Identification of the 1PL Model with Guessing Parameter: Parametric and Semi-parametric Results

Psychometrika

Volumn 78, Issue 2, 2013, Pages 341-379

  San Martín, Ernesto ^a,b,c   Rolin, Jean Marie ^d   Castro, Luis M ^e  

^a PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE   (Chile)

^b Measurement Center MIDE UC   (Chile)

^c CEPPE   (Chile)

^d UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

^e UNIVERSITY OF CONCEPCIÓN   (Chile)

Author keywords

2PL model; 3PL model; fixed effects; Gibbs sampler; Hilbert space; identified parameter; location scale distributions; measurable separability; parameters of interest; random effects

Indexed keywords

HUMAN; PROCEDURES; PSYCHOMETRY; STATISTICAL MODEL;

HUMANS; MODELS, STATISTICAL; PSYCHOMETRICS;

EID: 84875379712     PISSN: 00333123     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11336-013-9322-8     Document Type: Article

References (59)

1. Adams, R. J., Wilson, M. R., & Wang, W. (1997). The multidimensional random coefficients multinomial logit model. Applied Psychological Measurement, 21, 1-23.
2. Adams, R. J., & Wu, M. L. (2007). The mixed-coefficients multinomial logit model: a generalization form of the Rasch model. In M. von Davier & C. H. Carstensen (Eds.), Multivariate and mixture distribution Rasch models (pp. 57-75). Berlin: Springer.
3. Andersen, E. B. (1980). Discrete statistical models with social sciences applications. Amsterdam: North-Holland.
4. Béguin, A. A., & Glas, C. A. W. (2001). MCMC estimation and some model-fit analysis of multidimensional IRT models. Psychometrika, 66, 541-562.
5. Berti, P., Pratelli, L., & Riggo, P. (2008). Trivial intersection of σ-fields and Gibbs sampling. Annals of Probability, 36, 2215-2234.
6. Berti, P., Pratelli, L., & Riggo, P. (2010). Atomic intersection of σ-fields and some of its consequences. Probability Theory and Related Fields, 148, 269-283.
7. Birnbaum, A. (1968). Some latent trait models and their use in inferring any examinee's ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 395-479). Reading: Addison-Wesley.
8. Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: application of an EM algorithm. Psychometrika, 46, 443-459.
9. Bock, R. D., & Zimowski, M. F. (1997). Multiple group IRT. In W. J. van der Linden & R. K. Hambleton (Eds.), Handbook of modern item response theory (pp. 433-448). Berlin: Springer.
10. Carlin, B. P., & Louis, T. A. (2000). Bayes and empirical Bayes methods for data analysis (2nd ed.). London: Chapman & Hall/CRC.
11. De Boeck, P., & Wilson, M. (2004). Explanatory item response models. A generalized linear and nonlinear approach. Berlin: Springer.
12. Del Pino, G., San Martín, E., González, J., & De Boeck, P. (2008). On the relationships between sum score based estimation and joint maximum likelihood estimation. Psychometrika, 73, 145-151.
13. Eberly, E. E., & Carlin, B. P. (2000). Identifiability and convergence issues for Markov chain Monte Carlo fitting of spatial data. Statistics in Medicine, 19, 2279-2294.
14. Embretson, S. E., & Reise, S. P. (2000). Item response theory for psychologists. Mahwah: Lawrence Erlbaum Associates.
15. Florens, J.-P., & Mouchart, M. (1982). A note on noncausality. Econometrica, 50, 583-591.
16. Florens, J.-P., Mouchart, M., & Rolin, J.-M. (1990). Elements of Bayesian statistics. New York: Dekker.
17. Florens, J.-P., & Rolin, J.-M. (1984). Asymptotic sufficiency and exact estimability. In J.-P. Florens, M. Mouchart, J.-P. Raoult, & L. Simar (Eds.), Alternative approaches to time series analysis (pp. 121-142). Bruxelles: Publications des Facultés Universitaires Saint-Louis.
18. Gabrielsen, A. (1978). Consistency and identifiability. Journal of Econometrics, 8, 261-263.
19. Gelfand, A. E., & Sahu, S. K. (1999). Identifiability, improper priors, and Gibbs sampling for generalized linear models. Journal of the American Statistical Association, 94, 247-253.
20. Gelman, A., & Rubin, R. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457-511.
21. Geweke, J. (1992). Bayesian statistics: Vol. 4. Evaluating the accuracy of sampling-based approaches to calculating posterior moments. Oxford: Clarendon Press.
22. Ghosh, M., Ghosh, A., Chen, M.-H., & Agresti, A. (2000). Noninformative priors for one-parameter item response models. Journal of Statistical Planning and Inference, 88, 99-115.
23. Gustafson, P. (2005). On model expansion, model contraction, identifiability and prior information: two illustrative scenarios involving mismeasured variables (with discussion). Statistical Sciences, 20, 111-140.
24. Halmos, P. (1951). Introduction to Hilbert space, and the theory of spectral multiplicity. New York: Chelsea.
25. Hambleton, R. K., Swaminathan, H., & Rogers, H. J. (1991). Fundamentals of item response theory. Thousand Oaks: Sage.
