Estimating Ability and Item Parameters
In its simplest form, the parameter estimation problem is the following. We are given a matrix U ≡ ∥ Uia ∥ consisting of the responses (uia = 0 or 1) of each of N examinees to each of n items. We assume that these responses arise from a certain model such as Eq. (2-1) or (2-2). We need to infer the parameters of the model: ai, bi, ci (i = 1, 2, . . . , n) and θa (a = 1, 2,. . . . , N).
As noted in Section 4.10 and illustrated for one θ in Fig. 4.9.1, the maximum
likelihood estimates are the parameter values that maximize the likelihood L(Uθ;
a, b, c) given the observations U. Maximum likelihood estimates are usually
found from the roots of the likelihood equations (4-30), which set the derivatives
of the log likelihood equal to zero. The likelihood equations (4-30) are
For the three-parameter logistic model,
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Publication information: Book title: Applications of Item Response Theory to Practical Testing Problems. Contributors: Frederic M. Lord - Author. Publisher: Lawrence Erlbaum Associates. Place of publication: Hillsdale, NJ. Publication year: 1980. Page number: 179.
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