Rasch Model derived from Consistent Stochastic Guttman Ordering

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,

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.

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,

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 derived from Consistent Stochastic Guttman Ordering, E Roskam & P Jansen … Rasch Measurement Transactions, 1992, 6:3 p. 232

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