Estimation of Item Response Models Using the EM Algorithm for Finite Mixtures
A C T Researcli Report SeriesEstimation of Item Response Models Using the EM Algorithm for Finite MixturesDavid J. Woodruff Bradley A. Hanson For additional copies write:AC T Research Report Series P.O. Box 168Iowa City, Iowa 52243-0168© 1996 by ACT, Inc. All rights reserved Estimation for Item Response Models using the EM Algorithm for Finite MixturesDavid Woodruff Bradley A. Hanson AbstractThis 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 di- chotomous and polytomous item response models. The relation between the EM estimates and Bock-Aitken marginal maximum likelihood estimates is discussed. Estimation for Item Response Models using the EM Algorithm for Finite MixturesThe purpose of this paper is to present a fairly simple and unified treatment of how the general EM algorithm can be used to obtain maximum likelihood estimates (MLEs) of both the item parameters and the probability distribution of the latent ability variable for item response models. The approach taken in this paper is to assume the latent ability variable being measured by the items is discrete. When the latent ability variable is discrete the distribution of the observed data is a finite mixture (Titterington, Smith, and Makov, 1985). With a discrete latent ability variable the EM algorithm for finding maximum likelihood estimates for finite mixtures can be used (Dempster, Laird, and Rubin, 1977; Titterington, Smith, and Makov, 1985).This paper clarifies previously established results using a finite mixture approach. A complete, self-contained description of maximum likelihood parameter estimates of item response models for dichotomous and polytomous items using the EM algorithm for finite mixtures is presented. The use of the finite mixture model allows a variety of previously disparate results to be consolidated using a single relatively simple approach that allows a straight-forward presentation with pedagogic value.Versions of the results in this paper have been presented by others for a variety of specific item response models. Maximum likelihood estimates of item parameters using the EM algorithm have been presented for a variety of item response models for dichotomous items (Bock and Aitken, 1981; Thissen, 1982; Rigdon and Tsutakawa, 1983; Tsutakawa, 1984; Bartholomew, 1987; Harwell, Baker, and Zwarts, 1988; Baker, 1992) and polytomous items (Thissen and Steinberg, 1984; Bartholomew, 1987; Muraki, 1992; Wilson and Adams, 1993). The EM algorithm for finite mixtures has been applied in estimating parameters for the Rasch model by De Leeuw h Verhelst (1986) and Follmann (1988). The maximum likelihood estimates of the probabilities of the discrete latent ability distribution presented here were given by Bock and Aitken (1981), Mislevy (1984), and Titterington, Smith, and Makov (1985).The data to be modeled are the responses of i = 1,..., N examinees, randomly sampled from a population of examinees, to a fixed non-random set of j = 1,..., n items. The responses of the N examinees to the n items are contained in a n x TV matrix Y made up of n x 1 column vectors y i,... ,y*7..., yw that contain the responses of the ith randomly sampled examinee to the n fixed items. The matrix Y is given byY = [yi,... ,y*,... ,yiv] • (1)The jth element of y * (the response of the ith randomly sampled examinee to item j) is denoted . It is assumed that the set of responses to each item is finite. If the responses are dichotomous then the possible values of yij axe 0 and 1. If the responses are polytomous then the possible values of are taken to be the integers 0,1,..., Lj — 1 (item j has Lj response categories). In practical applications values of the polytomous items need not be integers or even ordered. Note that different items may have different numbers of response categories.Associated with item j is a set of Vj item parameters denoted by the i/j x 1 column vector, 6j. The parameters for all n items are represented by A , the collection of all 6j column vectors, that is A = [6i,..., 6j1..., £n]. When the number of item response categories is the same for every item (e.g., dichotomous items) then the number of parameters will typically be the same for every item so that v3 — v for all j.In addition to
