Draxler
and Zessin [1] derived the power function for a class of conditional tests of
assumptions of a psychometric model known as the Rasch model and suggested an
MCMC approach developed by Verhelst [2] for the
numerical approximation of the power of the tests. In this contribution, the
precision of the Verhelst approach is investigated and compared with an exact
sampling procedure proposed by Miller and Harrison [3] for which the discrete probability distribution to be sampled from is
exactly known. Results show no substantial differences between the two
numerical procedures and quite accurate power computations. Regarding the
question of computing time the Verhelst approach will have to be considered
much more efficient.
References
[1]
Draxler, C. and Zessin, J. (2015) The Power Function of Conditional Tests of the Rasch Model. ASTA Advances in Statistical Analysis, 99, 367-378.
https://doi.org/10.1007/s10182-015-0249-5
[2]
Verhelst, N.D. (2008) An Efficient Mcmc Algorithm to Sample Binary Matrices with Fixed Marginals. Psychometrika, 73, 705-728.
https://doi.org/10.1007/s11336-008-9062-3
[3]
Miller, J.W. and Harrison, M.T. (2013) Exact Sampling and Counting for Fixed-Margin Matrices. The Annals of Statistics, 41, 1569-1592.
https://doi.org/10.1214/13-AOS1131
[4]
Rasch, G. (1960) Probabilistic Models for Some Intelligence and Attainment Tests. Danish Institute for Educational Research, Copenhagen.
[5]
Fischer, G.H. and Molenaar, I.W. (1995) Rasch Models. Foundations, Recent Developments and Applications. Springer, New York.
[6]
Draxler, C. (2011) A Note on a Discrete Probability Distribution Derived from the Rasch Model. Advances and Applications in Statistical Sciences, 6, 665-673.
[7]
Chen, Y., Diaconis, P., Holmes, S.P. and Liu J.S. (2005) Sequential Monte Carlo Methods for Statistical Analysis of Tables. Journal of the American Statistical Association, 100, 109-120. https://doi.org/10.1198/016214504000001303
[8]
Chen, Y. and Small, D. (2005) Exact Tests for the Rasch Model via Sequential Importance Sampling. Psychometrika, 70, 11-30.
https://doi.org/10.1007/s11336-003-1069-1
[9]
Neyman, J. and Pearson, E.S. (1933) On the Problem of the Most Efficient Tests of Statistical Hypotheses. Philosophical Transactions of the Royal Society A, 231, 289-337. https://doi.org/10.1098/rsta.1933.0009
[10]
Ponocny, I. (2001) Nonparametric Goodness-of-Fit Tests for the Rasch Model. Psychometrika, 66, 437-459. https://doi.org/10.1007/BF02294444
[11]
Draxler, C. (2018) Bayesian Conditional Inference for Rasch Models. ASTA Advances in Statistical Analysis, 102, 245-262.
https://doi.org/10.1007/s10182-017-0303-6
[12]
Gale, D. (1957) A Theorem on Flows in Networks. Pacific Journal of Mathematics, 7, 1073-1082. https://doi.org/10.2140/pjm.1957.7.1073
[13]
Ryser, H.J. (1957) Combinatorial Properties of Matrices of Zeros and Ones. Canadian Journal of Mathematics, 9, 371-377. https://doi.org/10.4153/CJM-1957-044-3
[14]
Verhelst, N.D., Hatzinger, R. and Mair, P. (2007) The Rasch Sampler. Journal of Statistical Software, 20, 1-14. https://doi.org/10.18637/jss.v020.i04
[15]
Draxler, C. (2010) Sample Size Determination for Rasch Model Tests. Psychometrika, 75, 708-724. https://doi.org/10.1007/s11336-010-9182-4
[16]
Andersen, E.B. (1973) A Goodness of Fit Test for the Rasch Model. Psychometrika, 38, 123-140. https://doi.org/10.1007/BF02291180
[17]
Glas, C.A. and Verhelst, N.D. (1995) Testing the Rasch Model. Rasch Models, Springer, New York, 69-95. https://doi.org/10.1007/978-1-4612-4230-7_5
[18]
Draxler, C. and Alexandrowicz, R.W. (2015) Sample Size Determination within the Scope of Conditional Maximum Likelihood Estimation with Special Focus on Testing the Rasch Model. Psychometrika, 80, 897-919.
https://doi.org/10.1007/s11336-015-9472-y
[19]
Draxler, C. and Kubinger, K. (2018) Power and Sample Size Considerations in Psychometrics. In: Pilz, J., Rasch, D., Melas, V. and Moder, K., Eds., Statistics and Simulation. IWS 2015. Springer Proceedings in Mathematics & Statistics, Vol 231, Springer, Cham.
