Inferences from Bootstrap Method for Ability Parameters in 2-Parameter Logistic Model

Abstract

Ability parameter of persons/examinees estimates can be obtained using Joint Maximum Likelihood (JML) estimation method in Item Response Theory (IRT). However, JML estimates can be biased in some cases. Although Bootstrap method has been considered for JML, existing studies remain far from satisfactory with respect to the ability parameter estimation. This research evaluate the performances of JML and Bootstrap estimates of ability parameter in terms of Standard Error Measurement (SEM) in 2-Parameter Logistic (2-PL) model conducting a detailed Monte Carlo simulation study. According to the results, the average SEM estimates of Bootstrap method are less than the average SEM estimates of JML in terms of the ability parameter.

Keywords

• ability parameter,difficulty parameter,joint maximum likelihood estimation,discrimination parameter,two-parameter logistic model

References

1. Rasch, G. Probabilistic Models for Some Intelligence and Attainment Tests; Chicago: MESA; 1960.
2. 2. Hambleton, RK, Jones, RW. 1993. Comparison of classical test theory and item response theory and their applications to test development. Educational Measurement: Issues and Practice; 12(3): 38-47.
3. 3. Baker, FB. The basis of item response theory. ERIC. 2001.
4. 4. Birnbaum, A. Some latent trait models and their use in inferring an examinee’s ability, In Lord, FM, Novick, MR (Eds.), Statistical Theories of Mental Test Scores; 1968.
5. 5. Paolino, JP. Penalized joint maximum likelihood estimation applied to two parameters logistic item. Columbia University Graduate School of Arts and Sciences; 2013.
6. 6. McCulloch, CE, Searle, SR. Generalized, linear, and mixed models. John Wiley & Sons, New York; 2001.
7. 7. Harris, D. Comparison of 1-,2- and 3-parameter IRT models, instructional topics in educational measurement, An NCME Instructional Module on; 1989.
8. 8. Liou, M., Yu, L. 1991. Assesing statistical accuracy in ability estimation: bootstrap approach. Psychometrika; 56(1): 55-67.

9. 9. Atanasov, D. 2009. Estimation of IRT parameters over a small sample: Bootstrapping of the item responses. Pliska Studia Mathematica Bulgaria; 19: 58-68.
10. 10. Heene, M, Draxler, C, Ziegler, M, Bühner, M. 2011. Performance of the bootstrap Rasch model test under violations of non-intersecting item response functions. Psychological Test and Assessment Modeling; 53:283–294. 11. Wolfe, EW, McGill, MT. Comparison of asymptotic and bootstrap item fit indices in identifying misfit to the Rasch model. National Conference on Measurement in Education New Orleans; 2011.
11. 12. Patton, JM, Cheng, Y, Yuan, KH, Diao, Q. 2014. Bootstrap standard errors for maximum likelihood ability estimates when item parameters are unknown. Educational and Psychological Measurement; 74(4): 697-712.
12. 13. Olmuş, H., Nazman, E. 2017. An evaluation of the two parameter (2-PL) IRT models through a simulation study. Gazi University Journal of Science; 30(1): 235-249.
13. 14. Liu, Y, Yang, JS. 2018. Bootstrap-calibrated interval estimates for latent variable scores in item response theory. Psychometrika; 83(2): 333-354.
14. 15. Liu, Y., Hu, G., Cao, L.Wang, X., Chen, M.H. 2019. A comparison of Monte Carlo methods for computing marginal likelihoods of item response theory models. Journal of the Korean Statistical Society; 48:503-512.
15. 16. Chen, S., Haziza, D., Leger, C., Mashreghi, Z. 2019. Pseudo-population bootstrap methods for imputed survey data. Biometrica; 106(2):369-384.
16. 17. Baker, FB, Kim, SH. Item Response Theory: Parameter Estimation Techniques. Marcel Dekker, Inc; 2004.
17. 18. Partchev, I. 2004. A visual guide to item response theory, Friedrich-Schiller-Universitat Jena. https://www.metheval.uni-jena.de/irt/ VisualIRT.pdf.
18. 19. Hesterberg, T, Monaghan, S, Moore, DS, Clipson, A, Epstein, R. Bootstrap method and permutation tests. W.H. Freeman and Company New York; 2003.
19. 20. Baur, T, Lukes, D. 2009. An Evaluation of the IRT models through monte carlo simulation. Journal of Undergraduate Research XII:1-7. Clearinghouse on Assessment and Evaluation.
20. 21. Toribio, SG. Bayesian model checking strategies for dichotomous item response theory models. Graduate College of Bowling Green State University; 2006.