Skip to main content

Considerations on the Impact of Ill-Conditioned Configurations in the CML Approach

  • Conference paper
  • First Online:
Advances in Data Analysis, Data Handling and Business Intelligence
  • 2888 Accesses

Abstract

Considering complete designs, the configurations of non-existence of the Maximum Likelihood (ML) estimates for the Partial Credit Model are known in the Joint (JML) approach: null categories and ill-conditioned patterns are the only two sources of trouble. In the Conditional (CML) approach, apart from datasets with null categories, the other “anomalous” configurations are not known. In this paper, the impact of ill-conditioned patterns in the conditional approach, as well as the incidence of CML-anomalous configurations, are both studied by a systematic analysis on small-dimensional data matrices. Obtained results emphasize the presence of a large number of additional CML configurations of non-existence, compared to those valid in the JML case.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Andersen, E. B. (1995). Polytomous Rasch models and their estimation. In G. H. Fischer, & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 271–291). New York: Springer.

    Google Scholar 

  • Barndorff-Nielsen, O. (1978). Information and exponential families in statistical theory. New York: Wiley.

    MATH  Google Scholar 

  • Bertoli-Barsotti, L. (2002). On a condition for the existence of the maximum likelihood estimate for concave log-likelihood functions. In Studi in Onore di Angelo Zanella (pp. 25–42). Milan: Vita & Pensiero.

    Google Scholar 

  • Bertoli-Barsotti, L. (2005). On the existence and uniqueness of JML estimates for the partial credit model. Psychometrika, 70(3), 517–531.

    Article  MathSciNet  Google Scholar 

  • Bertoli-Barsotti, L. (2008). Parametrizzazione del PCM e ordinamento (Technical report). DMSIA, University of Bergamo.

    Google Scholar 

  • Fischer, G. H. (1981). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46(1), 59–77.

    Article  MATH  MathSciNet  Google Scholar 

  • Fischer, G. H. (1995). Derivations of the Rasch model. In G. H. Fischer, & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 15–38). New York: Springer.

    Google Scholar 

  • Jacobsen, M. (1989). Existence and unicity of MLEs in discrete exponential family distributions. Scandinavian Journal of Statistics, 16, 335–349.

    MathSciNet  Google Scholar 

  • Linacre, J. M. (2004). Rasch model estimation: Further topics. Journal of Applied Measurement, 5(1), 95–110.

    Google Scholar 

  • Mair, P., & Hatzinger, R. (2007). Extended Rasch modeling: The eRm package for the application of IRT models in R. Journal of Statistical Software, 20(9), 1–20.

    Google Scholar 

  • Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149–174.

    Article  MATH  Google Scholar 

  • Masters, G. N., & Wright, B. D. (1997). The partial credit model. In W. J. van der Linden, & R. K. Hambleton (Eds.), Handbook of modern item response theory (pp. 101–121). New York: Springer.

    Google Scholar 

  • Neyman, J., & Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16(1), 1–32.

    Article  MathSciNet  Google Scholar 

  • Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: The Danish Institute of Educational Research.

    Google Scholar 

  • Rasch, G. (1977). On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements. The Danish Yearbook of Philosophy, 14(4), 58–93.

    Google Scholar 

  • Wilson, M., & Masters, G. N. (1993). The partial credit model and null categories. Psychometrika, 58(1), 87–99.

    Article  Google Scholar 

  • Wright, B. D., & Masters, G. N. (1982). Rating scale analysis. Chicago: Mesa.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Antonio Punzo .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Punzo, A. (2009). Considerations on the Impact of Ill-Conditioned Configurations in the CML Approach. In: Fink, A., Lausen, B., Seidel, W., Ultsch, A. (eds) Advances in Data Analysis, Data Handling and Business Intelligence. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01044-6_52

Download citation

Publish with us

Policies and ethics