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Parameter estimation routines for logistic Item Characteristic Curves (ICC) from Item Response Theory (IRT)

Project description

IRT Parameter Estimation routines

This package implements parameter estimation for logistic Item Characteristic Curves (ICC) from Item Response Theory (IRT).

Maximum likelihood estimation (MLE) fitting routines for the following logistic models are implemented:

  • 1PL - 1 parameter logistic (Rausch) model
    • b (difficulty)
  • 2PL - 2 parameter logistic model
    • a (discrimination) and b (difficulty)
  • 3PL - 3 parameter logistic (Birnbaum) model
    • a (discrimination), b (difficulty) and c (pseudo-guessing)

In addition, fitting using the zeta/lamdba/c formulation is also implemented. The difference here boils down to the logistic exponent. The conversion is:

a (θj - b) = ζ + λ θj

a * (theta_j - b) = zeta + lambda * theta_j

This seemingly insignificant change has drastic effects on the convergence properties (especially in the 2PL case, but also the 3PL case).

Many of the methods in this package are derived from work by Frank B. Baker and Seock-Ho Kim:

Item Response Theory: Parameter Estimation Techniques

The exception is the 3 parameter zeta/lambda/c implementation which to our knowledge has not been derived or documented before. For this reason, we include the mathematical derivations here:


The original BASIC code that work was derived from can be downloaded here:

The original python port of these hand-coded iterative schemes can be found in the file (imported as "baker"). These are mainly useful for comparative purposes (for instance, the 2PL version matches 1-1 with the original routine's published values).

The main routines in this package ( and use scipy.optimize.root instead for greater efficiency and accuracy. zlc uses a hybrid (zeta, lambda, c) formulation which makes the 2PL and 3PL systems converge much more much stably. This version is the one imported at the top level. This version also includes an "abc emulation" mode where zlc is still used internally, but automatic conversion is used so that the function takes a/b/c as arguments.

abc (in uses the (a,b,c) formulation, which may also be useful (try both for 3PL!). If you want to really dig into this code, it is very informative to use a side-by-side difference tool (like Vimdiff, KDiff3, WinMerge, FileMerge, or Meld) to compare with

All common utilities are found in


Updated 2021

To see a sample usage, check out the QuickTutorial notebook

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