ACER ConQuest (Adams, Wu, and Wilson, 2012) has been popularly used for analysing testing and assessment data. Two of the most common estimation methods for Rasch measurement models (Rasch, 1960/1980) are available in this software, marginal maximum likelihood estimation (MML) and joint maximum likelihood estimation (JML).

Under the JML method, as developed by Birnbaum (1968), and Wright and Pachapakesan (1969), all item parameters and all person parameters are regarded as fixed unknowns to be estimated. Therefore, the parameters involved in the estimation procedure of this method are all of the case parameters and all of the item parameters.

The MML method was developed by Bock and Lieberman (1970), and Bock and Aitken (1981). Under this method, item parameters are considered as “structural”, while ability parameters are “incidental”. It is assumed that individual’s positions on the latent variable are sampled from distributions of possible values. As a result, in its estimation procedure, the MML includes the item parameters and the population parameters but not case parameters. Although the distribution can be of any type, with a limit on the number of parameters, normal densities are most frequently used (see Mislevy, 1984). Additionally, it can be applied with a discrete distribution where a fixed set of grid points is assumed and a weight is estimated at each grid point (Adams and Wilson, 1996).

From a theoretical perspective however JML has some shortcomings. Andersen (1973) showed that JML estimates of the item parameters for Rasch models are not consistent if the number of items is fixed and the size N ->infinity. To deal with the bias in JML, Wright and Douglas (1978) proposed a correction of, (K-1)/K, where K is the number of items. They argued that this correction removed most of the bias for K>20. For tests of fewer than 15 items, van den Wollenberg, Wierda, and Jansen (1998) suggested that this bias correction is inappropriate since the bias was dependent not only on the number of items, but also on the skewness of the item difficulty distribution.

A second potential shortcoming of JML is that in many of its potential applications the goal is to make inferences concerning populations. If JML is used for estimating the measurement model then a two-step analysis is required. First the case parameters are all estimated with JML and then the population parameters are estimated from individual case estimates. A number of researchers have illustrated that the use of case parameter estimates as though they were true values in a two-step analysis can lead to quite misleading outcomes.

The MML method can overcome these disadvantages of the JML method. Particularly, if both the item response models and the assumed population distributions are correct the MML item parameter estimates are consistent (Bock and Aitkin, 1981). Additionally, population parameters are estimated directly from the observed responses to avoid the problems associated with estimating population characteristics using fallible case parameter estimates in a two-step process.

However, the application of MML approach is often restricted to the assumption of a distribution for the population when this may not be a desirable assumption. Some empirical studies demonstrate that MML estimators loose accuracy and efficiency when the prior assumption of the latent distribution is violated (Yen, 1987; Drasgow, 1989; Seong, 1990; Harwell and Janosky, 1991; Stone, 1992).

This study is concerned with item parameter recovery for the dichotomous Rasch model. Our primary focus is on comparing JML and MML when the assumptions of MML are violated, that is the abilities are not sampled from the distribution that is assumed in the estimation.

Adams, R. J., and Wilson, M. R. (1996). A random coefficients multinomial logit: A generalized approach to fitting Rasch models. In Objective Measurement III: Theory into Practice, G. Engelhard and M. Wilson (eds.), pp 143–166. Norwood, New Jersey: Ablex. Adams, R.J., Wu, M.L., & Wilson, M.R. (2012) ACER ConQuest 3.0. [computer program]. Melbourne: ACER. Andersen, E. B. (1973). Conditional inference in multiple choice questionnaires. British Journal of Mathematical and Statistical Psychology, 26, 31–44. Birnbaum, A. (1968). Some Latent trait models and their use in inferring an examinee’s ability. In F. M. Lord and M. R. Novick (Eds.) Statistical Theories of Mental Scores (pp. 397–472). Reading, MA: Addition-Wesley. Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: An application of the EM algorithm. Psychometrika, 46, 443–459. Bock, R. D., & Lieberman, M. (1970). Fitting a response model for dichotomously scored items. Psychometrika, 35, 179–187. Drasgow, F. (1989). An evaluation of marginal maximum likelihood estimation for the two-parameter logistic model. Applied Psychological Measurement, 13, 77–90. Harwell, M. R., & Janosky, J. E. (1991). An empirical study of the effects of small datasets and varying prior variances on item parameter estimation in BILOG. Applied Psychological Measurement, 15, 279–291. Mislevy, R. J. (1984). Estimating latent distributions. Psychometrika, 49, 359–381. Rasch, G. (1980). Probabilistic models for some intelligence and attainment tests. (expanded ed). Chicago. The University of Chicago Press. (Original work published 1960). Seong, T. (1990). Sensitivity of Marginal Maximum Likelihood Estimation of Item and Ability Parameters to the Characteristics of the Prior Ability Distributions. Applied Psychological Measurement , 14, 299–311. Stone, C. A. (1992). Recovery of marginal maximum likelihood estimates in the two-parameter logistic response model: An evaluation of MULTILOG. Applied Psychological Measurement, 16, 1–16. van den Wollenberg, A. L., Wierda, F. W., & Jansen, P. G. W. (1998). Consistency of Rasch Model Parameter Estimation: A Simulation Study. Applied Psychological Measurement, 12, 307–313. Wright, B. D., & Panchapakesan, N. A. (1969). A procedure for sample-free item analysis. Educational and Psychological Measurement, 29, 23–48. Wright, B. D., & Douglas, G. A. (1978). Better procedures for sample-free item analysis. MESA Memorandum No. 20. Chicago, IL: University of Chicago, Department of Education. Yen, W. M. (1987). A comparison of the efficiency and accuracy of BILOG and LOGIST. Psychometrika, 52, 275–291.