## Estimation of item response models using the EM algorithm for finite mixtures

David Woodruff and Bradley A. Hanson

Paper presented at the Annual
Meeting of the Psychometric Society (Gatlinburg, Tennessee, June, 1997).

This paper is a revised version of ACT Research Report 96-6

**Abstract**: This paper presents a detailed description of maximum likelihood parameter estimation for item response models using the general EM algorithm. In this paper the models are specified using a univariate discrete latent ability variable. When the latent ability variable is discrete the distribution of the observed item responses is a finite mixture, and the EM algorithm for finite mixtures can be used. Maximum likelihood estimates of the item parameters and of the discrete probabilities of the latent ability distribution are given using the EM algorithm for finite mixtures. Results are presented in general for both dichotomous and polytomous item response models. The relation between the EM estimates and Bock-Aitken marginal maximum likelihood estimates is discussed.

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**Links**: I wrote a paper that describes using the EM algorithm to compute Bayes modal estimates of a discrete latent varible distribution, and a short note that describes using the EM algorithm to compute maximum likelihood estimates of a continuous parametric latent variable distribution.

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