The accuracy of the Gibbs sampling Markov chain monte carlo procedure was examined for estimating item and person (.) parameters in the one-parameter logistic model. Four datasets were analyzed using the Gibbs sampling method, conditional maximum likelihood, marginal maximum likelihood, and joint maximum likelihood. Maximum likelihood and expected a posteriori. estimation methods were used with marginal maximum likelihood estimation of item parameters. Item parameter estimates from the four methods were almost identical;. estimates from Gibbs sampling were similar to those obtained from the expected a posteriori method.
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References
1.
Albert, J. H. (1992). Bayesian estimation of normal ogive item response curves using Gibbs sampling. Journal of Educational Statistics, 17, 251–269.
2.
Andersen, E. B. (1970). Asymptotic properties of conditional maximum likelihood estimators. Journal of the Royal Statistics Society, Series B, 32, 283–301.
3.
Andersen, E. B. (1972). The numerical solution of a set of conditional estimation equations. Journal of the Royal Statistical Society, Series B, 34, 42–54.
4.
Andersen, E. B. (1980). Discrete statistical models with social science applications. Amsterdam: North-Holland.
5.
Baker, F. B. (1998). An investigation of the item parameter recovery characteristics of a Gibbs sampling approach. Applied Psychological Measurement, 22, 153–169.
6.
Bernardo, J. M. , & Smith, A. F. M. (1994). Bayesian theory. Chichester, England: Wiley.
7.
Best, N. G. , Cowles, M. K., & Vines, S. K. (1997). CODA: Convergence diagnosis and output analysis software for Gibbs sampling output (Version 0.4) [Computer program]. Cambridge, UK: University of Cambridge, Institute of Public Health, Medical Research Council Biostatistics Unit.
8.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F. M. Lord & M. R. Novick, Statistical theories of mental test scores (pp. 395–479). Reading MA: Addison-Wesley.
9.
Bock, R. D. , & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Applications of anEMalgorithm. Psychometrika, 46, 443–459; 47–47, 369–369 (Errata).
10.
Bock, R. D. , & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.
11.
Carlin, B. P. , & Louis, T. A. (1996). Bayes and empirical Bayes methods for data analysis. London: Chapman and Hall.
12.
Cowles, M. K. , & Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: A comparative review. Journal of the American Statistical Association, 91, 883–904.
13.
Fischer, G. H. , & Molenaar, I. W. (Eds.). (1995). Rasch models: Foundations, recent developments, and applications. New York: Springer-Verlag.
14.
Gelfand, A. E. , & Smith, A. F. M. (1990). Samplingbased approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409.
15.
Gelman, A. (1996). Inference and monitoring convergence. In W. R. Gilks, S. Richardson, & D. J. Spiegelhalter (Eds.), Markov chain Monte Carlo in practice (pp. 131–143). London: Chapman and Hall.
16.
Gelman, A. , Carlin, J. B., Stern, H. S., & Rubin, D. B. (1995). Bayesian data analysis. London: Chapman and Hall.
17.
Gelman, A. ,& Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457–511.
18.
Geman, S. , & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.
19.
Ghosh, M. , Ghosh, A., Chen, M.-H., & Agresti, A. (1999). Bayesian estimation for item response models (Tech. Rep.). Gainsville FL: University of Florida, Department of Statistics.
20.
Gilks, W. R. (1996). Full conditional distribution. In W. R. Gilks, S. Richardson, & D. J. Spiegelhalter (Eds.), Markov chain Monte Carlo in practice (pp. 75–88). London: Chapman and Hall.
21.
Gilks, W. R. , Richardson, S., & Spiegelhalter, D. J. (Eds.). (1996). Markov chain Monte Carlo in practice. London: Chapman and Hall.
22.
Gilks, W. R. , & Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41, 337–348.
23.
Hasting, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.
24.
Hoijtink, H. , & Boomsma, A. (1995). On person parameter estimation in the dichotomous Rasch model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 53–68). New York: Springer-Verlag.
25.
Hoijtink, H. , & Boomsma, A. (1996). Statistical inference based on latent ability estimates. Psychometrika, 61, 313–330.
26.
Hulin, C. L. , Lissak, R. I., & Drasgow, F. (1982). Recovery of two- and three-parameter logistic item characteristic curves: A Monte Carlo study. Applied Psychological Measurement, 6, 249–260.
27.
Johnson, N. L. , Kotz, S., & Balakrishnan, N. (1994). Continuous univariate distributions (2nd ed., Vol. 1). New York: Wiley.
28.
