Item response function in antagonistic situations

The main characteristic of a binary test is the item response function (IRF) expressing the probability P (d, a) of an object under test (OUT), possessing ability a, to successfully overcome the test item (TI) of difficulty d. Each specific test requires its own definitions of TI difficulty and OUT ability and has its own P (d, a) describing the probability of “success” mentioned above. This is demonstrated on the basis of several examples taken from different areas of statistical engineering. A common feature is that they all relate to “antagonistic” situations, in which the “success” of one side may formally be considered as a “loss” to the opposite side. For such situations ability and difficulty are two interchangeable sides of the same coin and the corresponding IRFs are complementary, that is, P (d, a) = 1 − P(a, d), with all consequences and restrictions imposed by this property. A study shows that the family of feasible IRFs is limited and has a number of interesting properties, which are discussed in the article. The analysis provided should facilitate avoiding errors in decisions about an IRF adequately describing the studied test.