## Test Equating under the Multiple-Choice Model

Jee-Seon Kim, University of Illinois

Bradley A. Hanson, ACT, Inc.

Paper presented at the Annual Meeting of the American Educational Research Association (New Orleans, April, 2000)

**Revised**: May 12, 2000

**Abstract**: This paper presents a characteristic curve procedure for computing transformations of the item response theory (IRT) ability scale assuming the multiple-choice model (Thissen & Steinberg, 1984). The multiple-choice model provides a realistic and informative approach to analyzing multiple-choice items in two important ways. First, the probability of guessing is a decreasing function of ability rather than a constant across different ability levels as in the three-parameter logistic model. Second, the model utilizes information from incorrect answers as well as from correct answers. The multiple-choice model includes many well-known IRT models as special cases, such as Bock's (1972) nominal response model. Formulas needed to implement a characteristic curve method for scale transformation are presented for the multiple-choice model. Two moment methods of estimating a scale transformation for the multiple-choice model (the mean/mean and mean/sigma methods) are also presented. The use of the characteristic curve method for the multiple-choice model is illustrated in an example equating ACT mathematics tests, and is compared to the results from the mean/mean and mean/sigma methods. In the process of deriving the scale transformation procedure for the multiple-choice model, corrections were made in some of the formulas presented by Baker (1993) for computing a scale transformation for the nominal response model.

**Programs**: The programs used in this paper to compute the equating results for the examples are available for downloading.

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