Multidimensional test assembly based on Lagrangian relaxation techniques

In this paper, a mathematical programming approach is presented for the assembly of ability tests measuring multiple traits. The values of the variance functions of the estimators of the traits are minimized, while test specifications are met. The approach is based on Lagrangian relaxation techniques and provides good results for the two dimensional case in a small amount of time. Empirical examples of a test assembly problem from a two dimensional mathematics item pool illustrate the method. In the area of ability measurement, Item Response Theory (IRT) is usually used as a psychometric theory underlying the test assembly process. In this process, three steps can be distinguished. First, an IRT model has to be chosen and the items in the item bank have to be calibrated. From this item bank, many different tests can be assembled. Therefore, the second step consists of specifying the properties of the desired test. One could specify, for example, the test length, the desired amount of information, or the administration time of the test. The third step of the test assembly process is to formulate a model that selects items from the item bank so that test specifications are met. A mathematical programming approach is often used for this step.