## Parameter Estimation via MMLE and an EM Algorithm

ISince it was formulated in 1968 by Dr. Allan Birnbaum, the primary approach to item parameter estimation has been the joint maximum likelihood estimation (JMLE) paradigm. A distinguishing characteristic of this paradigm is that examinee abilities are unknown and, hence, must be estimated along with the item parameters. As discussed in Chapter 4, the item parameters are the "structural" parameters, which are fixed in number by the size of the test. The ability parameters of the examinees are the "incidental" parameters, the number of which depends on the sample size. From a theoretical point of view, this paradigm has an inherent problem first recognized in another context by Neyman and Scott (1948) (see also Little & Rubin, 1983). Neyman and Scott showed that, when structural parameters are estimated simultaneously with the incidental parameters, the maximum likelihood estimates of the former need not be consistent as sample size increases. If sufficient statistics are available for the incidental parameters, the conditional maximum likelihood estimation procedure (Andersen, 1972) can be established for the consistent estimation of the structural parameters. As explained in Chapter 5, such conditional estimation can be established only for the one-parameter (Rasch) model. As a result, an estimation procedure for two-and three-parameter IRT models that avoids the problem of inconsistent estimates of structural parameters has considerable value. The basic paper in this regard was due to Bock and Lieberman (1970), who developed a marginal maximum likelihood procedure for estimating item parameters. Unfortunately, the Bock and Lieberman approach posed a formidable computational task and was practical for only very short tests. A subsequent reformulation of this marginal maximum likelihood estimation (MMLE) approach by Bock and Aitkin (1981) has resulted in a procedure that is both theoretically acceptable and computationally feasible. Their reformulation, under certain conditions, is an instance of an expectation-maximization (EM) algorithm. As a result, MMLE/EM will be used to identify the Bock and Atkin

procedure for estimating item parameters (see Section 6.4.1 for a definition of EM).