The Annals of Statistics

Nonlinear sequential designs for logistic item response theory models with applications to computerized adaptive tests

Hua-Hua Chang and Zhiliang Ying

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Computerized adaptive testing is becoming increasingly popular due to advancement of modern computer technology. It differs from the conventional standardized testing in that the selection of test items is tailored to individual examinee’s ability level. Arising from this selection strategy is a nonlinear sequential design problem. We study, in this paper, the sequential design problem in the context of the logistic item response theory models. We show that the adaptive design obtained by maximizing the item information leads to a consistent and asymptotically normal ability estimator in the case of the Rasch model. Modifications to the maximum information approach are proposed for the two- and three-parameter logistic models. Similar asymptotic properties are established for the modified designs and the resulting estimator. Examples are also given in the case of the two-parameter logistic model to show that without such modifications, the maximum likelihood estimator of the ability parameter may not be consistent.

Article information

Ann. Statist., Volume 37, Number 3 (2009), 1466-1488.

First available in Project Euclid: 10 April 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L05: Sequential design
Secondary: 62P15: Applications to psychology

Sequential design computerized adaptive testing item response theory Rasch model logistic models Fisher information maximum likelihood recursion martingale local convergence consistency asymptotic normality


Chang, Hua-Hua; Ying, Zhiliang. Nonlinear sequential designs for logistic item response theory models with applications to computerized adaptive tests. Ann. Statist. 37 (2009), no. 3, 1466--1488. doi:10.1214/08-AOS614.

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