What Do You Mean by a Difficult Item? On the Interpretation of the Difficulty Parameter in a Rasch Model

  • Ernesto San MartínEmail author
  • Paul De Boeck
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 89)


Three versions of the Rasch model are considered: the fixed-effects model, the random-effects model with normal distribution, and the random-effects model with unspecified distribution. For each of the three, we discuss the meaning of the difficulty parameter starting each time from the corresponding likelihood and the resulting identified parameters. Because the likelihood and the identified parameters are different depending on the model, the identification of the parameter of interest is also different, with consequences for the meaning of the item difficulty. Finally, for all the three models, the item difficulties are monotonically related to the marginal probabilities of a correct response.


Likelihood Function Marginal Probability Item Difficulty Item Parameter Difficulty Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This research was partially funded by the ANILLO Project SOC1107 Statistics for Public Policy in Education from the Chilean Government.


  1. Andersen EB (1980) Discrete statistical models with social science applications. North Holland, AmsterdamzbMATHGoogle Scholar
  2. Bamber D, Van Santen JPH (2000) How to asses a model’s testability and identifiability. J Math Psychol 44:20–40CrossRefzbMATHGoogle Scholar
  3. Cox DR, Hinkley DV (1974) Theoretical statistics. Chapman and Hall, LondonCrossRefzbMATHGoogle Scholar
  4. Fisher RA (1922) On the mathematical foundations of theoretical statistics. Philos Trans R Soc Lond A 222:309–368CrossRefzbMATHGoogle Scholar
  5. Ghosh M (1995) Inconsistent maximum likelihood estimators for the Rasch model. Stat Probab Lett 23:165–170CrossRefzbMATHGoogle Scholar
  6. Lancaster T (2000) The incidental parameter problem since 1948. J Econom 95:391–413CrossRefzbMATHMathSciNetGoogle Scholar
  7. McCullagh P (2002) What is a statistical model? (with Discussion). Ann Stat 30:1225–1310CrossRefzbMATHMathSciNetGoogle Scholar
  8. Molenberghs G, Verbeke G, Demetrio CGB, Vieira A (2010) A family of generalized linear models for repeated measures with normal and cojugate random effects. Stat Sci 25:325–347CrossRefMathSciNetGoogle Scholar
  9. San Martín E, Rolin J-M (2013) Identification of parametric Rasch-type models. J Stat Plan Inference 143:116–130CrossRefzbMATHGoogle Scholar
  10. San Martín E, Jara A, Rolin J-M, Mouchart M (2011) On the Bayesian nonparametric generalization of IRT-types models. Psychometrika 76:385–409CrossRefzbMATHMathSciNetGoogle Scholar
  11. San Martín E, Rolin J-M, Castro M (2013) Identification of the 1PL model with guessing parameter: parametric and semi-parametric results. Psychometrika 78:341–379CrossRefzbMATHMathSciNetGoogle Scholar
  12. Woods CM (2006) Ramsay-curve item response theory (RC-IRT) to detect and correct for nonnormal latent variables. Psychol Methods 11:235–270CrossRefGoogle Scholar
  13. Woods CM, Thissen D (2006) Item response theory with estimation of the latent population distribution using spline-based densities. Psychometrika 71:281–301CrossRefzbMATHMathSciNetGoogle Scholar
  14. Zeger SL, Liang K-Y, Albert PS (1988) Models for longitudinal data: a generalized estimating equation approach. Biometrics 44:1049–1060CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics and Faculty of EducationPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Center for Operations Research and Econometrics CORELouvain-la-NeuveBelgium
  3. 3.Department of PsychologyThe Ohio State UniversityColumbusUSA

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