1, 2, 3-Parameter Logistic Rasch and Birnbaum Models and Item Analysis

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The item difficulty statistic is an appropriate choice for achievement or aptitude tests when the items are scored dichotomously (i.e., they are either correct or incorrect). Several methods were developed by G. Rasch and A. Birnbaum to estimate test reliability. The One-Parameter Logistic Model (1PLM) is


, or ,

where is the parameter describing the ability of the person being tested, is the probability of getting a correct response, and is the parameter describing the difficulty of item .

The Two-Parameter Logistic Model (2PLM) is


where is the slope parameter; when , this is the same as 1PLM.

The Three-Parameter Logistic Model (3PLM, or Birnbaum's Model):


The parameter is used when item is constructed so that guessing the correct answer is possible.

The Demonstration plots the 1PLM, 2PLM, and 3PLM item characteristic curves in red, blue, and green, respectively.


Contributed by: Olexandr Eugene Prokopchenko (March 2011)
Open content licensed under CC BY-NC-SA



G. Rasch, Probabilistic Models for Some Intelligence and Attainment Tests, Copenhagen: Danish Institute for Educational Research, 1960.

A. Birnbaum, "Some Latent Trait Models and Their Use in Inferring an Examinee's Ability," in F. M. Lord and M. R. Novick, Statistical Theories of Mental Test Scores, Reading, MA: Addison–Wesley, 1968.

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