A Bayesian analysis of a model for misreadings is undertaken. The model is essentially the Poisson model with multiplicative parameter structure first proposed and experimentally justified by Rasch. Conditional and unconditional posterior densities, predictive densities and reliabilities are presented. Some results for point and interval estimation and examination design are derived from the basic results.

Apart from the widely known logistic model for binary scored items, Rasch developed several other models for the analysis of achievement test data. The model considered in this paper is the so-called multiplicative Poisson model for misreadings. This model assumes that the number of reading errors that an individual makes when reading a text is approximately Poisson distributed with an intensity parameter λ, which depends on the ratio of two other parameters, one pertaining to the ability of the individual and one to the difficulty of the words in the text. In this paper a Bayesian procedure is developed for the simultaneous estimation of the ability and difficulty parameters. Similar procedures have been applied to the estimation of the parameters of the logistic model. According to several criteria, the Bayesian estimates are better than comparable maximum likelihood estimates. The method is illustrated by an empirical example. Some evidence on the potential usefulness of the method, based on Monte Carl...

A bayesian analysis of rasch's multiplicative poisson model for misreadings

A Bayesian Version of Rasch’s Multiplicative Poisson Model for the Number of Errors of an Achievement Test:

https://onlinelibrary.wiley.com/doi/pdf/10.1002/j.2333-8504.1969.tb00742.x

Probabilistic Models for Some Intelligence and Attainment Tests

This is a reprint of the orginal book released in 1968. Our primary goal in this book is to sharpen the skill, sophistication, and in- tuition of the reader in the interpretation of mental test data, and in the construction and use of mental tests both as instruments of psychological theory and as tools in the practical problems of selection, evaluation, and guidance. We seek to do this by exposing the reader to some psychologically meaningful statistical theories of mental test scores. Although this book is organized in terms of test-score theories and models, the practical applications and limitations of each model studied receive substantial emphasis, and these discussions are presented in as nontechnical a manner as we have found possible. Since this book catalogues a host of test theory models and formulas, it may serve as a reference handbook. Also, for a limited group of specialists, this book aims to provide a more rigorous foundation for further theoretical research than has heretofore been available.One aim of this book is to present statements of the assumptions, together with derivations of the implications, of a selected group of statistical models that the authors believe to be useful as guides in the practices of test construction and utilization. With few exceptions we have given a complete proof for each major result presented in the book. In many cases these proofs are simpler, more complete, and more illuminating than those originally offered. When we have omitted proofs or parts of proofs, we have generally provided a reference containing the omitted argument. We have left some proofs as exercises for the reader, but only when the general method of proof has already been demonstrated. At times we have proved only special cases of more generally stated theorems, when the general proof affords no additional insight into the problem and yet is substantially more complex mathematically.

A bayesian analysis of rasch's multiplicative poisson model for misreadings

Citations

A Bayesian Version of Rasch’s Multiplicative Poisson Model for the Number of Errors of an Achievement Test:

References

Probabilistic Models for Some Intelligence and Attainment Tests

Statistical Theories of Mental Test Scores

Related Papers (5)

A Bayesian Version of Rasch’s Multiplicative Poisson Model for the Number of Errors of an Achievement Test:

On the Bayesian nonparametric generalization of IRT-type models

Maximum Likelihood Estimation in Generalized Rasch Models

Bayesian inference for Poisson and multinomial log-linear models

Bayesian Analysis of Structural Equation Model with Fixed Covariates.