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Slice-Gibbs sampling algorithm for estimating the parameters of a multilevel item response model. (English) Zbl 1396.62266

Summary: In a fully Bayesian framework, a novel slice-Gibbs algorithm is developed to estimate a multilevel item response theory (IRT) model. The advantage of this algorithm is that it can recover parameters well based on various types of prior distributions of the item parameters, including informative and non-informative priors. In contrast to the traditional Metropolis-Hastings (M-H) within Gibbs algorithm, the slice-Gibbs algorithm is faster and more efficient, due to its drawing the sample with acceptance probability as one, rather than tuning the proposal distributions to achieve the reasonable acceptance probabilities, especially for the logistic model without conjugate distribution. In addition, based on the Markov chain Monte Carlo (MCMC) output, two model assessment methods are investigated concerning the goodness of fit between models. The information criterion method on the basis of marginal likelihood is proposed to assess the different structural multilevel models, and the cross-validation method is used to evaluate the overall multilevel IRT models. The feasibility and effectiveness of the slice-Gibbs algorithm are investigated in simulation studies. An application using a real data involving students’ mathematics test achievements is reported.

MSC:

62P15 Applications of statistics to psychology

Software:

Stata; GLLAMM; PRMLT
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References:

[1] Adams, R. J.; Wilson, M.; Wu, M., Multilevel item response models: an approach to errors in variables regression, Journal of Educational and Behavioral Statistics, 22, 47-76, (1997)
[2] Akaike, H., Information theory and an extension of the maximum likelihood principle, (Petrov, B. N.; Csaki, F., Second International symposium on information theory, (1973), Akademiai Kiado Budapest), 267-281 · Zbl 0283.62006
[3] Akaike, H., Factor analysis and AIC, Psychometrika, 52, 317-332, (1987) · Zbl 0627.62067
[4] Albert, J. H., Bayesian estimation of normal ogive item response curves using Gibbs sampling, Journal of Educational Statistics, 17, 251-269, (1992)
[5] Ando, T., Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models, Biometrika, 94, 443-458, (2007) · Zbl 1132.62005
[6] Ando, T., Bayesian model selection and statistical modeling, (2010), CRC Press · Zbl 1303.62006
[7] Ando, T., Predictive Bayesian model selection, American Journal of Mathematical and Management Sciences, 31, 13-38, (2011)
[8] Azevedo, C. L.N.; Fox, J.-P.; Andrade, D. F., Bayesian longitudinal item response modeling with restricted covariance pattern structures, Statistics and Computing, 26, 443-460, (2016) · Zbl 1342.62181
[9] Béguin, A. A.; Glas, C. A.W., MCMC estimation of multidimensional IRT models, Psychometrika, 66, 541-561, (2001) · Zbl 1293.62234
[10] Bishop, C. M., Slice sampling, (Pattern recognition and machine learning, (2006), Springer New York)
[11] Bock, R. D.; Aitkin, M., Marginal maximum likelihood estimation of item parameters: application of an EM algorithm, Psychometrika, 46, 443-459, (1981)
[12] Bock, R. D.; Schilling, S. G., High-dimensional full-information item factor analysis, (Berkane, M., Latent variable modelling and applications to causality, (1997), Springer New York), 164-176 · Zbl 0919.62058
[13] Bozdogan, H., Model selection and akaike’s information criterion (AIC): the general theory and its analytical extensions, Psychometrika, 52, 345-370, (1987) · Zbl 0627.62005
[14] Browne, W. J.; Draper, D., A comparison of Bayesian and likelihood-based methods for Fitting multilevel models, Bayesian Analysis, 1, 473-514, (2006) · Zbl 1331.62125
[15] Chen, M.-H.; Shao, Q.-M.; Ibrahim, J. G., Monte Carlo methods in Bayesian computation, (2000), Springer New York · Zbl 0949.65005
[16] Chib, S.; Greenberg, E., Understanding the metropolis-Hastings algorithm, The American Statistician, 49, 327-335, (1995)
[17] Clayton, D., Generalized linear mixed models, (Gilks, W. R.; Richardson, S.; Spiegelhalter, D. J., Markov Chain Monte Carlo methods in practice, (1996), Chapman-Hall London), 275-303 · Zbl 0841.62059
