ON HOLLAND DUTCH IDENTITY CONJECTURE

The manifest probabilities of observed examinee response patterns resu lting from marginalization with respect to the latent ability distribu tion produce the marginal likelihood function in item response theory. Under the conditions that the posterior distribution of examinee abil ity given some test response pattern is normal and the item logit func tions are linear, Holland (1990a) gives a quadratic form for the log-m anifest probabilities by using the Dutch Identity. Further, Holland co njectures that this special quadratic form is a limiting one for all ' 'smooth'' unidimensional item response models as test length tends to infinity. The purpose of this paper is to give three counterexamples t o demonstrate that Holland's Dutch Identity conjecture does not hold i n general. The counterexamples suggest that only under strong assumpti ons can it be true that the limits of log-manifest probabilities are q uadratic. Three propositions giving sets of such strong conditions are given.