Abstract
Item response theory (IRT) models with random person parameters have become a common choice among practitioners in the field of educational and psychological measurement. Though initially the choice for such models was motivated by an attempt to get rid of the statistical problems inherent in the incidental nature of the person parameters (Bock & Lieberman, 1970), the insight soon emerged that such models more adequately represent cases where the focus is not on the measurement of individual persons but on the estimation of characteristics of populations. Early examples of models with random person parameters in the literature are those proposed by Andersen and Madsen (1977) and Sanathanan and Blumenthal (1978), who were interested in estimates of the mean and variance in a population of persons, and by Mislevy (1991), who provided tools for inference from a response model with a regression structure on the person parameters introduced to account for sampling persons differing background variables.
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References
Albers, W., Does, R. J. M. M., Imbos, T. & Janssen, M. P. E. (1989). A stochastic growth model applied to repeated tests of academic knowledge. Psychometrika, 54, 451–466.
Albert, J. H. (1992). Bayesian estimation of normal-ogive item response curves using Gibbs sampling. Journal of Educational and Behavioral Statistics, 17, 251–269.
Andersen, E. B. & Madsen, M. (1977). Estimating the parameters of the latent population distribution. Psychometrika, 42, 357–374.
Béguin, A. A. & Glas, C. A. W. (2001). MCMC estimation and some model-fit analysis of multidimensional IRT models. Psychometrika, 66, 541–562.
Bejar, I. I. (1993). A generative approach to psychological and educational measurement. In N. Frederiksen, R. J. Mislevy & I. I. Bejar (Eds.), Test theory for a new generation of tests (pp. 323–357). Hillsdale, NJ: Lawrence Erlbaum Associates.
Berger, M. P. F. (1997). Optimal designs for latent variable models: A review. In J. Rost & R. Langeheine (Eds.), Applications of latent trait and latent class models in the social sciences (pp. 71–79). Münster, Germany: Waxmann.
Bock, R. D. & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: application of an EM-algorithm. Psychometrika, 46, 443–459.
Bock, R. D. & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.
Box, G. E. P. & Tiao, G. C. (1973). Bayesian inference in statistical analysis. Reading, MA: Addison-Wesley.
Bradlow, E. T., Wainer, H. & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153–168.
de Boeck, P. (2008). Random item IRT models. Psychometrika, 73, 533–559.
de Jong, M. G., Steenkamp, J. B. E. M. & Fox, J. P. (2007). Relaxing measurement invariance in cross-national consumer research using a hierarchical IRT model. Journal of Consumer Research, 34, 260–278.
Efron, B. (1977). Discussion on maximum likelihood from incomplete data via the EM algorithm (by A. P. Demster, N. M. Laird and D. B. Rubin). Journal of the Royal Statistical Society (Series B), 39, 1–38.
Fox, J. P. & Glas, C. A. W. (2001). Bayesian estimation of a multilevel IRT model using Gibbs sampling. Psychometrika, 66, 271–288.
Geerlings, H., van der Linden, W. J. & Glas, C. A. W. (2009). Modeling rule-based item generation. Manuscript submitted for publication.
Gelfand, A. E. & Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409.
Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. (1995). Bayesian data analysis. London: Chapman and Hall.
Glas, C. A. W. (1992). A Rasch model with a multivariate distribution of ability. In M. Wilson (Ed.), Objective measurement: Theory into practice (Vol. 1; pp. 236–258). Norwood, NJ: Ablex Publishing Corporation.
Glas, C. A. W. (1998). Detection of differential item functioning using Lagrange multiplier tests. Statistica Sinica, 8, 647–667.
Glas, C. A. W. (2000). Item calibration and parameter drift. In W. J. van der Linden & C. A. W. Glas (Eds.), Computerized adaptive testing: Theory and practice (pp. 183–199). Boston: Kluwer-Nijhof Publishing.
Glas, C. A. W. & van der Linden, W. J. (2001). Modeling variability in item parameters in item response models. (Research Rep. 01-11). Enschede, the Netherlands: University of Twente.
