×

Identifying Guttman structures in incomplete Rasch datasets. (English) Zbl 1462.62145

Summary: In applications of IRT, it often happens that many examinees omit a substantial proportion of item responses. This can occur for various reasons, though it may well be due to no more than the simple fact of design incompleteness. In such circumstances, literature not infrequently refers to various types of estimation problem, often in terms of generic “convergence problems” in the software used to estimate model parameters. With reference to the Partial Credit Model and the instance of data missing at random, this article demonstrates that as their number increases, so does that of anomalous datasets, intended as those not corresponding to a finite estimate of (the vector parameter that identifies) the model. Moreover, the necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood estimation of the Partial Credit Model (and hence, in particular, the Rasch model) in the case of incomplete data are given – with reference to the model in its more general form, the number of response categories varying according to item. A taxonomy of possible cases of anomaly is then presented, together with an algorithm useful in diagnostics.

MSC:

62F10 Point estimation
62B99 Sufficiency and information
91C05 Measurement theory in the social and behavioral sciences

Software:

Mathematica; SPSS; Matlab
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams R. J., Quest: The Interactive Test Analysis System (1998)
[2] DOI: 10.1002/0471249688 · doi:10.1002/0471249688
[3] DOI: 10.1080/01621459.1970.10481160 · doi:10.1080/01621459.1970.10481160
[4] Andersen E. B., Discrete Statistical Models with Social Sciences Applications (1980) · Zbl 0423.62001
[5] Andersen E. B., Principals of Modern Psychological Measurement pp 117– (1983)
[6] DOI: 10.1007/BF02293656 · Zbl 0384.62088 · doi:10.1007/BF02293656
[7] Andrich D., J. Appl. Measure. 4 pp 205– (2003)
[8] Assessment System Corporation . ( 2008 ). User’s Manual for the RASCAL–Rasch Analysis Program. St. Paul: Author .
[9] Barndorff-Nielsen O., Information and Exponential Families in Statistical Theory (1978) · Zbl 0387.62011
[10] DOI: 10.1007/s11336-001-0917-0 · Zbl 1306.62381 · doi:10.1007/s11336-001-0917-0
[11] Burket G. R., PARMATE (1995)
[12] Custer M., Appl. Measure. Educ. 19 pp 143– (2006)
[13] De Ayala R. J., J. Appl. Measure. 7 pp 278– (2006)
[14] DOI: 10.1007/978-1-4757-3990-9 · doi:10.1007/978-1-4757-3990-9
[15] DOI: 10.1177/014662168801200201 · doi:10.1177/014662168801200201
[16] DOI: 10.1207/S15324818AME1501_02 · doi:10.1207/S15324818AME1501_02
[17] DOI: 10.1016/j.jsc.2005.04.003 · Zbl 1120.62015 · doi:10.1016/j.jsc.2005.04.003
[18] DOI: 10.1016/j.jspi.2007.03.022 · Zbl 1119.62053 · doi:10.1016/j.jspi.2007.03.022
[19] DOI: 10.1007/BF02293919 · Zbl 0465.62103 · doi:10.1007/BF02293919
[20] DOI: 10.1007/978-1-4612-4230-7_8 · doi:10.1007/978-1-4612-4230-7_8
[21] DOI: 10.1007/BF02294324 · Zbl 0862.62086 · doi:10.1007/BF02294324
[22] Fischer G. H., Handbook of Statistics. Psychometrics pp 515– (2007)
[23] Furlow C. F., J. Appl. Measure. 8 pp 388– (2007)
[24] DOI: 10.1214/aos/1176343941 · Zbl 0368.62019 · doi:10.1214/aos/1176343941
[25] Kolakovski D., LOGOG. Maximum Likelihood Item Analysis and Test Scoring: Logistic Model for Multiple Item Responses (1973)
[26] Lee O. K., Rasch Measure. Trans. 5 pp 172– (1991)
[27] Lee O. K., Rasch Measure. Trans. 6 pp 202– (1992)
[28] Lee O. K., J. Appl. Measure. 4 pp 10– (2003)
[29] Lehmann E. L., Theory of Point Estimation., 2. ed. (1998) · Zbl 0916.62017
[30] Linacre J. M., Introduction to Rasch Measurement. Theory, Models and Applications pp 48– (2004)
[31] Linacre J. M., Facets. Rasch Measurement Computer Program (2009)
[32] Linacre J. M., WINSTEPSregistered. Rasch Measurement Computer Program (2009)
[33] Linacre J. M., A User’s Guide to BIGSTEPS: A Rasch-Model Computer Program (2006)
[34] DOI: 10.1002/9781119013563 · doi:10.1002/9781119013563
[35] DOI: 10.1007/978-1-4899-1292-3_2 · doi:10.1007/978-1-4899-1292-3_2
[36] Lord F. M., Applications of Item Response Theory to Practical Testing Problems (1980)
[37] Luo G., J. Appl. Measure. 6 pp 128– (2005)
[38] DOI: 10.1007/BF02296272 · Zbl 0493.62094 · doi:10.1007/BF02296272
[39] Mathworks Inc. ( 2007 ).MATLAB 7.0[Computer Software]. Natik, MA .
[40] Mislevy , R. J. , Wu , P. K. ( 1996 ). Missing Responses and IRT Ability Estimation: Omits, Choice, Time Limits, Adaptive Testing. ETS Research Report RR-96–30-ONR, Princeton, NJ: Educational Testing Service .
[41] DOI: 10.1007/978-1-4612-4230-7_3 · doi:10.1007/978-1-4612-4230-7_3
[42] OECD . ( 2009a ).PISA Data Analysis Manual–SPSS, 2nd ed. Paris: OECD .
[43] PISA 2009–Assessment Framework (2009)
[44] DOI: 10.3102/00346543074004525 · doi:10.3102/00346543074004525
[45] Rockafellar R. T., Convex Analysis (1970) · Zbl 0193.18401 · doi:10.1515/9781400873173
[46] DOI: 10.1007/BF02294084 · doi:10.1007/BF02294084
[47] DOI: 10.1201/9781439821862 · doi:10.1201/9781439821862
[48] DOI: 10.1037/1082-989X.7.2.147 · doi:10.1037/1082-989X.7.2.147
[49] von Davier M., Multivariate and Mixture Distribution Rasch Models. Extensions and Applications (2007) · Zbl 1117.62133 · doi:10.1007/978-0-387-49839-3
[50] DOI: 10.1007/BF02294473 · Zbl 04512703 · doi:10.1007/BF02294473
[51] Wingersky M. S., LOGIST User’s Guide (1999)
[52] Wolfram Research Inc., Mathematica 7: Version 7.0.1 (2009) · Zbl 1205.00034
[53] Wright B. D., Rating Scale Analysis (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.