Abstract: A new parameter-estimation method, the minimum c2/EM algorithm for unknown parameters of the 2PLM, was proposed. The new estimation paradigm was based on careful considerations of the differences between item response theory (IRT) and classical test theory (CTT). Specifically, it is derived from a modified version of the minimum c2 algorithm originally proposed by Berkson (1955).
The starting point of the minimum c2 algorithm is the Pearson c2. Given ability score level, examinees can be classified into categories; the congruence of the sample and the expected distribution can be measured by c2 statistic. The subsequent estimation procedure is to seek appropriate item parameters to minimize c2. Because true ability scores are unobservable, most of the time, examinees are classified according to observed scores. We believe this practice is based on the point of view of CTT, which assumes that the examinees with the same observed scores have the same ability scores.
As we all know, the posterior distribution of ability parameter is affected by item parameters. Thus, the new method takes the posterior distribution of ability parameter into account and introduces artificial data in the EM algorithm for estimating the unknown parameters in IRT models. The new method redefines , (the observed proportion of correct responses and incorrect responses) of Berkson’s minimum c2 algorithm, and replaces it with artificial datum and respectively. The statistical reasoning and operations behind this method can be intuitively explained as the following:
In the minimum c2 algorithm, the observed proportion of responses is fixed and the theoretical distribution is changed with the new estimated value of the unknown parameters. In other words, the algorithm draws the theoretical distribution closer to the observed distribution and, as a consequence, the estimating speed slows down. In order to accelerate estimation, the new method connects artificial data to the item parameters through the EM algorithm so that the theoretical and the observed distribution represented by the artificial data can change simultaneously. Because item parameters are the so called “structural” parameters, whereas ability parameters of examinees are “incidental” parameters. In order to remove the effect of item parameters’ estimation to ability parameters, the new method also factored out ability parameters during estimation.
With the new estimation procedure, examinees can be classified just according to the posterior distribution of the ability parameter. After arriving at the final item parameters, examinee ability scores can be estimated using Bayesian EAP method. Through these procedures, the new method overcomes the restriction that the ability parameters must be known before estimation and expands the application range.
The results of a Monte Carlo simulation test demonstrated that the new method was not restricted by either the number of items or the number of examinees. It is also more effective and more robust than BILOG in terms of ability parameters recovery. When the number of examinees exceeded 2000, the new method was also much more effective than BILOG for item parameter recovery. The best advantage of the new method is that the ABS (the absolute value of the difference between the true and the estimated parameters) of item parameters were smaller than 0.08 when the number of examinee was 2000; the value decreased further with an increase in the number of examinees

Key words: IRT, parameters estimate, EM algorithm