Kim, S.-H. (1997). Software review: BILOG 3 for Windows: Item analysis and test scoring with binary logistic models. Applied Psychological Measurement, 21, 371–376.
29.
Kim, S.-H. , Cohen, A. S., Baker, F. B., Subkoviak, M. J., & Leonard, T. (1994). An investigation of hierarchical Bayes procedures in item response theory. Psychometrika, 59, 405–421.
30.
Lauritzen, S. L. , Dawid, A. P., Larsen, B. N., & Leimer, H.-G. (1990). Independence properties of directed Markov fields. Networks, 20, 491–505.
31.
Lord, F. M. (1980). Applications of item response theory to practical testing problems. Hillsdale NJ: Erlbaum.
32.
Lord, F. M. (1986). Maximum likelihood and Bayesian parameter estimation in item response theory. Journal of Educational Measurement, 23, 157–162.
Metropolis, N. , Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21, 1087–1092.
35.
Metropolis, N. , & Ulam, S. (1949). The Monte Carlo method. Journal of the American Statistical Association, 44, 335–341.
36.
Mislevy, R. J. (1986). Bayes modal estimation in item response models. Psychometrika, 51, 177–195.
37.
Mislevy, R. J. , & Bock, R. D. (1990). BILOG 3: Item analysis and test scoring with binary logistic models [Computer program]. Mooresville IN: Scientific Software.
38.
Molenaar, I. W. (1990). PML user’s manual, PC version [Computer program]. Groningen, The Netherlands: ProGAMMA.
39.
Molenaar, I. W. (1995). Estimation of item parameters. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 39–51). New York: Springer-Verlag.
40.
Patz, R. J. , & Junker, B. W. (1999). A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, 24, 146–178.
41.
Raftery, A. E. (1996). Hypothesis testing and model selection. In W. R. Gilks, S. Richardson, & D. J. Spiegelhalter (Eds.), Markov chain Monte Carlo in practice (pp. 163–187). London: Chapman and Hall.
42.
Rasch, G. (1960/1980). Probabilistic models for some intelligence and attainment tests. (Copenhagen: Danish Institute for Educational Research). Expanded edition (1980), with foreword and afterword by B. D. Wright. Chicago: University of Chicago Press.
43.
Spiegelhalter, D. J. , Best, N. G., Gilks, W. R., & Inskip, H. (1996). Hepatitis B: A case study in MCMC methods. In W. R. Gilks, S. Richardson, & D. J. Spiegelhalter (Eds.), Markov chain Monte Carlo in practice (pp. 21–43). London: Chapman and Hall.
44.
Spiegelhalter, D. J. , Thomas, A., Best, N. G., & Gilks, W. R. (1996). BUGS 0.5 examples (Vol. 1, Version i). Cambridge, UK: University of Cambridge, Institute of Public Health, Medical Research Council Biostatistics Unit.
45.
Spiegelhalter, D. J. , Thomas, A., Best, N. G., & Gilks, W. R. (1997). BUGS: Bayesian inference using Gibbs sampling (Version 0.6) [Computer program]. Cambridge, UK: University of Cambridge, Institute of Public Health, Medical Research Council Biostatistics Unit.
46.
Swaminathan, H. , & Gifford, J. A. (1982). Bayesian estimation in the Rasch model. Journal of Educational Statistics, 7, 175–191.
47.
Swaminathan, H. , & Gifford, J. A. (1985). Bayesian estimation in the two-parameter logistic model. Psychometrika, 50, 349–364.
48.
Swaminathan, H. , & Gifford, J. A. (1986). Bayesian estimation in the three-parameter logistic model. Psychometrika, 51, 581–601.
49.
Tanner, M. A. (1996). Tools for statistical inference: Methods for the exploration of posterior distributions and likelihood functions (2nd ed.). New York: Springer-Verlag.
50.
The MathWorks, Inc. (1996). MATLAB: The language of technical computing [Computer program]. Natick MA: Author.
51.
Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika, 47, 175–186.
52.
Tsutakawa, R. K. , & Johnson, J. C. (1990). The effect of uncertainty of item parameter estimation on ability estimates. Psychometrika, 55, 371–390.
53.
Tsutakawa, R. K. , & Lin, H. Y. (1986). Bayesian estimation of item response curves. Psychometrika, 51, 251–267.
54.
Wright, B. D. , Linacre, J. M., & Schultz, M. (1989). A user’s guide to BIGSCALE: Rasch-model rating scale analysis computer program [Computer program]. Chicago: MESA Press.
55.
Wright, B. D. , & Stone, M. H. (1979). Best test design. Chicago: MESA Press.