[18] Congdon, P., Bayesian statistical modelling, (2006), Wiley Chichester · Zbl 1193.62034
[19] Damien, P.; Wakefield, J.; Walker, S., Gibbs sampling for Bayesian non-conjugate and hierarchical models by auxiliary variables, Journal of the Royal Statistical Society. Series B., 61, 331-344, (1999) · Zbl 0913.62028
[20] Fox, J. P., Multilevel IRT using dichotomous and polytomous items, The British Journal of Mathematical and Statistical Psychology, 58, 145-172, (2005)
[21] Fox, J. P., (Bayesian item response modeling: Theory and applications, Series: Statistics for social and behavioral sciences, (2010), Springer New York)
[22] Fox, J. P.; Glas, C. A.W., Bayesian estimation of a multilevel IRT model using Gibbs sampling, Psychometrika, 66, 271-288, (2001) · Zbl 1293.62242
[23] Geisser, S.; Eddy, W., A predictive approach to model selection, Journal of the American Statistical Association, 74, 153-160, (1979) · Zbl 0401.62036
[24] Gelfand, A. E., Model determination using sampling-based methods, (Gilks, W. R.; Richardson, S.; Spiegelhalter, D. J., Markov Chain Monte Carlo methods in practice, (1996), Chapman-Hall London), 145-161 · Zbl 0840.62003
[25] Gelfand, A. E.; Dey, D. K., Bayesian model choice: asymptotics and exact calculations, Journal of the Royal Statistical Society. Series B., 56, 501-514, (1994) · Zbl 0800.62170
[26] Gelfand, A. E.; Smith, A. F.M., Sampling-based approaches to calculating marginal densities, Journal of the American Statistical Association, 85, 398-409, (1990) · Zbl 0702.62020
[27] Gelman, A.; Rubin, D. B., Inference from iterative simulation using multiple sequences, Statistical Science, 457-472, (1992) · Zbl 1386.65060
[28] Geman, S.; Geman, D., Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721-741, (1984) · Zbl 0573.62030
[29] Gilks, W. R.; Wild, P., Adaptive rejection sampling for Gibbs sampling, Applied Statistics, 41, 337-348, (1992) · Zbl 0825.62407
[30] Goldstein, H., Multilevel statistical models, (2011), Edward Arnold London · Zbl 1274.62006
[31] Goldstein, H.; McDonald, R. P., A general model for the analysis of multilevel data, Psychometrika, 53, 455-467, (1987) · Zbl 0718.62158
[32] Hastings, W. K., Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57, 97-109, (1970) · Zbl 0219.65008
[33] Hedeker, D.; Gibbons, R. D., A random-effects ordinal regression model for multilevel analysis, Bioemtrics, 50, 933-944, (1994) · Zbl 0826.62049
[34] Hoijtink, H., Posterior inference in the random intercept model based on samples obtained with Markov chain Monte Carlo methods, Computational Statistics, 15, 3, 315-336, (2000) · Zbl 1038.65006
[35] Kamata, A., Item analysis by the hierarchical generalized linear model, Journal of Educational Measurement, 38, 79-93, (2001)
[36] Kass, R. E.; Raftery, A. E., Bayes factors, Journal of the American Statistical Association, 90, 773-795, (1995) · Zbl 0846.62028
[37] Kuk, A. Y.C., Laplace importance sampling for generalized linear mixed models, Journal of Statistical Computation and Simulation, 63, 143-158, (1999) · Zbl 0956.62052
[38] Longford, N. T.; Muthén, B. O., Factor analysis for clustered populations, Psychometrika, 57, 581-597, (1992) · Zbl 0825.62531
[39] Lord, F. M., Applications of item response theory to practical testing problems, (1980), Lawrence Erlbaum Associates Hillsdale, NJ
[40] Maier, K. S., A rasch hierarchical measurement model, Journal of Educational and Behavioral Statistics, 26, 307-330, (2001)
[41] Masters, G. M., A rasch model for partial credit scoring, Psychometrika, 47, 149-174, (1982) · Zbl 0493.62094
[42] McCulloch, C. E.; Searle, S. R., Generalized, linear, and mixed models, (2001), Wiley New York · Zbl 0964.62061
[43] Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E., Equations of state calculations by fast computing machines, Journal of Chemical Physics, 21, 1087-1092, (1953)
[44] Mislevy, R. J.; Bock, R. D., A hierarchical item-response model for educational testing, (Bock, R. D., Multilevel analysis of educational data, (1989), Academic Press San Diego), 57-74
[45] Muraki, E., A generalized partial credit model: application of an EM algorithm, Applied Psychological Measurement, 16, 159-176, (1992)