Glas, C. A. W. & van der Linden, W. J. (2003). Computerized adaptive testing with item cloning. Applied Psychological Measurement, 27, 247–261.
Glas, C. A. W., Wainer, H. & Bradlow, E. T. (2000). MML and EAP estimates for the testlet response model. In W. J. van der Linden & C. A. W. Glas (Eds.), Computerized adaptive testing: Theory and practice (pp. 271–287). Boston: Kluwer-Nijhof Publishing.
Janssen, R., Tuerlinckx, F., Meulders, M. & de Boeck, P. (2000). A hierarchical IRT model for criterion-referenced measurement. Journal of Educational and Behavioral Statistics, 25, 285–306.
Johnson, M. S. & Sinharay, S. (2005). Calibration of polytomous item families using Bayesian hierarchical modeling. Applied Psychological Measurement, 29, 369–400.
Kiefer, J. & Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Annals of Mathematical Statistics, 27, 887–906.
Lord, F. M. & Novick, M. R. (1968). Statistical theories of mental test scores. Reading, MA: Addison-Wesley.
Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B, 44, 226–233.
Millman, J. (1973). Passing score and test lengths for domain-referenced measures. Review of Educational Research, 43, 205–216.
Millman, J. & Westman, R. S. (1989). Computer-assisted writing of achievement test items: Toward a future technology. Journal of Educational Measurement, 26, 177–190.
Mislevy, R. J. (1986). Bayes modal estimation in item response models. Pychometrika, 51, 177–195.
Mislevy, R. J. (1991). Randomization-based inferences about latent variables from complex samples. Pychometrika, 56, 177–196.
Neyman, J. & Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1–32.
Patz, R. J. & Junker, B. W. (1999a). A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, 24, 146–178.
Patz, R. J. & Junker, B. W. (1999b). Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses. Journal of Educational and Behavioral Statistics, 24, 342–366.
Robert, C. P. & Casella, G. (1999). Monte Carlo statistical methods. New York: Springer-Verlag.
Roid, G. & Haladyna, T. (1982). A technology for test-item writing. New York: Academic Press.
Sanathanan, L. & Blumenthal, S. (1978). The logistic model and estimation of latent structure. Journal of the American Statistical Association, 73, 794–799.
Shi, J. Q. & Lee, S. Y. (1998). Bayesian sampling based approach for factor analysis models with continuous and polytomous data. British Journal of Mathematical and Statistical Psychology, 51, 233–252.
Sinharay, S., Johnson, M. S. & Williamson, D. M. (2003). Calibrating item families and summarizing the results using family expected response functions. Journal of Educational and Behavioral Statistics, 28, 295–313.
Stocking, M. L. (1989). Empirical estimation errors in item response theory as a function of test properties (Research Report 89-5). Princeton, NJ: Educational Testing Service.
van der Linden, W. J. (1994). Optimum design in item response theory: Test assembly and item calibration. In G. H. Fischer and D. Laming (Eds.), Contributions to mathematical psychology, psychometrics, and methodology (pp. 305–318). New York: Springer-Verlag.
Wainer, H., Bradlow, E. T. & Du, Z. (2000). Testlet response theory: An analog for the 3PL model useful in testlet-based adaptive testing. In W. J. van der Linden & C. A. W. Glas (Eds.), Computerized adaptive testing: Theory and practice (pp. 245–269). Boston: Kluwer-Nijhof Publishing.
Wingersky, M. S. & Lord, F. M. (1984). An investigation of methods for reducing sampling error in certain IRT procedures. Applied Psychological Measurement, 8, 347–364.
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Glas, C.A.W., van der Linden, W.J., Geerlings, H. (2009). Estimation of the Parameters in an Item-Cloning Model for Adaptive Testing. In: van der Linden, W., Glas, C. (eds) Elements of Adaptive Testing. Statistics for Social and Behavioral Sciences. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85461-8_15
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