Unbiased Estimators of Ability Parameters, of Their Variance, and of Their Parallel-Forms Reliability

Abstract
This paper is primarily concerned with determining the statistical bias in the maximum likelihood estimate theta-carat of the examinee ability parameter Theta in item response theory (IRT) (Lord, 1980); also of certain functions of such parameters. We will deal only with unidimensional tests composed of dichotomously scored items. We assume the item response function is three-parameter logistic. Available results for the sampling variance of theta- carat are currently limited to the case where the item parameters are known; the present derivations are limited to this case also. This limitation is tolerable in situations where the item parameters are predetermined, as in item banking and tailored testing. In the absence of a prior distribution for theta, it is well known that examinees with perfect scores have Theta-carat = Infinity; also that examinees who perform near or below the chance level on multiple-choice items may be given large negative values of theta-carat. This (correctly) suggests that theta-carat is positively biased for high-ability examinees and negatively biased for low-ability examinees. Will a correction of theta-carat for bias be helpful in such cases? The methods used to derive formulas for correction for bias are presented here.

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