Maximum Clique Algorithm for Uniform Test Forms Assembly Takatoshi Ishii 1 , Pokpong Songmuang 2 , and Maomi Ueno 1 1 University of Electro-Communications, Tokyo, Japan {ishii,ueno}@ai.is.uec.ac.jp 2 Thammasat University, Pratumthani, Thailand Abstract. Educational assessments occasionally require “uniform test forms” for which each test form consists of a different set of items, but the forms meet equivalent test specifications (i.e., qualities indicated by test information functions based on item response theory). We propose two maximum clique algorithms (MCA) for uniform test forms assembly. The proposed methods can assemble uniform test forms with allowance of overlapping items among uniform test forms. First, we propose an ex- act method that maximizes the number of uniform test forms from an item pool. However, the exact method presents computational cost prob- lems. To relax those problems, we propose an approximate method that maximizes the number of uniform test forms asymptotically. Accord- ingly, the proposed methods can use the item pool more efficiently than traditional methods can. We demonstrate the efficiency of the proposed methods using simulated and actual data. Keywords: test assembly, uniform test forms, maximum clique prob- lem, item response theory. 1 Introduction Educational assessments occasionally require “uniform test forms” for which each form consists of a different set of items but which still must have equiv- alent specifications (e.g., equivalent amounts of test information based on item response theory, equivalent average test score, equivalent time limits). For ex- ample, uniform test forms are necessary when a testing organization administers a test in different time slots. To achieve this, uniform test forms are assembled in which all forms have equivalent qualities so that examinees who have taken different test forms can be evaluated objectively using the same scale. Recently, automatic assembly for test forms has become popular. Automatic assembly assembles test forms to satisfy given test constraints (e.g., number of test items, amount of test information, average test score) to provide equivalent qualities [16,22,9,3,1,2,14,4,24,7,23,8,21,20,6]. In these studies, a test assembly is formalized as a combinational optimization problem. For example, van der Linden [23] proposed the big-shadow-test method using linear programming (LP). This method sequentially assembles uniform K. Yacef et al. (Eds.): AIED 2013, LNAI 7926, pp. 451–462, 2013. c  Springer-Verlag Berlin Heidelberg 2013