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 b j ’s known where r i = # of items i th examinee answers correctly Estimate by ML

Bayes set-up

Posterior calculation Need to  wrt  2 and 

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 b j ’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 8 th grade n=25, N = ? = 10 = 5,8,15,25 Conclusions Estimators similar except at extremes of ability/difficulty Bayes allows estimation of ability for perfect score