## Bayes modal estimates of a discrete latent variable
distribution in item response models using the EM algorithm

Bradley A. Hanson

Paper presented at the Annual Meeting of the American Educational Research
Association (San Diego, April, 1998)

**Abstract**: There are typically two sets of parameters used in models of the observed distribution of item responses in a population: 1) item parameters, and 2) parameters of the distribution of the latent variable measured by the items. Several authors have presented Bayesian estimates of the item parameters. This paper presents Bayes modal estimates of a discrete latent variable distribution for item response models. The EM algorithm for computing maximum likelihood estimates of a discrete latent variable distribution is first reviewed, followed by a presentation of the EM algorithm for computing Bayes modal estimates using a Dirichlet prior for the discrete probabilities. Examples are presented comparing the maximum likelihood and Bayes modal estimates. A small simulation study is performed to examinee the performance of the Bayes modal estimates of a discrete latent variable distribution as applied to estimating average domain scores. The results of the simulation illustrate that whether Bayes modal estimates will have less error than maximum likelihood estimates depends on the tradeoff between the higher bias and lower variance of the Bayes modal estimates relative to the higher variance and lower bias of the maximum likelihood estimates. Bayes modal estimates are more likely to perform well relative to maximum likelihood estimates when sample sizes are small and the variance of the maximum likelihood estimates is large.

Download paper in PDF format. Version 3.0 or later of Adobe Acrobat Reader (which is available for free) is needed to view this paper.

**Links**: A paper I wrote with David Woodruff describes using the EM algorithm to estimate item parameters as well as a discrete distribution of the latent variable in item response models. I wrote a short note that describes using the EM algorithm to compute maximum likelihood estimates of a continuous parametric latent variable distribution.

Papers by Brad Hanson
Brad Hanson's Home Page