Roskam, E.E., & Jansen, P.G.W. (1984). A new derivation of the Rasch model. In E. Degreef & J. van Bruggenhaut (Eds.), Trends in mathematical psychology (pp. 293-307). Amsterdam: North-Holland.
Consistent stochastic ordering of items and subjects is analogous to the ordering derived from composite transitivity in Guttman's scalogram structure. Guttman ordering is equivalent to the deterministic condition that
iRj =: nSj & -nSi, _n
This means that if item i is harder than item j [iRj] for some subject n, a condition implied by the fact that [=:] subject n succeeds on item j [nSj] and [&] subject n does not succeed on item i [-nSi], then item order is i then j for any n [_ n], i.e., item ordering is independent of subject.
Differential item ordering iRj can only be observed for subject n who succeeds on one item and fails on the other, i.e., whose number of successes on the two items, Rn, is 1. Therefore, the equivalent stochastic ordering is
p(iRj) =: p(nSj & -nSi ! Rn =1), _n
This means that the probability that (item i is more difficult than item j) is the probability that subject n, who succeeds on only one of the items, succeeds on item j and does not succeed on item i.
Proceeding algebraically, and specifying local independence,
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Reparameterize p(nSi) as fi, p(nSj) as fj, where f is some continuously differentiable and nowhere equal to zero item-dependent function of z, a subject-dependent, but item-independent parameter.
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But, for subject-independent stochastic ordering, the probability that item i is more difficult than item j must be independent of the subject forming the basis of the comparison, i.e., of the value of zn for subject n. Thus,
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Thus the Rasch model follows from the requirement of stochastically consistent item orders, and so is the probabilistic counterpart of Guttman's scalogram rule. Since a scalogram is symmetrical in its treatment of subjects and items, the Rasch model is also obtained by considering stochastically consistent subject ordering.
Rasch Model from Consistent Stochastic Guttman Ordering, E Roskam & P Jansen … Rasch Measurement Transactions, 1992, 6:3 p. 232
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