26. Heidelberg, P., & Welch, P. (1992). Simulation run length control in the presence of an initial transient. Operations Research, 31, 1109-1144.
27. Hutschinson, T. P. (1991). Ability, parameter information, guessing: statistical modelling applied to multiple-choice tests. Rundle Mall: Rumsby Scientific Publishing.
28. Karabatsos, G., & Walker, S. (2009). Coherent psychometric modelling with Bayesian nonparametrics. British Journal of Mathematical & Statistical Psychology, 62, 1-20.
29. Kass, R., Carlin, B., Gelman, A., & Neal, R. (1998). Markov chain Monte Carlo in practice: a roundtable discussion. American Statistician, 52, 93-100.
30. Koopmans, T. C., & Reiersøl, O. (1950). The identification of structural characteristics. The Annals of Mathematical Statistics, 21, 165-181.
31. Lancaster, T. (2000). The incidental parameter problem since 1948. Journal of Econometrics, 95, 391-413.
32. van der Linden, W., & Hambleton, R. K. (1997). Handbook of modern item response theory. Berlin: Springer.
33. Lindley, D. V. (1971). Bayesin statistics: a review. Philadelphia: Society for Industrial and Applied Mathematics.
34. Maris, G., & Bechger, T. (2009). On interpreting the model parameters for the three parameter logistic model. Measurement Interdisciplinary Research & Perspective, 7, 75-86.
35. McDonald, R. P. (1999). Test theory: a unified treatment. Hillsdale: Erlbaum.
36. Millsap, R., & Maydeu-Olivares, A. (2009). Quantitative methods in psychology. Thousand Oaks: Sage.
37. Miyazaki, K., & Hoshino, T. (2009). A Bayesian semiparametric item response model with Dirichlet process priors. Psychometrika, 74, 375-393.
38. Molenaar, I. W. (1995). Estimation of item parameters. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models. Foundations, recent developments and applications. New York: Springer (Chapter 3).
39. Mouchart, M. (1976). A note on Bayes theorem. Statistica, 36, 349-357.
40. Poirier, D. J. (1998). Revising beliefs in nonidentified models. Econometric Theory, 14, 483-509.
41. R Development Core Team (2006). R: a language and environment for statistical computing [Computer software manual]. Vienna, Austria. http://www. R-project. org (ISBN 3-900051-07-0).
42. Rao, M. M. (2005). Conditional measures and applications (2nd ed.). London: Chapman & Hall/CRC.
43. Rizopoulos, D. (2006). ltm: an R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1-25. http://www. jstatsoft. org/v17/i05/.
44. Roberts, G. O., & Rosenthal, J. (1998). Markov-chain Monte Carlo: some practical implications of theoretical results. Canadian Journal of Statistics, 26(1), 5-20.
45. San Martín, E., Del Pino, G., & De Boeck, P. (2006). IRT models for ability-based guessing. Applied Psychological Measurement, 30, 183-203.
46. San Martín, E., & González, J. (2010). Bayesian identifiability: contributions to an inconclusive debate. Chilean Journal of Statistics, 1, 69-91.
47. San Martín, E., González, J., & Tuerlinckx, F. (2009). Identified parameters, parameters of interest and their relationships. Measurement Interdisciplinary Research & Perspective, 7, 95-103.
48. San Martín, E., Jara, A., Rolin, J.-M., & Mouchart, M. (2011). On the Bayesian nonparametric generalization of IRT-type models. Psychometrika, 76, 385-409.
49. San Martín, E., Mouchart, M., & Rolin, J. M. (2005). Ignorable common information, null sets and Basu's first theorem. Sankhyā, 67, 674-698.
50. San Martín, E., & Quintana, F. (2002). Consistency and identifiability revisited. Brazilian Journal of Probability and Statistics, 16, 99-106.
51. Shiryaev, A. N. (1995). Probability (2nd ed.). Berlin: Springer.
52. Spivak, M. (1965). Calculus on manifolds: a modern approach to classical theorems of advanced calculus. Cambridge: Perseus Book Publishing.
53. Swaminathan, H., & Gifford, J. A. (1986). Bayesian estimation in the three-parameter logistic model. Psychometrika, 51, 589-601.
54. Thissen, D. (2009). On interpreting the parameters for any item response model. Measurement Interdisciplinary Research & Perspective, 7, 104-108.
55. Thissen, D., & Wainer, H. (2001). Item response models for items scored in two categories. Berlin: Springer.
56. Woods, C. M. (2006). Ramsay-curve item response theory (RC-IRT) to detect and correct for nonnormal latent variables. Psychological Methods, 11, 253-270.
57. Woods, C. M. (2008). Ramsay-curve item response theory for the three-parameter logistic item response model. Applied Psychological Measurement, 32, 447-465.
58. Woods, C. M., & Thissen, D. (2006). Item response theory with estimation of the latent population distribution using spline-based densities. Psychometrika, 71, 281-301.
59. Xie, Y., & Carlin, B. P. (2006). Measure of Bayesian learning and identifiability in hierarchical models. Journal of Statistcal Planning and Inference, 136, 3458-3477.