Ability Distributions Pattern Probabilities and Quasidensities
ILLINOIS UNIV AT URBANA MODEL BASED MEASUREMENT LAB
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The quasidensity is a useful surrogate for the ability density in item response theory analyses of multiple choice tests. Like the ability density, it can be used to calculate the probability of sampling an examinee with a specified pattern of responses. Sometimes it is preferable to the density because it is continuous densities need not be continuous, it is unique two very different densities can give exactly the same pattern probabilities and all other expected values of random variables that are functions of item responses, it always exists a discontinuous ability distributions has a quasidensity, but it does not have a density. Some large sample results are proven for quasidensity estimation. It is shown that the maximum likelihood quasidensity estimate mle is strongly consistent. The asymptotic distribution of the mle is derived. Some new results on the relation between latent class and latent trait models are also presented. It is shown that every item response model with a smooth density and smooth item response functions is isomorphic to the latent class model obtained with the same item response functions and a discrete ability distribution. An upper bound for the number of points is derived.
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