Summary of Bayesian Estimation in the Rasch Model

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Summary of Bayesian Estimation in the Rasch Model

• H. Swaminathan and J. Gifford
• Journal of Educational Statistics (1982)

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Problem

• Estimate ability of each of N standardized test
  takers, based on a performance on a set of n test
  items

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Rasch model

• Model used in psychometrics relating performance
  on a series of test items to ability
• It is a logistic regression model with a single
  parameter describing each test item

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Estimating N ability parameters, assuming bjs
known

• where ri of items ith examinee answers
  correctly
• Estimate by ML

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Bayes set-up
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Posterior calculation
Need to ? wrt s2 and m
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Posterior (cont)
No known distribution
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Computation

• In 1983, this joint posterior was too complicated
  to compute and use
• Authors suggested using modes as estimators
• Find maxima using single-valued Newton-Raphson
  i.e.,

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Estimating N ability parameters, and n difficulty
parameters

• Same idea as before, except add hierarchical and
  prior structure for bjs
• Same structure as for ability parameters
• Can compute joint posterior

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Specification of priors

• Authors want prior to be proper and to have
  variance defined ? ? gt 4
• Recommend 5 ? ? ? 15
• Set (?)

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Simulation Studies 12

• Artificial data was generated according to
  logistic model
• Ability and difficulty parameters generated as
  uniform
• Conducted factorial simulation experiments (1) n
  x N (2) n x N x (?b and ??)
• Calculated Bayes and ML estimators

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Conclusions

• MSE smaller for Bayes estimators
• Varying ? has little effect except in smallest
  cases

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Example NAEP Math 8th grade

• n25, N ?
• l 10
• ? 5,8,15,25
• Conclusions
• Estimators similar except at extremes of
  ability/difficulty
• Bayes allows estimation of ability for perfect
  score