In this article the authors focus on the issue of the nonuniqueness of the maximum likelihood (ML) estimator of proficiency level in item response theory (with special attention to logistic models). The usual maximum a posteriori (MAP) method offers a good alternative within that framework; however, this article highlights some drawbacks of its use. The authors then propose an iteratively based MAP estimator (IMAP), which can be useful in detecting multiple local likelihood maxima. The efficiency of the IMAP estimator is studied and is compared to the ML and MAP methods by means of a simulation study.
Birnbaum, A. ( 1968). Some latent trait models and their use in inferring an examinee’s ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores. Reading, MA : Addison-Wesley.
2.
Birnbaum, A. ( 1969). Statistical theory for logistic mental test models with a prior distribution of ability. Journal of Mathematical Psychology , 6, 258-276.
3.
Bock, R.D., & Mislevy, R.J. ( 1982). Adaptive EAP estimation of ability in a micro computer environment. Applied Psychological Measurement, 6, 431-444.
4.
Chang, H.H., & Ying, Z. ( 2008). To weight or not to weight? Balancing influence of initial items in adaptive testing. Psychometrika, 73, 441-450.
5.
Lord, F.M. ( 1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum.
6.
Magis, D. ( 2007). Influence, information and item response theory in discrete data analysis. Unpublished doctoral dissertation, Department of Mathematics, University of Liège, Belgium .
7.
R Development Core Team. (2007). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing .
8.
Raîche, G., & Blais, J.-G. ( 2002, April). Practical considerations about expected a posteriori estimation in adaptive testing: Adaptive a priori, adaptive correction for bias, and adaptive integration interval. Paper presented at the Biennial International Objective Measurement Workshop, New Orleans, LA.
9.
Rulison, K.L., & Loken, E. ( 2009). I’ve fallen and I can’t get up: Can high-ability students recover from early mistakes in CAT?Applied Psychological Measurement, 33, 83-101.
10.
Samejima, F. ( 1973). A comment on Birnbaum’s three-parameter logistic model in the latent trait theory. Psychometrika, 38, 221-223.
