Hypergeometric family and item overlap rates in computerized adaptive testing

A computerized adaptive test (CAT) is usually administered to small groups of examinees at frequent time intervals. It is often the case that examinees who take the test earlier share information with examinees who will take the test later, thus increasing the risk that many items may become known. Item overlap rate for a group of examinees refers to the number of overlapping items encountered by these examinees divided by the test length. For a specific item pool, different item selection algorithms may yield different item overlap rates. An important issue in designing a good CAT item selection algorithm is to keep item overlap rate below a preset level. In doing so, it is important to investigate what the lowest rate could be for all possible item selection algorithms. In this paper we rigorously prove that if every item has an equal possibility to be selected from the pool in a fixed-length CAT, the number of overlapping items among any alpha randomly sampled examinees follows the hypergeometric distribution family for alpha greater than or equal to 1. Thus, the expected values of the number of overlapping items among any randomly sampled alpha examinees can be calculated precisely. These values may serve as benchmarks in controlling item overlap rates for fixed-length adaptive tests.