6026
12120.65.1112031
28788.29.77088107
3240228.114.5200228256
4361366.017.1333366399
5303294.817.6261295330


Figure 2. Observed and Expected Frequencies

Flexibility in Enhancing the Model

WinBUGS allows a great flexibility in modeling. For example, we could easily enhance the modeling of student abilities θi with other covariates Xti, if such information is available. One of the possible formulations could be:

θi ~ N(μi, σθ²)

where μi = β0 + ΣtβtXti and σθ²~IG(0.001,0.001).

The WinBUGS code above could be modified easily to incorporate such an enhancement. Parameter estimation in the enhanced model could be automatically taken care by WinBUGS.

Summary

As compared with the proprietary software, the advantages of using the WinBUGS include the following:

(1) the Rasch model can be displayed in a graphical display to facilitate communication and understanding;

(2) testing statistics for model checking could be tailored for the problem at hand; and

(3) a great flexibility in modeling is provided.

Dr. Fung Tze-ho
Manager-Assessment Technology & Research,
Hong Kong Examinations and Assessment Authority
www.hkeaa.edu.hk/en



Tze-ho F. (2009) Bayesian Estimation for the Rasch Model using WinBUGS, Rasch Measurement Transactions, 2009, 23:1, 1190-1

Please help with Standard Dataset 4: Andrich Rating Scale Model



Rasch Publications
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

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