On the sampling theory foundations of item response theory models (Q1174707)

If the test \(T\) is made up of \(J\) test questions and if \(x_ j\) is an indicator variable to denote right or wrong answers to item \(j\) of \(T\) (that is, \(x_ j=0\) or 1 if the item is answered incorrectly or correctly), then an examinee produces the response vector \(x=(x_ 1,\ldots,x_ J)\). There are \(2^ J\) possible values of \(x\). The probability that a randomly selected examinee will produce response vector \(x\) is equal to p(x). Section 2 of the paper discusses the history of studies of item response theory models and includes numerous and lengthy quotations from papers by different authors. Section 3 presents loglinear models for \(p(x)\). Section 4 is devoted to geometrical aspects of IRT-models, namely to the dimension of these models. In Section 5, the relationship between unconditional maximum likelihood, conditional, joint and marginal maximum likelihood is studied. Sections 6 and 7 deal with some related topics.