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# Applications of Item Response Theory to Practical Testing Problems

By: Frederic M. Lord | Book details

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Page 179

12
Estimating Ability and Item Parameters

12.1. MAXIMUM LIKELIHOOD

In its simplest form, the parameter estimation problem is the following. We are given a matrix U Uiaconsisting 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

(12-1a) (12-1b)

For the three-parameter logistic model,

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