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Some patterns of observables—for example, a given person’s pattern of responses—may be so improbable under the model as to cast doubt on using the model for that individual, even if the model fits well in general. 1) where Ψ (·) ≡ exp (·)/[1 + exp (·)] is the cumulative logistic probability distribution, θ is a one-dimensional measure of proficiency, bj is a difficulty parameter for Item j, and xj is 1 if right and 0 if wrong. Under the usual IRT assumption of conditional independence, the probability of a vector of responses to n items is 22 Robert Mislevy and Chun-Wei Huang n P(x1 , .

It is compelling to examine the items that differentiate the groups. Is it that background knowledge differs among different groups of people? Are different people using different strategies to solve items? 3. It may be found that the RM fits well within the classes determined by partitioning persons and responses on the basis of w. In these circumstances one again obtains measurement models in the sense of probabilistic versions of conjoint measurement. 2 Mixtures of Rasch Models The not-uncommon finding of DIF among manifest groups raises the possibility that this phenomenon may be occurring even when the analyst does not happen to know persons’ values on the appropriate grouping variable.

7) For s = 1, we assume the RM, so the responses on the first subtest are used as a base line for θ. In practice, a response pattern is too short to test whether more than two ability parameters are involved. Therefore, the response pattern is usually partitioned into two parts, that is, S = 2. Further, assume that the item parameters are known. Then the log-likelihood is given by k xvi log Pi (θv ) + (1 − xvi ) log(1 − Pi (θv )). 8) i=1 Taking the first-order derivative of the likelihood of the response pattern with respect to θs results in h= ∂ log L = ∂θs [xvi − Pi (θ)].

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