[46] Muraki, E., Information functions of the generalized partial credit model, Applied Psychological Measurement, 17, 351-363, (1993)
[47] Muthén, B. O., Latent variable modeling in heterogeneous populations, Psychometrika, 54, 557-585, (1989)
[48] Natesan, P., Estimation of two-parameter multilevel item response models with predictor variables: Simulation and substantiation for an urban school district, (2007), Texas A & M University, (Ph. D thesis)
[49] Neal, R., Slice sampling, The Annals of Statistics, 31, 705-767, (2003) · Zbl 1051.65007
[50] Newton, M. A.; Raftery, A. E., Approximate Bayesian inference with the weighted likelihood bootstrap, Journal of the Royal Statistical Society. Series B., 56, 3-48, (1994) · Zbl 0788.62026
[51] Patz, R. J.; Junker, B. W., A straightforward approach to Markov chain Monte Carlo methods for item response models, Journal of Educational and Behavioral Statistics, 24, 2, 146-178, (1999)
[52] Rabe-Hesketh, S.; Skrondal, A.; Pickles, A., Reliable estimation of generalized linear mixed models using adaptive quadrature, The Stata Journal, 2, 1-21, (2002)
[53] Rabe-Hesketh, S.; Skrondal, A.; Pickles, A., Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects, Journal of Econometrics, 128, 301-323, (2005) · Zbl 1336.62079
[54] Raftery, A. E., Bayesian model selection in social research, (Raftery, A. E., Sociological methodology, (1995), Blackwell Oxford), 111-164
[55] Raudenbush, S. W.; Bryk, A. S., Hierarchical linear models: Applications and data analysis methods, (2002), Sage Thousand Oaks, CA · Zbl 1001.62004
[56] Raudenbush, S. W.; Yang, M.-L.; Yosef, M., Maximum likelihood for generalized linear models with nested random effects via high-order, multivariate Laplace approximation, Journal of Computational and Graphical Statistics, 9, 141-157, (2000)
[57] Schwarz, G., Estimating the dimension of a model, The Annals of Statistics, 6, 461-464, (1978) · Zbl 0379.62005
[58] Skaug, H. J., Automatic differentiation to facilitate maximum likelihood estimation in nonlinear random effects models, Journal of Computational and Graphical Statistics, 11, 458-470, (2002)
[59] Snijders, T. A.B.; Bosker, R. J., Multilevel analysis: An introduction to basic and advanced multilevel modeling, (2012), Sage Los Angeles, CA · Zbl 1296.62008
[60] Spiegelhalter, D. J.; Best, N. G.; Carlin, B. P.; van der Linde, A., Bayesian measures of model complexity and fit (with discussion), Journal of the Royal Statistical Society. Series B., 64, 583-639, (2002) · Zbl 1067.62010
[61] Spiegelhalter, D. J.; Best, N. G.; Carlin, B. P.; van der Linde, A., The deviance information criterion: 12 years on (with discussion), Journal of the Royal Statistical Society. Series B., 76, 485-493, (2014)
[62] Tanner, M. A.; Wong, W. H., The calculation of posterior distributions by data augmentation, Journal of the American Statistical Association, 82, 528-550, (1987) · Zbl 0619.62029
[63] Tierney, L., Markov chains for exploring posterior distributions (with discussions), The Annals of Statistics, 22, 1701-1762, (1994) · Zbl 0829.62080
[64] Tsutakawa, R. K., Estimation of two-parameter logistic item response curves, Journal of Educational Statistics, 9, 4, 263-276, (1984)
[65] Tsutakawa, R. K.; Soltys, M. J., Approximation for Bayesian ability estimation, Journal of Educational Statistics, 13, 117-130, (1988)
[66] Vaida, F.; Blanchard, S., Conditional Akaike information for mixed-effects models, Biometrika, 92, 351-370, (2005) · Zbl 1094.62077
[67] van der Linden, W. J.; Hambleton, R. K., Handbook of modern item response theory, (1997), Springer-Verlag New York · Zbl 0872.62099
[68] Wang, C.; Xu, G., A mixture hierarchical model for response times and response accuracy, The British Journal of Mathematical and Statistical Psychology, 68, 456-477, (2015)
[69] Wasserman, L., Bayesian model selection and model averaging, Journal of Mathematical Psychology, 44, 92-107, (2000) · Zbl 0946.62032
[70] Zeger, S. L.; Karim, M. R., Generalized linear models with random effects: A Gibbs sampling approach, Journal of the American Statistical Association, 86, 79-86, (1991)
[71] Zucchini, W., An introduction to model selection, Journal of Mathematical Psychology, 44, 41-61, (2000) · Zbl 0949.62092
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