Abstract
In this chapter, we introduce a family of IRT models where there is an assumption about the shape of the population distribution of abilities.
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References
Adams RJ, Wilson M, Wu M (1997) Multilevel item response models: an approach to errors in variables regression. J Educ Behav Stat 22:47–76
Baker FB, Kim S-H (2004) Item response theory: parameter estimation techniques, 2nd edn. Marcel Dekker, New York
Mislevy RJ, Beaton AE, Kaplan B, Sheehan KM (1992) Estimating population characteristics from sparse matrix samples of item response. J Educ Meas 29:133–161
OECD (2009a). PISA 2009 Technical report. PISA, OECD Publishing. http://dx.doi.org/10.1787/9789264167872-en
OECD (2009b) PISA data analysis manual. OECD publishing, PISA
Wu M (2005) The role of plausible values in large-scale surveys. Stud Educ Eval 31:114–128
Further Reading
Bock RD, Aitkin M (1981) Marginal maximum likelihood estimation of item parameters: application of an EM algorithm. Psychometrika 46(4):443–459
Fox (2010) covers a comprehensive discussion of Bayesian item response modelling with an emphasis on sampling-based estimation methods such as MCMC (Markov chain Monte Carlo)
Fox J-P (2010) Bayesian item response modelling: theory and applications. Springer, New York
For marginal maximum likelihood estimation methods, there are many important papers including Bock & Aitken (1981)
For plausible values, see Mislevy, Beaton, Kaplan and Sheehan (1992) for an application to NAEP (National Assessment of Educational Progress) data. For a non-technical explanation of plausible values, see Wu (2005)
For a more technical discussion on latent regression IRT models, see Mislevy and Sheehan (1989) as well as Adams, Wilson and Wu (1997)
Mislevy RJ, Sheehan KM (1989) The role of collateral information about examinees in item parameter estimation. Psychometrika 54:661–679
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Appendices
Discussion Points
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(1)
From prior data, it has been found that drivers under 40 years-old have considerable higher accident rates than drivers over 40 years-old. Driver A is 25 and Driver B is 45 years-old. Both drivers have not had an accident in the past 3 years. An insurance company sets premium rates based on age as well as individual driver’s record of accidents over a three-year period. If a Bayesian approach is used by the insurance company, would Drivers A and B pay the same premium? Discuss this from the point of view of the insurance company, and also from the point of view of the drivers.
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(2)
The choice of an IRT model depends on the purposes of an assessment. Surveys such as PISA and TIMSS mainly focus on the performance of a country. In contrast, state-wide testing programs are often interested in measuring individual students. Discuss the relative merits between Bayesian and non-Bayesian IRT models in relation to the purposes of an assessment.
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(3)
Discuss which ability estimates (WLE, EAP, plausible values) are most suitable for representing the abilities of individuals.
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(4)
Discuss why plausible values are useful in large scaled educational assessment studies.
Exercises
Q1. Indicate whether you agree or disagree with each of the following statements
Two basketball players both scored 5/10 in goal shooting. Player A comes from the local high school while Player B comes from the state basketball team. If we use a Bayesian approach to estimate the long-term goal-shooting rates, Player B will have a higher estimated rate | Agree/disagree |
In a Bayesian IRT model, the prior is a normal distribution with mean 0. If a student’s WLE ability estimate under JML (non-Bayesian) is 0.8, the student’s EAP ability estimate will be less than 0.8 | Agree/disagree |
EAP ability estimates are more accurate than WLE ability estimates in that the bias in the EAP estimates is smaller | Agree/disagree |
The variance of a posterior distribution for an individual student will be smaller if the test length is longer | Agree/disagree |
2000 plausible values for a student are generated. The mean of these plausible values can be used as the EAP estimate for the student | Agree/disagree |
When plausible values across all students are collected, we have the prior distribution | Agree/disagree |
The variable “test-delivery mode (computer versus paper)” should be regarded as a facet term and not as a regressor term | Agree/disagree |
The variable “grade (years in school)” should be regarded as a facet term and not as a regressor term | Agree/disagree |
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Wu, M., Tam, H.P., Jen, TH. (2016). Bayesian IRT Models (MML Estimation). In: Educational Measurement for Applied Researchers. Springer, Singapore. https://doi.org/10.1007/978-981-10-3302-5_14
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DOI: https://doi.org/10.1007/978-981-10-3302-5_14
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