Abstract
Because the actual values of the parameters of the items in a test are unknown, one of the tasks performed when a test is analyzed under item response theory is to estimate these parameters. The obtained item parameter estimates then provide information as to the technical properties of the test items. To keep matters simple in the following presentation, the parameters of a single item will be estimated under the assumption that the examinees ability scores are known. In reality, these scores are not known, but it is easier to explain how item parameter estimation is accomplished if this assumption is made.
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The likelihood, L, that is the probability of observing a set of r g values from f g given item parameters is maximized as if it is a function of the item parameters: \(L = _{f_{g}}C_{r_{g}}\prod _{g=1}^{G}P(\theta _{g})^{r_{g}}Q(\theta _{g})^{(f_{g}-r_{g})}\), where C designates combination. Because item parameters that maximize L also maximize the logarithm of L, logL is used to find the estimates of item parameters. To find the values of item parameter estimates that maximize logL, the Newton-Raphson method can be employed. The partial derivatives as well as the second partial derivatives of logL with respect to every item parameter for an item are required in the Newton-Raphson method. Using a set of initial values of the item parameters the iteration in the Newton-Raphson method will be performed until a stable set of parameter estimates are obtained.
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Baker, F.B., Kim, SH. (2017). Estimating Item Parameters. In: The Basics of Item Response Theory Using R. Statistics for Social and Behavioral Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-54205-8_3
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DOI: https://doi.org/10.1007/978-3-319-54205-8_3
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-54205-8
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