A note on the equivalence of the graded response model and the sequential model

This paper concerns items that consist of several item steps to be responded to sequentially. The item scoreX is defined as the number of correct responses until the first failure. Samejima's graded response model states that each steph=1,...,m is characterized by a parameterb_h, and, for a subject with ability, Pr(Xh; )=F(–b_h). Tutz's general sequential model associates with each step a parameterdh, and it states that Pr(Xh;)=_r^=1hG(–d_r). Tutz's (1991, 1997) conjectures that the models are equivalent if and only ifF(x)=G(x) is an extreme value distribution. This paper presents a proof for this conjecture.