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What is the two parameter IRT model (2PL)?


Item response theory is the predominant psychometric paradigm for mid or large scale assessment.  As noted in my introductory blog post, it is actually a family of models.  In this post, we discuss the two parameter IRT model (IRT 2PL).

Consider the following 3PL equation (simplified from Hambleton & Swaminathan, 1985, Eq. 3.3).  The IRT 2PL simply removes the c and (1-c) elements, so that probability is only a function of a and b.

3PL irt equation

This equation is predicting the probability of a certain response based on the examinee trait/ability level, the item discrimination parameter a, and the item difficulty/location parameter b.  If the examinee’s trait level is higher than the item location, the person has more than a 50% chance of responding in the keyed direction.

This phrase “in the keyed direction” is one you might often hear with the IRT 2PL.  This is because it is not often used with education/knowledge/ability assessments where items usually have a correct answer and guessing is often possible.  The IRT 2PL is used more often in attitudinal or other psychological assessments where guessing is irrelevant and there is no correct answer.  For example, consider an Extroversion scale, where examinees are responding Yes/No to statements like “I love to go to parties” or “I prefer to read books in my free time.”  There is not much to guess here, and the sense of “correct” is not relevant.

However, it is quite clear that the first statement is keyed in the direction of extroversion while the second statement is the reverse.  In fact, you would get the 1 point of response for saying No to that statement rather than Yes.  This is often called reverse-scored.

There are other aspects that go into whether you should use the 2PL model, but this is one of the most important.  In addition, you should also examine model fit indices and take sample size into account.

How do I implement the two parameter IRT model?

Like other IRT models, the 2PL requires specialized software.  Not all statistical packages will do it.  And while you can easily calculate classical statistics in Excel, there is no way to do IRT (well, unless you want to write your own VBA programs to do so).  As mentioned in this article on the three parameter model, there are a lot of Irt software programs available, but not all meet the required standards.

You should evaluate cost and functionality.  If you are a fan of R, there are packages to estimate IRT there.  However, I recommend our Xcalibre program for both newbies and professionals.  For newbies, it is much easier to use, which means you spend more time learning the concepts of IRT and not fighting command code that might be 30 years old.  For professionals, Xcalibre saves you from having to create reports by copy and paste which is incredibly expensive.

Nathan Thompson, PhD

Nathan Thompson, PhD, is CEO and Co-Founder of Assessment Systems Corporation (ASC). He is a psychometrician, software developer, author, and researcher, and evangelist for AI and automation. His mission is to elevate the profession of psychometrics by using software to automate psychometric work like item review, job analysis, and Angoff studies, so we can focus on more innovative work. His core goal is to improve assessment throughout the world.

Nate was originally trained as a psychometrician, with an honors degree at Luther College with a triple major of Math/Psych/Latin, and then a PhD in Psychometrics at the University of Minnesota. He then worked multiple roles in the testing industry, including item writer, test development manager, essay test marker, consulting psychometrician, software developer, project manager, and business leader. He is also cofounder and Membership Director at the International Association for Computerized Adaptive Testing ( He’s published 100+ papers and presentations, but his favorite remains

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