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True-Score Theory--The Four-Parameter Beta Model With Binomial Errors

Lord, Frederic M.
Publication Year:
Report Number:
ETS Research Bulletin
Document Type:
Page Count:
Subject/Key Words:
Office of Naval Research, Beta Binomial Test Model, Mathematical Models, True Scores


A promising model for the relation between true score, zeta, and observed score, x, asserts that the conditional distribution of observed score when true score is fixed is a binomial with parameters zeta and n, where n is the number of items in the test; also that the distribution of zeta in the group tested is a four-parameter beta distribution. It was found that this model successfully fitted most observed-score distributions, as judged by a chi-square test, except for those with more than 100,000 examinees. The model is also capable of predicting the frequencies in the scatterplot between two tests of the same trait, using only the information given by the two marginal frequency distributions. When such predictions were tried out for six different scatterplots, the fits seemed visually good. However, five of six chi-squares were significant at the 5 percent level. Examination of the discrepancies between observed scatterplots and predicted scatterplots showed a simple pattern. A modification of the model to deal with this is the subject of a separate report.

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