Abstract
In this study, after defining the equating coefficients of the continuous response model (CRM, Samejima, 1973, 1974), we proposed three procedures of linking tests under the CRM in the context of both common examinees and items designs. One was for the common examinees design, and the other two were for the common items design. As for the common examinees design, we proposed a method for estimating the equating coefficients using the marginal maximum likelihood estimation with the EM algorithm, where each common examinee’s latent trait θ, which becomes a nuisance parameter, was integrated over the posterior distribution of θ. Under the common items design, we applied the weighted least squares method (Haebara, 1980) and the test characteristic curve method (Stocking & Lord, 1983) to the CRM after introducing the item response function of the CRM. We also confirmed the accuracy of the three proposed methods using simulation data and actual data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baker, S. G. (1992). A simple method for computing the observed information matrix when using the EM algorithm with categorical data. Journal of Computational and Graphical Statistics, 1, 63–76.
Dempster, A. P., Laird, N. M. & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, 1–38.
Haebara, T. (1980). Equating logistic ability scales by a weighted least squares method. Japanese Psychological Research, 23, 144–149.
Hambleton, R. K. & Swaminathan, H. (1985). Rem Response Theory. Boston: Kluwer-Nijhoff.
Lord, F. M. (1980). Applications of Rem Response Theory to Practical Testing Problems. Hillsdale, New Jersey: Lawrence Erlbaum Associates.
Jamshidian, M. & Jennrich, R. I. (2000). Standard errors for EM estimation. Journal of the Royal Statistical Society, Series B, 62, 257–270.
Louis, T. A. (1982). Finding observed information using the EM algorithm. Journal of the Royal Statistical Society, Series B, 44, 98–130.
Marco, G. L. (1977). Item characteristic curve solutions to three intractable testing problems. Journal of Educational Measurement, 14, 139–160.
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.
Mayekawa, S. & Suzuki, N. (1988). Simultaneous equating of several sets of separately calibrated item parameters. Research Bulletin of the National Center for University Entrance Examinations, 17, 249–271. Tokyo, (in Japanese)
Mislevy, R. J. (1984). Estimating latent distribution. Psychometrika, 49, 359–381.
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.
Muraki, E. (1992). RESGEN: Rem response generator. Princeton, NJ: Educational Testing Service.
Noguchi, H. (1983). An equating method for latent trait scales using subject’s estimated scale values. Japanese Journal of Educational Psychology, 31, 233–238. (in Japanese)
Noguchi, H. (1986). An equating method for latent trait scales using common subjects’ item response patterns. Japanese Journal of Educational Psychology, 34, 315–323. (in Japanese with English abstract)
Noguchi, H. (1990). Marginal maximum likelihood estimation of the equating coeficients for two IRT scales using common subjects’ design. Bulletin of the Faculty of Education, Nagoya University (Educational Psychology), 37, 191–198. (in Japanese)
Oakes, D. (1999). Direct calculation of the information matrix via the EM algorithm. Journal of the Royal Statistical Society, Series B, 61, 479–482.
Ogasawara, H. (2001a). Marginal maximum likelihood estimation of item response theory (IRT) equating coefficients for the common-examinee design. Japanese Psychological Research, 43, 72–82.
Ogasawara, H. (2001b). Standard errors of item response theory equating/linking by response function methods. Applied Psychological Measurement, 25, 53–67.
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, No. 17.
Samejima, F. (1973). Homogeneous case of the continuous response model. Psychometrika, 38, 203–219.
Samejima, F. (1974). Normal ogive model on the continuous response level in the multidimensional latent space. Psychometrika, 39, 111–121.
Stocking, M. L. & Lord, F. M. (1983). Developing a common metric in item response theory. Applied Psychological Measurement, 7, 201–210.
Tanner, M. A. (1993). Tools for statistical inference: Methods for the exploration of posterior distributions and likelihood functions. New York: Springer-Verlag.
Toyoda, H. (1986). An equating method of two latent ability scales by using subjects’ estimated scale values and test informations. Japanese Journal of Educational Psychology, 34, 163–167. (in Japanese with English abstract)
Wang, T. & Zeng, L. (1998). Item parameter estimation for a continuous response model using an EM algorithm. Applied Psychological Measurement, 22, 333–344.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Shojima, K. Linking Tests Under the Continuous Response Model. Behaviormetrika 30, 155–171 (2003). https://doi.org/10.2333/bhmk.30.155
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.2333/bhmk.30.155