Summary of Bayesian Estimation in the Rasch Model

Summary of Bayesian Estimation in the Rasch Model H. Swaminathan and J. Gifford Journal of Educational Statistics (1982)

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

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;

Estimating N ability parameters, assuming bj’s known where ri = # of items ith examinee answers correctly • Estimate by ML

Bayes set-up

Posterior calculation Need to  wrt s2 and m

Posterior (con’t) No known distribution…

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.,

Estimating N ability parameters, and n difficulty parameters • Same idea as before, except add hierarchical and prior structure for bj’s • Same structure as for ability parameters: • Can compute joint posterior

Specification of priors • Authors want prior to be proper and to have variance defined   > 4 • Recommend 5    15 • Set (?)

Simulation Studies 1&2 • 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 ө) • Calculated Bayes and ML estimators

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

Example: NAEP Math 8th grade • n=25, N = ? • l =10 •  = 5,8,15,25 • Conclusions • Estimators similar except at extremes of ability/difficulty • Bayes allows estimation of ability for perfect score

Summary of Bayesian Estimation in the Rasch Model