Functional Models for Test Items

Abstract

After our bank accounts and our taxes, it is hard to imagine data playing a more central role in our lives than the examinations, opinion surveys, attitude questionnaires, and psychological scales administered to ourselves, our children, and our students. These data may not on first impression appear to be functional, but we show that functional data analysis can reveal how both test takers and test items perform in test situations. To provide a concrete frame of reference, we look at the responses of 5000 examinees to 60 items in a test of mathematics achievement developed by the American College Testing Program. We apply functional principal components analysis to explore variation across test items, and we check the fairness of certain items by comparing male and female performance. Finally, we use a functional property of these data to develop a useful new way of describing the performance of individual examinees.