Approximation for Bayesian Ability Estimation

An approximation is proposed for the posterior mean and standard deviation of the ability parameter in an item response model. The procedure assumes that approximations to the posterior mean and covariance matrix of item parameters are available. It is based on the posterior mean of a Taylor series approximation to the posterior mean conditional on the item parameters. The method is illustrated for the two-parameter logistic model using data from an ACT math test with 39 items. A numerical comparison with the empirical Bayes method using n = 400 examinees shows that the point estimates are very similar but the standard deviations under empirical Bayes are about 2% smaller than those under Bayes. Moreover, when the sample size is decreased to n = 100, the standard deviation under Bayes is shown to increase by 14% in some cases.

[1]  Hsin Ying Lin,et al.  Bayesian estimation of item response curves , 1986 .

[2]  David Lindley,et al.  Bayes Empirical Bayes , 1981 .

[3]  Melvin R. Novick,et al.  Bayesian Full Rank Marginalization for Two-Way Contingency Tables , 1986 .

[4]  T. W. Anderson.  An Introduction to Multivariate Statistical Analysis , 1959 .

[5]  Robert K. Tsutakawa,et al.  Estimation of Two-Parameter Logistic Item Response Curves , 1984 .

[6]  Jeffrey E. Harris,et al.  Bayes Methods for Combining the Results of Cancer Studies in Humans and other Species , 1983 .

[7]  Frederic M. Lord,et al.  Unbiased estimators of ability parameters, of their variance, and of their parallel-forms reliability , 1983 .

[8]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[9]  F. Lord.  Statistical bias in maximum likelihood estimators of item parameters , 1983 .

[10]  D. Lindley,et al.  Approximate Bayesian methods , 1980 .

[11]  R. D. Bock,et al.  Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm , 1981 .

[12]  Steven E. Rigdon,et al.  Estimation in Latent Trait Models. , 1981 .

[13]  Michel Lejeune,et al.  A Simple Predictive Density Function , 1982 .

[14]  M. R. Novick,et al.  Statistical Theories of Mental Test Scores. , 1971 .

[15]  F. Lord.  Applications of Item Response Theory To Practical Testing Problems , 1980 .

[16]  S. Fienberg,et al.  Inference and Disputed Authorship: The Federalist , 1966 .

[17]  T. W. Anderson,et al.  An Introduction to Multivariate Statistical Analysis , 1959 .

[18]  R. Kass.  Bayes Methods for Combining the Results of Cancer Studies in Humans and Other Species: Comment , 1983 .