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

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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+